The vorticity equation of fluid dynamics describes evolution of the vorticity of a particle of a fluid as it moves with its flow, that is, the local rotation of the fluid (in terms of vector calculus this is the curl of the flow velocity).


where is the material derivative operator, is the flow velocity, is the local fluid density, is the local pressure, is the viscous stress tensor and represents the sum of the external body forces. The first source term on the right hand side represents vortex stretching.


The equation is valid in the absence of any concentrated torques and line forces, for a compressible Newtonian fluid.


In the case of incompressible (i.e. low Mach number) and isotropic fluids, with conservative body forces, the equation simplifies to the vorticity transport equation


where is the kinematic viscosity and is the Laplace operator.


Thus for an inviscid, barotropic fluid with conservative body forces, the vorticity equation simplifies to


Alternately, in case of incompressible, inviscid fluid with conservative body forces,


For a brief review of additional cases and simplifications, see also. For the vorticity equation in turbulence theory, in context of the flows in oceans and atmosphere, refer to.


The vorticity equation can be derived from the Navier–Stokes equation for the conservation of angular momentum. In the absence of any concentrated torques and line forces, one obtains


Now, vorticity is defined as the curl of the flow velocity vector. Taking the curl of momentum equation yields the desired equation.


The following identities are useful in derivation of the equation:


The vorticity equation can be expressed in tensor notation using Einstein's summation convention and the Levi-Civita symbol :


In the atmospheric sciences, the vorticity equation can be stated in terms of the absolute vorticity of air with respect to an inertial frame, or of the vorticity with respect to the rotation of the Earth. The absolute version is


Here, is the polar () component of the vorticity, is the atmospheric density, , , and w are the components of wind velocity, and is the 2-dimensional (i.e. horizontal-component-only) del.


The Vogel-Fulcher-Tammann equation, also known as Vogel-Fulcher-Tammann-Hesse equation or Vogel-Fulcher equation (abbreviated: VFT equation), is used to describe the viscosity of liquids as a function of temperature, and especially its strongly temperature dependent variation in the supercooled regime, upon approaching the glass transition. In this regime the viscosity of certain liquids can increase by up to 13 orders of magnitude within a relatively narrow temperature interval.


where formula_2 and formula_3 are empirical material-dependent parameters, and formula_4 is also an empirical fitting parameter, and typically lies about 50 °C below the glass transition temperature. These three parameters are normally used as adjustable parameters to fit the VFT equation to experimental data of specific systems.


The VFT equation is named after Hans Vogel, Gordon Scott Fulcher (1884–1971) and Gustav Tammann (1861–1938).


In fluid dynamics, the Boussinesq approximation for water waves is an approximation valid for weakly non-linear and fairly long waves. The approximation is named after Joseph Boussinesq, who first derived them in response to the observation by John Scott Russell of the wave of translation (also known as solitary wave or soliton). The 1872 paper of Boussinesq introduces the equations now known as the Boussinesq equations.


The Boussinesq approximation for water waves takes into account the vertical structure of the horizontal and vertical flow velocity. This results in non-linear partial differential equations, called Boussinesq-type equations, which incorporate frequency dispersion (as opposite to the shallow water equations, which are not frequency-dispersive). In coastal engineering, Boussinesq-type equations are frequently used in computer models for the simulation of water waves in shallow seas and harbours.


While the Boussinesq approximation is applicable to fairly long waves – that is, when the wavelength is large compared to the water depth – the Stokes expansion is more appropriate for short waves (when the wavelength is of the same order as the water depth, or shorter).


The essential idea in the Boussinesq approximation is the elimination of the vertical coordinate from the flow equations, while retaining some of the influences of the vertical structure of the flow under water waves. This is useful because the waves propagate in the horizontal plane and have a different (not wave-like) behaviour in the vertical direction. Often, as in Boussinesq's case, the interest is primarily in the wave propagation.


This elimination of the vertical coordinate was first done by Joseph Boussinesq in 1871, to construct an approximate solution for the solitary wave (or wave of translation). Subsequently, in 1872, Boussinesq derived the equations known nowadays as the Boussinesq equations.


The steps in the Boussinesq approximation are:


Thereafter, the Boussinesq approximation is applied to the remaining flow equations, in order to eliminate the dependence on the vertical coordinate.


As a result, the resulting partial differential equations are in terms of functions of the horizontal coordinates (and time).


As an example, consider potential flow over a horizontal bed in the ("x,z") plane, with "x" the horizontal and "z" the vertical coordinate. The bed is located at , where "h" is the mean water depth. A Taylor expansion is made of the velocity potential "φ(x,z,t)" around the bed level :


where "φb(x,t)" is the velocity potential at the bed. Invoking Laplace's equation for "φ", as valid for incompressible flow, gives:


since the vertical velocity is zero at the – impermeable – horizontal bed . This series may subsequently be truncated to a finite number of terms.


For water waves on an incompressible fluid and irrotational flow in the ("x","z") plane, the boundary conditions at the free surface elevation are:


Now the Boussinesq approximation for the velocity potential "φ", as given above, is applied in these boundary conditions. Further, in the resulting equations only the linear and quadratic terms with respect to "η" and "ub" are retained (with the horizontal velocity at the bed ). The cubic and higher order terms are assumed to be negligible. Then, the following partial differential equations are obtained:


This set of equations has been derived for a flat horizontal bed, "i.e." the mean depth "h" is a constant independent of position "x". When the right-hand sides of the above equations are set to zero, they reduce to the shallow water equations.


Under some additional approximations, but at the same order of accuracy, the above set A can be reduced to a single partial differential equation for the free surface elevation "η":


From the terms between brackets, the importance of nonlinearity of the equation can be expressed in terms of the Ursell number.


In dimensionless quantities, using the water depth "h" and gravitational acceleration "g" for non-dimensionalization, this equation reads, after normalization:


Water waves of different wave lengths travel with different phase speeds, a phenomenon known as frequency dispersion. For the case of infinitesimal wave amplitude, the terminology is "linear frequency dispersion". The frequency dispersion characteristics of a Boussinesq-type of equation can be used to determine the range of wave lengths, for which it is a valid approximation.


The linear frequency dispersion characteristics for the above set A of equations are:


The relative error in the phase speed "c" for set A, as compared with linear theory for water waves, is less than 4% for a relative wave number . So, in engineering applications, set A is valid for wavelengths "λ" larger than 4 times the water depth "h".


The linear frequency dispersion characteristics of equation B are:


The relative error in the phase speed for equation B is less than 4% for , equivalent to wave lengths "λ" longer than 7 times the water depth "h", called fairly long waves.


For short waves with equation B become physically meaningless, because there are no longer real-valued solutions of the phase speed.


The original set of two partial differential equations (Boussinesq, 1872, equation 25, see set A above) does not have this shortcoming.


The shallow water equations have a relative error in the phase speed less than 4% for wave lengths "λ" in excess of 13 times the water depth "h".


There are an overwhelming number of mathematical models which are referred to as Boussinesq equations. This may easily lead to confusion, since often they are loosely referenced to as "the" Boussinesq equations, while in fact a variant thereof is considered. So it is more appropriate to call them Boussinesq-type equations. Strictly speaking, "the" Boussinesq equations is the above-mentioned set B, since it is used in the analysis in the remainder of his 1872 paper.


Some directions, into which the Boussinesq equations have been extended, are:


While the Boussinesq equations allow for waves traveling simultaneously in opposing directions, it is often advantageous to only consider waves traveling in one direction. Under small additional assumptions, the Boussinesq equations reduce to:


Besides solitary wave solutions, the Korteweg–de Vries equation also has periodic and exact solutions, called cnoidal waves. These are approximate solutions of the Boussinesq equation.


For the simulation of wave motion near coasts and harbours, numerical models – both commercial and academic – employing Boussinesq-type equations exist. Some commercial examples are the Boussinesq-type wave modules in MIKE 21 and SMS. Some of the free Boussinesq models are Celeris, COULWAVE, and FUNWAVE. Most numerical models employ finite-difference, finite-volume or finite element techniques for the discretization of the model equations. Scientific reviews and intercomparisons of several Boussinesq-type equations, their numerical approximation and performance are e.g. , and .


In fluid dynamics, Batchelor vortices, first described by George Batchelor in a 1964 article, have been found useful in analyses of airplane vortex wake hazard problems.


The Batchelor vortex is an approximate solution to the Navier-Stokes equations obtained using a boundary layer approximation. The physical reasoning behind this approximation is the assumption that the axial gradient of the flow field of interest is of much smaller magnitude than the radial gradient.


The axial, radial and azimuthal velocity components of the vortex are denoted formula_1,formula_2 and formula_3 respectively and can be represented in cylindrical coordinates formula_4 as follows:<br>


The parameters in the above equations are


Note that the radial component of the velocity is zero and that the axial and azimuthal components depend only on formula_13.


We now write the system above in dimensionless form by scaling time by a factor formula_14. Using the same symbols for the dimensionless variables, the Batchelor vortex can be expressed in terms of the dimensionless variables as


where formula_16 denotes the free stream axial velocity and formula_17 is the Reynolds number.


If one lets formula_18 and considers an infinitely large swirl number then the Batchelor vortex simplifies to the Lamb–Oseen vortex for the azimuthal velocity:


When using the notation formula_1 for dynamic viscosity, formula_2 for the liquid-solid contact angle, formula_3 for surface tension , formula_4 for the fluid density, "t" for time, and "r" for the cross-sectional radius of the capillary and "x" for the distance the fluid has advanced, the Bosanquet equation of motion is


assuming that the motion is completely driven by surface tension, with no applied pressure at either end of the capillary tube.


The solution of the Bosanquet equation can be split into two timescales, firstly to account for the initial motion of the fluid by considering a solution in the limit of time approaching 0 giving the form


For the condition of short time this shows a meniscus front position proportional to time rather than the Lucas-Washburn square root of time, and the independence of viscosity demonstrates plug flow.


As time increases after the initial time of acceleration, the equation decays to the familiar Lucas-Washburn form dependent on viscosity and the square root of time.


The black-oil equations are a set of partial differential equations that describe fluid flow in a petroleum reservoir, constituting the mathematical framework for a black-oil reservoir simulator.


The term "black-oil" refers to the fluid model, in which water is modeled explicitly together with two hydrocarbon components, one (pseudo) oil phase and one (pseudo-)gas phase.


This is in contrast with a compositional formulation, in which each hydrocarbon component (arbitrary number) is handled separately


The equations of an extended black-oil model are


formula_4 is a porosity of the porous medium,


and vapor ("gas") phases in the reservoir,


Darcy velocities of the liquid phase, water phase and vapor phase in the reservoir.


The oil and gas at the surface (standard conditions) could be produced from both liquid and vapor phases existing at high pressure and temperature of reservoir conditions. This is characterized by the following quantities:


(ratio of some volume of reservoir liquid


to the volume of oil at standard conditions


obtained from the same volume of reservoir liquid),


formula_9 is a water formation volume factor


(ratio of volume of water at reservoir conditions to volume of water at standard conditions),


formula_10 is a gas formation volume factor


(ratio of some volume of reservoir vapor


to the volume of gas at standard conditions obtained from the same volume of reservoir vapor),


formula_11 is a solution of gas in oil phase


(ratio of volume of gas to the volume of oil at standard conditions


obtained from some amount of liquid phase at reservoir conditions),


formula_12 is a vaporized oil in gas phase


(ratio of volume of oil to the volume of gas at standard conditions


obtained from some amount of vapor phase at reservoir conditions).


In fluid dynamics, the Oseen equations (or Oseen flow) describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.


Oseen's work is based on the experiments of G.G. Stokes, who had studied the falling of a sphere through a viscous fluid. He developed a correction term, which included inertial factors, for the flow velocity used in Stokes' calculations, to solve the problem known as Stokes' paradox. His approximation leads to an improvement to Stokes' calculations.


The Oseen equations are, in case of an object moving with a steady flow velocity U through the fluid—which is at rest far from the object—and in a frame of reference attached to the object:


The boundary conditions for the Oseen flow around a rigid object are:


with "r" the distance from the object's center, and "p"∞ the undisturbed pressure far from the object.


A fundamental property of Oseen's equation is that the general solution can be split into "longitudinal" and "transversal" waves.


A solution formula_3 is a longitudinal wave if the velocity is irrotational and hence the viscous term drops out. The equations become


Velocity is derived from potential theory and pressure is from linearized Bernoulli's equations.


A solution formula_6 is a transversal wave if the pressure formula_7 is identically zero and the velocity field is solenoidal. The equations are


Then the complete Oseen solution is given by


a splitting theorem due to Horace Lamb. The splitting is unique if conditions at infinity (say formula_10) are specified.


For certain Oseen flows, further splitting of transversal wave into irrotational and rotational component is possible formula_11 Let formula_12 be the scalar function which satisfies formula_13 and vanishes at infinity and conversely let formula_14 be given such that formula_15, then the transversal wave is


where formula_12 is determined from formula_18 and formula_19 is the unit vector. Neither formula_20 or formula_21 are transversal by itself, but formula_22 is transversal. Therefore,


The only rotational component is being formula_21.


The fundamental solution due to a singular point force embedded in an Oseen flow is the Oseenlet. The closed-form fundamental solutions for the generalized unsteady Stokes and Oseen flows associated with arbitrary time-dependent translational and rotational motions have been derived for the Newtonian and micropolar fluids.


Using the Oseen equation, Horace Lamb was able to derive improved expressions for the viscous flow around a sphere in 1911, improving on Stokes law towards somewhat higher Reynolds numbers. Also, Lamb derived—for the first time—a solution for the viscous flow around a circular cylinder.


The solution to the response of a singular force formula_25 when no external boundaries are present be written as


If formula_27, where formula_28 is the singular force concentrated at the point formula_29 and formula_30 is an arbitrary point and formula_31 is the given vector, which gives the direction of the singular force, then in the absence of boundaries, the velocity and pressure is derived from the fundamental tensor formula_32 and the fundamental vector formula_33


Now if formula_25 is arbitrary function of space, the solution for an unbounded domain is


where formula_37 is the infinitesimal volume/area element around the point formula_29.


Without loss of generality formula_39 taken at the origin and formula_40. Then the fundamental tensor and vector are


where formula_43 is the modified Bessel function of the second kind of order zero.


Without loss of generality formula_44 taken at the origin and formula_45. Then the fundamental tensor and vector are


Oseen considered the sphere to be stationary and the fluid to be flowing with a flow velocity (formula_48) at an infinite distance from the sphere. Inertial terms were neglected in Stokes’ calculations. It is a limiting solution when the Reynolds number tends to zero. When the Reynolds number is small and finite, such as 0.1, correction for the inertial term is needed. Oseen substituted the following flow velocity values into the Navier-Stokes equations.


Inserting these into the Navier-Stokes equations and neglecting the quadratic terms in the primed quantities leads to the derivation of Oseen's approximation:


Since the motion is symmetric with respect to formula_51 axis and the divergence of the vorticity vector is always zero we get:


the function formula_53 can be eliminated by adding to a suitable function in formula_51, is the vorticity function, and the previous function can be written as:


and by some integration the solution for formula_12 is:


thus by letting formula_51 be the "privileged direction" it produces:


then by applying the three boundary conditions we obtain


the new improved drag coefficient now become:


and finally, when Stokes' solution was solved on the basis of Oseen's approximation, it showed that the resultant drag force is given by


The force from Oseen's equation differs from that of Stokes by a factor of


In the far field formula_71 ≫ 1, the viscous stress is dominated by the last term. That is:


The inertia term is dominated by the term:


The error is then given by the ratio:


This becomes unbounded for formula_71 ≫ 1, therefore the inertia cannot be ignored in the far field. By taking the curl, Stokes equation gives formula_76 Since the body is a source of vorticity, formula_77 would become unbounded logarithmically for large formula_78 This is certainly unphysical and is known as Stokes' paradox.


Solution for a moving sphere in incompressible fluid.


Consider the case of a solid sphere moving in a stationary liquid with a constant velocity.


The liquid is modeled as an incompressible fluid (i.e. with constant density), and being stationary means that its velocity tends towards zero as the distance from the sphere approaches infinity.


For a real body there will be a transient effect due to its acceleration as it begins its motion; however after enough time it will tend towards zero, so that the fluid velocity everywhere will approach the one obtained in the hypothetical case in which the body is already moving for infinite time.


Thus we assume a sphere of radius "a" moving at a constant velocity formula_79, in an incompressible fluid that is at rest at infinity. We will work in coordinates formula_80 that move along with the sphere with the coordinate center located at the sphere's center. We have:


Since these boundary conditions, as well as the equation of motions, are time invariant (i.e. they are unchanged by shifting the time formula_82) when expressed in the formula_80 coordinates, the solution depends upon the time only through these coordinates.


The equations of motion are the Navier-Stokes equations defined in the resting frame coordinates formula_84. While spatial derivatives are equal in both coordinate systems, the time derivative that appears in the equations satisfies:


where the derivative formula_86 is with respect to the moving coordinates formula_80. We henceforth omit the "m" subscript.


Oseen's approximation sums up to neglecting the term non-linear in formula_88. Thus the incompressible Navier-Stokes equations become:


for a fluid having density ρ and kinematic viscosity ν = μ/ρ (μ being the dynamic viscosity). "p" is the pressure.


Due to the continuity equation for incompressible fluid formula_90, the solution can be expressed using a vector potential formula_91. This turns out to be directed at the formula_92 direction and its magnitude is equivalent to the stream function used in two-dimensional problems. It turns out to be:


where formula_94 is Reynolds number for the flow close to the sphere.


Note that in some notations formula_95 is replaced by formula_96 so that the derivation of formula_88 from formula_98 is more similar to its derivation from the stream function in the two-dimensional case (in polar coordinates).


The vector Laplacian of a vector of the type formula_104 reads:


where we have used the vanishing of the divergence of formula_91 to relate the vector laplacian and a double curl.


The equation of motion's left hand side is the curl of the following:


We calculate the derivative separately for each term in formula_95.


Taking the curl, we find an expression that is equal to formula_116 times the gradient of the following function, which is the pressure:


where formula_118 is the pressure at infinity, formula_119.is the polar angle originated from the opposite side of the front stagnation point (formula_120 where is the front stagnation point).


Also, the velocity is derived by taking the curl of formula_91:


These "p" and "u" satisfy the equation of motion and thus constitute the solution to Oseen's approximation.


One may question, however, whether the correction term was chosen by chance, because in a frame of reference moving with the sphere, the fluid near the sphere is almost at rest, and in that region inertial force is negligible and Stokes' equation is well justified. Far away from the sphere, the flow velocity approaches "u" and Oseen's approximation is more accurate. But Oseen's equation was obtained applying the equation for the entire flow field. This question was answered by Proudman and Pearson in 1957, who solved the Navier-Stokes equations and gave an improved Stokes' solution in the neighborhood of the sphere and an improved Oseen's solution at infinity, and matched the two solutions in a supposed common region of their validity. They obtained:


The method and formulation for analysis of flow at a very low Reynolds number is important. The slow motion of small particles in a fluid is common in bio-engineering. Oseen's drag formulation can be used in connection with flow of fluids under various special conditions, such as: containing particles, sedimentation of particles, centrifugation or ultracentrifugation of suspensions, colloids, and blood through isolation of tumors and antigens. The fluid does not even have to be a liquid, and the particles do not need to be solid. It can be used in a number of applications, such as smog formation and atomization of liquids.


Blood flow in small vessels, such as capillaries, is characterized by small Reynolds and Womersley numbers. A vessel of diameter of with a flow of , viscosity of for blood, density of and a heart rate of , will have a Reynolds number of 0.005 and a Womersley number of 0.0126. At these small Reynolds and Womersley numbers, the viscous effects of the fluid become predominant. Understanding the movement of these particles is essential for drug delivery and studying metastasis movements of cancers.


In fluid dynamics, stream thrust averaging is a process used to convert three-dimensional flow through a duct into one-dimensional uniform flow. It makes the assumptions that the flow is mixed adiabatically and without friction. However, due to the mixing process, there is a net increase in the entropy of the system. Although there is an increase in entropy, the stream thrust averaged values are more representative of the flow than a simple average as a simple average would violate the second Law of Thermodynamics.


Solving for formula_5 yields two solutions. They must both be analyzed to determine which is the physical solution. One will usually be a subsonic root and the other a supersonic root. If it is not clear which value of velocity is correct, the second law of thermodynamics may be applied.


The values formula_10 and formula_11 are unknown and may be dropped from the formulation. The value of entropy is not necessary, only that the value is positive.


One possible unreal solution for the stream thrust averaged velocity yields a negative entropy. Another method of determining the proper solution is to take a simple average of the velocity and determining which value is closer to the stream thrust averaged velocity.


In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behaviour of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system. The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics.


There are two main descriptions of motion: dynamics and kinematics. Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term "dynamics" refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.


However, kinematics is simpler. It concerns only variables derived from the positions of objects and time. In circumstances of constant acceleration, these simpler equations of motion are usually referred to as the SUVAT equations, arising from the definitions of kinematic quantities: displacement (), initial velocity (), final velocity (), acceleration (), and time ().


Equations of motion can therefore be grouped under these main classifiers of motion. In all cases, the main types of motion are translations, rotations, oscillations, or any combinations of these.


A differential equation of motion, usually identified as some physical law and applying definitions of physical quantities, is used to set up an equation for the problem. Solving the differential equation will lead to a general solution with arbitrary constants, the arbitrariness corresponding to a family of solutions. A particular solution can be obtained by setting the initial values, which fixes the values of the constants.


To state this formally, in general an equation of motion is a function of the position of the object, its velocity (the first time derivative of , ), and its acceleration (the second derivative of , ), and time . Euclidean vectors in 3D are denoted throughout in bold. This is equivalent to saying an equation of motion in is a second-order ordinary differential equation (ODE) in ,


where is time, and each overdot denotes one time derivative. The initial conditions are given by the "constant" values at ,


The solution to the equation of motion, with specified initial values, describes the system for all times after . Other dynamical variables like the momentum of the object, or quantities derived from and like angular momentum, can be used in place of as the quantity to solve for from some equation of motion, although the position of the object at time is by far the most sought-after quantity.


Sometimes, the equation will be linear and is more likely to be exactly solvable. In general, the equation will be non-linear, and cannot be solved exactly so a variety of approximations must be used. The solutions to nonlinear equations may show chaotic behavior depending on how "sensitive" the system is to the initial conditions.


Kinematics, dynamics and the mathematical models of the universe developed incrementally over three millennia, thanks to many thinkers, only some of whose names we know. In antiquity, priests, astrologers and astronomers predicted solar and lunar eclipses, the solstices and the equinoxes of the Sun and the period of the Moon. But they had nothing other than a set of algorithms to guide them. Equations of motion were not written down for another thousand years.


Medieval scholars in the thirteenth century — for example at the relatively new universities in Oxford and Paris — drew on ancient mathematicians (Euclid and Archimedes) and philosophers (Aristotle) to develop a new body of knowledge, now called physics.


At Oxford, Merton College sheltered a group of scholars devoted to natural science, mainly physics, astronomy and mathematics, who were of similar stature to the intellectuals at the University of Paris. Thomas Bradwardine extended Aristotelian quantities such as distance and velocity, and assigned intensity and extension to them. Bradwardine suggested an exponential law involving force, resistance, distance, velocity and time. Nicholas Oresme further extended Bradwardine's arguments. The Merton school proved that the quantity of motion of a body undergoing a uniformly accelerated motion is equal to the quantity of a uniform motion at the speed achieved halfway through the accelerated motion.


Discourses such as these spread throughout Europe, shaping the work of Galileo Galilei and others, and helped in laying the foundation of kinematics. Galileo deduced the equation in his work geometrically, using the Merton rule, now known as a special case of one of the equations of kinematics.


The term "inertia" was used by Kepler who applied it to bodies at rest. (The first law of motion is now often called the law of inertia.)


Galileo did not fully grasp the third law of motion, the law of the equality of action and reaction, though he corrected some errors of Aristotle. With Stevin and others Galileo also wrote on statics. He formulated the principle of the parallelogram of forces, but he did not fully recognize its scope.


Galileo also was interested by the laws of the pendulum, his first observations of which were as a young man. In 1583, while he was praying in the cathedral at Pisa, his attention was arrested by the motion of the great lamp lighted and left swinging, referencing his own pulse for time keeping. To him the period appeared the same, even after the motion had greatly diminished, discovering the isochronism of the pendulum.


More careful experiments carried out by him later, and described in his Discourses, revealed the period of oscillation varies with the square root of length but is independent of the mass the pendulum.


Thus we arrive at René Descartes, Isaac Newton, Gottfried Leibniz, et al.; and the evolved forms of the equations of motion that begin to be recognized as the modern ones.


Later the equations of motion also appeared in electrodynamics, when describing the motion of charged particles in electric and magnetic fields, the Lorentz force is the general equation which serves as the definition of what is meant by an electric field and magnetic field. With the advent of special relativity and general relativity, the theoretical modifications to spacetime meant the classical equations of motion were also modified to account for the finite speed of light, and curvature of spacetime. In all these cases the differential equations were in terms of a function describing the particle's trajectory in terms of space and time coordinates, as influenced by forces or energy transformations.


However, the equations of quantum mechanics can also be considered "equations of motion", since they are differential equations of the wavefunction, which describes how a quantum state behaves analogously using the space and time coordinates of the particles. There are analogs of equations of motion in other areas of physics, for collections of physical phenomena that can be considered waves, fluids, or fields.


From the instantaneous position , instantaneous meaning at an instant value of time , the instantaneous velocity and acceleration have the general, coordinate-independent definitions;


Notice that velocity always points in the direction of motion, in other words for a curved path it is the tangent vector. Loosely speaking, first order derivatives are related to tangents of curves. Still for curved paths, the acceleration is directed towards the center of curvature of the path. Again, loosely speaking, second order derivatives are related to curvature.


The rotational analogues are the "angular vector" (angle the particle rotates about some axis) , angular velocity , and angular acceleration :


where is a unit vector in the direction of the axis of rotation, and is the angle the object turns through about the axis.


The following relation holds for a point-like particle, orbiting about some axis with angular velocity :


where is the position vector of the particle (radial from the rotation axis) and the tangential velocity of the particle. For a rotating continuum rigid body, these relations hold for each point in the rigid body.


The differential equation of motion for a particle of constant or uniform acceleration in a straight line is simple: the acceleration is constant, so the second derivative of the position of the object is constant. The results of this case are summarized below.


Constant translational acceleration in a straight line.


These equations apply to a particle moving linearly, in three dimensions in a straight line with constant acceleration. Since the position, velocity, and acceleration are collinear (parallel, and lie on the same line) – only the magnitudes of these vectors are necessary, and because the motion is along a straight line, the problem effectively reduces from three dimensions to one.


Equations [1] and [2] are from integrating the definitions of velocity and acceleration, subject to the initial conditions and ;


which breaks into the radial acceleration , centripetal acceleration , Coriolis acceleration , and angular acceleration .


Special cases of motion described be these equations are summarized qualitatively in the table below. Two have already been discussed above, in the cases that either the radial components or the angular components are zero, and the non-zero component of motion describes uniform acceleration.


In 3D space, the equations in spherical coordinates with corresponding unit vectors , and , the position, velocity, and acceleration generalize respectively to


In the case of a constant this reduces to the planar equations above.


The first general equation of motion developed was Newton's second law of motion. In its most general form it states the rate of change of momentum of an object equals the force acting on it,


The force in the equation is "not" the force the object exerts. Replacing momentum by mass times velocity, the law is also written more famously as


since is a constant in Newtonian mechanics.


Newton's second law applies to point-like particles, and to all points in a rigid body. They also apply to each point in a mass continuum, like deformable solids or fluids, but the motion of the system must be accounted for; see material derivative. In the case the mass is not constant, it is not sufficient to use the product rule for the time derivative on the mass and velocity, and Newton's second law requires some modification consistent with conservation of momentum; see variable-mass system.


It may be simple to write down the equations of motion in vector form using Newton's laws of motion, but the components may vary in complicated ways with spatial coordinates and time, and solving them is not easy. Often there is an excess of variables to solve for the problem completely, so Newton's laws are not always the most efficient way to determine the motion of a system. In simple cases of rectangular geometry, Newton's laws work fine in Cartesian coordinates, but in other coordinate systems can become dramatically complex.


The momentum form is preferable since this is readily generalized to more complex systems, such as special and general relativity (see four-momentum). It can also be used with the momentum conservation. However, Newton's laws are not more fundamental than momentum conservation, because Newton's laws are merely consistent with the fact that zero resultant force acting on an object implies constant momentum, while a resultant force implies the momentum is not constant. Momentum conservation is always true for an isolated system not subject to resultant forces.


For a number of particles (see many body problem), the equation of motion for one particle influenced by other particles is


where is the momentum of particle , is the force on particle by particle , and is the resultant external force due to any agent not part of system. Particle does not exert a force on itself.


Euler's laws of motion are similar to Newton's laws, but they are applied specifically to the motion of rigid bodies. The Newton–Euler equations combine the forces and torques acting on a rigid body into a single equation.


Newton's second law for rotation takes a similar form to the translational case,


by equating the torque acting on the body to the rate of change of its angular momentum . Analogous to mass times acceleration, the moment of inertia tensor depends on the distribution of mass about the axis of rotation, and the angular acceleration is the rate of change of angular velocity,


Again, these equations apply to point like particles, or at each point of a rigid body.


Likewise, for a number of particles, the equation of motion for one particle is


where is the angular momentum of particle , the torque on particle by particle , and is resultant external torque (due to any agent not part of system). Particle does not exert a torque on itself.


Some examples of Newton's law include describing the motion of a simple pendulum,


and a damped, sinusoidally driven harmonic oscillator,


For describing the motion of masses due to gravity, Newton's law of gravity can be combined with Newton's second law. For two examples, a ball of mass thrown in the air, in air currents (such as wind) described by a vector field of resistive forces ,


where is the gravitational constant, the mass of the Earth, and is the acceleration of the projectile due to the air currents at position and time .


The classical -body problem for particles each interacting with each other due to gravity is a set of nonlinear coupled second order ODEs,


where labels the quantities (mass, position, etc.) associated with each particle.


Using all three coordinates of 3D space is unnecessary if there are constraints on the system. If the system has degrees of freedom, then one can use a set of generalized coordinates , to define the configuration of the system. They can be in the form of arc lengths or angles. They are a considerable simplification to describe motion, since they take advantage of the intrinsic constraints that limit the system's motion, and the number of coordinates is reduced to a minimum. The time derivatives of the generalized coordinates are the "generalized velocities"


where the "Lagrangian" is a function of the configuration and its time rate of change (and possibly time )


Setting up the Lagrangian of the system, then substituting into the equations and evaluating the partial derivatives and simplifying, a set of coupled second order ODEs in the coordinates are obtained.


is a function of the configuration and conjugate ""generalized" momenta"


in which is a shorthand notation for a vector of partial derivatives with respect to the indicated variables (see for example matrix calculus for this denominator notation), and possibly time ,


Setting up the Hamiltonian of the system, then substituting into the equations and evaluating the partial derivatives and simplifying, a set of coupled first order ODEs in the coordinates and momenta are obtained.


is "Hamilton's principal function", also called the "classical action" is a functional of . In this case, the momenta are given by


Although the equation has a simple general form, for a given Hamiltonian it is actually a single first order "non-linear" PDE, in variables. The action allows identification of conserved quantities for mechanical systems, even when the mechanical problem itself cannot be solved fully, because any differentiable symmetry of the action of a physical system has a corresponding conservation law, a theorem due to Emmy Noether.


All classical equations of motion can be derived from the variational principle known as Hamilton's principle of least action


stating the path the system takes through the configuration space is the one with the least action .


In electrodynamics, the force on a charged particle of charge is the Lorentz force:


Combining with Newton's second law gives a first order differential equation of motion, in terms of position of the particle:


The same equation can be obtained using the Lagrangian (and applying Lagrange's equations above) for a charged particle of mass and charge :


where and are the electromagnetic scalar and vector potential fields. The Lagrangian indicates an additional detail: the canonical momentum in Lagrangian mechanics is given by:


instead of just , implying the motion of a charged particle is fundamentally determined by the mass and charge of the particle. The Lagrangian expression was first used to derive the force equation.


Alternatively the Hamiltonian (and substituting into the equations):


The above equations are valid in flat spacetime. In curved spacetime, things become mathematically more complicated since there is no straight line; this is generalized and replaced by a "geodesic" of the curved spacetime (the shortest length of curve between two points). For curved manifolds with a metric tensor , the metric provides the notion of arc length (see line element for details). The differential arc length is given by:


and the geodesic equation is a second-order differential equation in the coordinates. The general solution is a family of geodesics:


where is a Christoffel symbol of the second kind, which contains the metric (with respect to the coordinate system).


Given the mass-energy distribution provided by the stress–energy tensor , the Einstein field equations are a set of non-linear second-order partial differential equations in the metric, and imply the curvature of spacetime is equivalent to a gravitational field (see equivalence principle). Mass falling in curved spacetime is equivalent to a mass falling in a gravitational field - because gravity is a fictitious force. The "relative acceleration" of one geodesic to another in curved spacetime is given by the "geodesic deviation equation":


where is the separation vector between two geodesics, ("not" just ) is the covariant derivative, and is the Riemann curvature tensor, containing the Christoffel symbols. In other words, the geodesic deviation equation is the equation of motion for masses in curved spacetime, analogous to the Lorentz force equation for charges in an electromagnetic field.


For flat spacetime, the metric is a constant tensor so the Christoffel symbols vanish, and the geodesic equation has the solutions of straight lines. This is also the limiting case when masses move according to Newton's law of gravity.


In general relativity, rotational motion is described by the relativistic angular momentum tensor, including the spin tensor, which enter the equations of motion under covariant derivatives with respect to proper time. The Mathisson–Papapetrou–Dixon equations describe the motion of spinning objects moving in a gravitational field.


Unlike the equations of motion for describing particle mechanics, which are systems of coupled ordinary differential equations, the analogous equations governing the dynamics of waves and fields are always partial differential equations, since the waves or fields are functions of space and time. For a particular solution, boundary conditions along with initial conditions need to be specified.


Sometimes in the following contexts, the wave or field equations are also called "equations of motion".


Equations that describe the spatial dependence and time evolution of fields are called "field equations". These include


This terminology is not universal: for example although the Navier–Stokes equations govern the velocity field of a fluid, they are not usually called "field equations", since in this context they represent the momentum of the fluid and are called the "momentum equations" instead.


Equations of wave motion are called "wave equations". The solutions to a wave equation give the time-evolution and spatial dependence of the amplitude. Boundary conditions determine if the solutions describe traveling waves or standing waves.


From classical equations of motion and field equations; mechanical, gravitational wave, and electromagnetic wave equations can be derived. The general linear wave equation in 3D is:


where is any mechanical or electromagnetic field amplitude, say:


and is the phase velocity. Nonlinear equations model the dependence of phase velocity on amplitude, replacing by . There are other linear and nonlinear wave equations for very specific applications, see for example the Korteweg–de Vries equation.


In quantum theory, the wave and field concepts both appear.


In quantum mechanics, in which particles also have wave-like properties according to wave–particle duality, the analogue of the classical equations of motion (Newton's law, Euler–Lagrange equation, Hamilton–Jacobi equation, etc.) is the Schrödinger equation in its most general form:


where is the wavefunction of the system, is the quantum Hamiltonian operator, rather than a function as in classical mechanics, and is the Planck constant divided by 2. Setting up the Hamiltonian and inserting it into the equation results in a wave equation, the solution is the wavefunction as a function of space and time. The Schrödinger equation itself reduces to the Hamilton–Jacobi equation when one considers the correspondence principle, in the limit that becomes zero.


Throughout all aspects of quantum theory, relativistic or non-relativistic, there are various formulations alternative to the Schrödinger equation that govern the time evolution and behavior of a quantum system, for instance:


In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. The partition function is dimensionless, it is a pure number.


Each partition function is constructed to represent a particular statistical ensemble (which, in turn, corresponds to a particular free energy). The most common statistical ensembles have named partition functions. The canonical partition function applies to a canonical ensemble, in which the system is allowed to exchange heat with the environment at fixed temperature, volume, and number of particles. The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential. Other types of partition functions can be defined for different circumstances; see partition function (mathematics) for generalizations. The partition function has many physical meanings, as discussed in Meaning and significance.


Initially, let us assume that a thermodynamically large system is in thermal contact with the environment, with a temperature "T", and both the volume of the system and the number of constituent particles are fixed. A collection of this kind of system comprises an ensemble called a canonical ensemble. The appropriate mathematical expression for the canonical partition function depends on the degrees of freedom of the system, whether the context is classical mechanics or quantum mechanics, and whether the spectrum of states is discrete or continuous.


For a canonical ensemble that is classical and discrete, the canonical partition function is defined as


The exponential factor formula_7 is otherwise known as the Boltzmann factor.


In classical mechanics, the position and momentum variables of a particle can vary continuously, so the set of microstates is actually uncountable. In "classical" statistical mechanics, it is rather inaccurate to express the partition function as a sum of discrete terms. In this case we must describe the partition function using an integral rather than a sum. For a canonical ensemble that is classical and continuous, the canonical partition function is defined as


To make it into a dimensionless quantity, we must divide it by "h", which is some quantity with units of action (usually taken to be Planck's constant).


For a gas of formula_15 identical classical particles in three dimensions, the partition function is


The reason for the factorial factor "N"! is discussed below. The extra constant factor introduced in the denominator was introduced because, unlike the discrete form, the continuous form shown above is not dimensionless. As stated in the previous section, to make it into a dimensionless quantity, we must divide it by "h"3"N" (where "h" is usually taken to be Planck's constant).


For a canonical ensemble that is quantum mechanical and discrete, the canonical partition function is defined as the trace of the Boltzmann factor:


The dimension of formula_32 is the number of energy eigenstates of the system.


For a canonical ensemble that is quantum mechanical and continuous, the canonical partition function is defined as


In systems with multiple quantum states "s" sharing the same energy "Es", it is said that the energy levels of the system are degenerate. In the case of degenerate energy levels, we can write the partition function in terms of the contribution from energy levels (indexed by "j") as follows:


where "gj" is the degeneracy factor, or number of quantum states "s" that have the same energy level defined by "Ej" = "Es".


The above treatment applies to "quantum" statistical mechanics, where a physical system inside a finite-sized box will typically have a discrete set of energy eigenstates, which we can use as the states "s" above. In quantum mechanics, the partition function can be more formally written as a trace over the state space (which is independent of the choice of basis):


where "Ĥ" is the quantum Hamiltonian operator. The exponential of an operator can be defined using the exponential power series.


The classical form of "Z" is recovered when the trace is expressed in terms of coherent states


and when quantum-mechanical uncertainties in the position and momentum of a particle


are regarded as negligible. Formally, using bra–ket notation, one inserts under the trace for each degree of freedom the identity:


where "x", "p" is a normalised Gaussian wavepacket centered at


A coherent state is an approximate eigenstate of both operators formula_44 and formula_45, hence also of the Hamiltonian "Ĥ", with errors of the size of the uncertainties. If Δ"x" and Δ"p" can be regarded as zero, the action of "Ĥ" reduces to multiplication by the classical Hamiltonian, and "Z" reduces to the classical configuration integral.


For simplicity, we will use the discrete form of the partition function in this section. Our results will apply equally well to the continuous form.


Consider a system "S" embedded into a heat bath "B". Let the total energy of both systems be "E". Let "pi" denote the probability that the system "S" is in a particular microstate, "i", with energy "Ei". According to the fundamental postulate of statistical mechanics (which states that all attainable microstates of a system are equally probable), the probability "pi" will be proportional to the number of microstates of the total closed system ("S", "B") in which "S" is in microstate "i" with energy "Ei". Equivalently, "pi" will be proportional to the number of microstates of the heat bath "B" with energy "E" − "Ei":


Assuming that the heat bath's internal energy is much larger than the energy of "S" ("E" ≫ "Ei"), we can Taylor-expand formula_47 to first order in "Ei" and use the thermodynamic relation formula_48, where here formula_49, formula_50 are the entropy and temperature of the bath respectively:


Since the total probability to find the system in "some" microstate (the sum of all "pi") must be equal to 1, we know that the constant of proportionality must be the normalization constant, and so, we can define the partition function to be this constant:


In order to demonstrate the usefulness of the partition function, let us calculate the thermodynamic value of the total energy. This is simply the expected value, or ensemble average for the energy, which is the sum of the microstate energies weighted by their probabilities:


Incidentally, one should note that if the microstate energies depend on a parameter λ in the manner


then the expected value of "A" is


This provides us with a method for calculating the expected values of many microscopic quantities. We add the quantity artificially to the microstate energies (or, in the language of quantum mechanics, to the Hamiltonian), calculate the new partition function and expected value, and then set "λ" to zero in the final expression. This is analogous to the source field method used in the path integral formulation of quantum field theory.


In this section, we will state the relationships between the partition function and the various thermodynamic parameters of the system. These results can be derived using the method of the previous section and the various thermodynamic relations.


As we have already seen, the thermodynamic energy is


The variance in the energy (or "energy fluctuation") is


In general, consider the extensive variable X and intensive variable Y where X and Y form a pair of conjugate variables. In ensembles where Y is fixed (and X is allowed to fluctuate), then the average value of X will be:


The sign will depend on the specific definitions of the variables X and Y. An example would be X = volume and Y = pressure. Additionally, the variance in X will be


In the special case of entropy, entropy is given by


where "A" is the Helmholtz free energy defined as "A" = "U" − "TS", where "U" = "E" is the total energy and "S" is the entropy, so that


Furthermore, the heat capacity can be expressed as


Suppose a system is subdivided into "N" sub-systems with negligible interaction energy, that is, we can assume the particles are essentially non-interacting. If the partition functions of the sub-systems are "ζ"1, "ζ"2, ..., "ζ"N, then the partition function of the entire system is the "product" of the individual partition functions:


If the sub-systems have the same physical properties, then their partition functions are equal, ζ1 = ζ2 = ... = ζ, in which case


However, there is a well-known exception to this rule. If the sub-systems are actually identical particles, in the quantum mechanical sense that they are impossible to distinguish even in principle, the total partition function must be divided by a "N"! ("N" factorial):


This is to ensure that we do not "over-count" the number of microstates. While this may seem like a strange requirement, it is actually necessary to preserve the existence of a thermodynamic limit for such systems. This is known as the Gibbs paradox.


It may not be obvious why the partition function, as we have defined it above, is an important quantity. First, consider what goes into it. The partition function is a function of the temperature "T" and the microstate energies "E"1, "E"2, "E"3, etc. The microstate energies are determined by other thermodynamic variables, such as the number of particles and the volume, as well as microscopic quantities like the mass of the constituent particles. This dependence on microscopic variables is the central point of statistical mechanics. With a model of the microscopic constituents of a system, one can calculate the microstate energies, and thus the partition function, which will then allow us to calculate all the other thermodynamic properties of the system.


The partition function can be related to thermodynamic properties because it has a very important statistical meaning. The probability "Ps" that the system occupies microstate "s" is


Thus, as shown above, the partition function plays the role of a normalizing constant (note that it does "not" depend on "s"), ensuring that the probabilities sum up to one:


This is the reason for calling "Z" the "partition function": it encodes how the probabilities are partitioned among the different microstates, based on their individual energies. The letter "Z" stands for the German word "Zustandssumme", "sum over states". The usefulness of the partition function stems from the fact that it can be used to relate macroscopic thermodynamic quantities to the microscopic details of a system through the derivatives of its partition function. Finding the partition function is also equivalent to performing a Laplace transform of the density of states function from the energy domain to the β domain, and the inverse Laplace transform of the partition function reclaims the state density function of energies.


We can define a grand canonical partition function for a grand canonical ensemble, which describes the statistics of a constant-volume system that can exchange both heat and particles with a reservoir. The reservoir has a constant temperature "T", and a chemical potential "μ".


The grand canonical partition function, denoted by formula_71, is the following sum over microstates


Here, each microstate is labelled by formula_73, and has total particle number formula_74 and total energy formula_75. This partition function is closely related to the grand potential, formula_76, by the relation


This can be contrasted to the canonical partition function above, which is related instead to the Helmholtz free energy.


It is important to note that the number of microstates in the grand canonical ensemble may be much larger than in the canonical ensemble, since here we consider not only variations in energy but also in particle number. Again, the utility of the grand canonical partition function is that it is related to the probability that the system is in state formula_73:


An important application of the grand canonical ensemble is in deriving exactly the statistics of a non-interacting many-body quantum gas (Fermi–Dirac statistics for fermions, Bose–Einstein statistics for bosons), however it is much more generally applicable than that. The grand canonical ensemble may also be used to describe classical systems, or even interacting quantum gases.


The grand partition function is sometimes written (equivalently) in terms of alternate variables as


where formula_81 is known as the absolute activity (or fugacity) and formula_82 is the canonical partition function.


In combustion, the Williams spray equation, also known as the Williams–Boltzmann equation, describes the statistical evolution of sprays contained in another fluid, analogous to the Boltzmann equation for the molecules, named after Forman A. Williams, who derived the equation in 1958.


The sprays are assumed to be spherical with radius formula_1, even though the assumption is valid for solid particles(liquid droplets) when their shape has no consequence on the combustion. For liquid droplets to be nearly spherical, the spray has to be dilute(total volume occupied by the sprays is much less than the volume of the gas) and the Weber number formula_2, where formula_3 is the gas density, formula_4 is the spray droplet velocity, formula_5 is the gas velocity and formula_6 is the surface tension of the liquid spray, should be formula_7.


The equation is described by a number density function formula_8, which represents the probable number of spray particles (droplets) of chemical species formula_9 (of formula_10 total species), that one can find with radii between formula_1 and formula_12, located in the spatial range between formula_13 and formula_14, traveling with a velocity in between formula_4 and formula_16, having the temperature in between formula_17 and formula_18 at time formula_19. Then the spray equation for the evolution of this density function is given by


A simplified model for liquid propellant rocket.


This model for the rocket motor was developed by Probert, Williams and Tanasawa. It is reasonable to neglect formula_31, for distances not very close to the spray atomizer, where major portion of combustion occurs. Consider a one-dimensional liquid-propellent rocket motor situated at formula_32, where fuel is sprayed. Neglecting formula_33(density function is defined without the temperature so accordingly dimensions of formula_34 changes) and due to the fact that the mean flow is parallel to formula_35 axis, the steady spray equation reduces to


where formula_37 is the velocity in formula_35 direction. Integrating with respect to the velocity results


The contribution from the last term (spray acceleration term) becomes zero (using Divergence theorem) since formula_40 when formula_41 is very large, which is typically the case in rocket motors. The drop size rate formula_42 is well modeled using vaporization mechanisms as


where formula_44 is independent of formula_1, but can depend on the surrounding gas. Defining the number of droplets per unit volume per unit radius and average quantities averaged over velocities,


If further assumed that formula_48 is independent of formula_1, and with a transformed coordinate


If the combustion chamber has varying cross-section area formula_51, a known function for formula_52 and with area formula_53 at the spraying location, then the solution is given by


where formula_55 are the number distribution and mean velocity at formula_32 respectively.


The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stated by Benoît Paul Émile Clapeyron in 1834 as a combination of the empirical Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. The ideal gas law is often written in an empirical form:


where formula_2, formula_3 and formula_4 are the pressure, volume and temperature; formula_5 is the amount of substance; and formula_6 is the ideal gas constant. It is the same for all gases.


It can also be derived from the microscopic kinetic theory, as was achieved (apparently independently) by August Krönig in 1856 and Rudolf Clausius in 1857.


Note that this law makes no comment as to whether a gas heats or cools during compression or expansion. An ideal gas may not change temperature, but most gases like air are not ideal and follow the Joule–Thomson effect.


The state of an amount of gas is determined by its pressure, volume, and temperature. The modern form of the equation relates these simply in two main forms. The temperature used in the equation of state is an absolute temperature: the appropriate SI unit is the kelvin.


In SI units, "p" is measured in pascals, "V" is measured in cubic metres, "n" is measured in moles, and "T" in kelvins (the Kelvin scale is a shifted Celsius scale, where 0.00 K = −273.15 °C, the lowest possible temperature). "R" has the value 8.314 J/(K·mol) ≈ 2 cal/(K·mol), or 0.0821 l·atm/(mol·K).


How much gas is present could be specified by giving the mass instead of the chemical amount of gas. Therefore, an alternative form of the ideal gas law may be useful. The chemical amount ("n") (in moles) is equal to total mass of the gas ("m") (in kilograms) divided by the molar mass ("M") (in kilograms per mole):


By replacing "n" with "m"/"M" and subsequently introducing density "ρ" = "m"/"V", we get:


Defining the specific gas constant "R"specific(r) as the ratio "R"/"M",


This form of the ideal gas law is very useful because it links pressure, density, and temperature in a unique formula independent of the quantity of the considered gas. Alternatively, the law may be written in terms of the specific volume "v", the reciprocal of density, as


It is common, especially in engineering and meteorological applications, to represent the specific gas constant by the symbol "R". In such cases, the universal gas constant is usually given a different symbol such as formula_21 or formula_22 to distinguish it. In any case, the context and/or units of the gas constant should make it clear as to whether the universal or specific gas constant is being referred to.


In statistical mechanics the following molecular equation is derived from first principles


where is the absolute pressure of the gas, is the number density of the molecules (given by the ratio = , in contrast to the previous formulation in which is the "number of moles"), is the absolute temperature, and is the Boltzmann constant relating temperature and energy, given by:


From this we notice that for a gas of mass , with an average particle mass of times the atomic mass constant, , (i.e., the mass is u) the number of molecules will be given by


and since , we find that the ideal gas law can be rewritten as


In SI units, is measured in pascals, in cubic metres, in kelvins, and


Combining the laws of Charles, Boyle and Gay-Lussac gives the combined gas law, which takes the same functional form as the ideal gas law save that the number of moles is unspecified, and the ratio of formula_27 to formula_4 is simply taken as a constant:


where formula_2 is the pressure of the gas, formula_3 is the volume of the gas, formula_4 is the absolute temperature of the gas, and formula_33 is a constant. When comparing the same substance under two different sets of conditions, the law can be written as


According to assumptions of the kinetic theory of ideal gases, we assume that there are no intermolecular attractions between the molecules of an ideal gas. In other words, its potential energy is zero. Hence, all the energy possessed by the gas is in the kinetic energy of the molecules of the gas.


This is the kinetic energy of "n" moles of a monatomic gas having 3 degrees of freedom; "x", "y", "z".


The table below essentially simplifies the ideal gas equation for a particular processes, thus making this equation easier to solve using numerical methods.


A thermodynamic process is defined as a system that moves from state 1 to state 2, where the state number is denoted by subscript. As shown in the first column of the table, basic thermodynamic processes are defined such that one of the gas properties ("P", "V", "T", "S", or "H") is constant throughout the process.


For a given thermodynamics process, in order to specify the extent of a particular process, one of the properties ratios (which are listed under the column labeled "known ratio") must be specified (either directly or indirectly). Also, the property for which the ratio is known must be distinct from the property held constant in the previous column (otherwise the ratio would be unity, and not enough information would be available to simplify the gas law equation).


In the final three columns, the properties ("p", "V", or "T") at state 2 can be calculated from the properties at state 1 using the equations listed.


a. In an isentropic process, system entropy ("S") is constant. Under these conditions, "p"1 "V"1"γ" = "p"2 "V"2"γ", where "γ" is defined as the heat capacity ratio, which is constant for a calorifically perfect gas. The value used for "γ" is typically 1.4 for diatomic gases like nitrogen (N2) and oxygen (O2), (and air, which is 99% diatomic). Also "γ" is typically 1.6 for mono atomic gases like the noble gases helium (He), and argon (Ar). In internal combustion engines "γ" varies between 1.35 and 1.15, depending on constitution gases and temperature.


b. In an isenthalpic process, system enthalpy ("H") is constant. In the case of free expansion for an ideal gas, there are no molecular interactions, and the temperature remains constant. For real gasses, the molecules do interact via attraction or repulsion depending on temperature and pressure, and heating or cooling does occur. This is known as the Joule–Thomson effect. For reference, the Joule–Thomson coefficient μJT for air at room temperature and sea level is 0.22 °C/bar.


Deviations from ideal behavior of real gases.


A residual property is defined as the difference between a real gas property and an ideal gas property, both considered at the same pressure, temperature, and composition.


The empirical laws that led to the derivation of the ideal gas law were discovered with experiments that changed only 2 state variables of the gas and kept every other one constant.


All the possible gas laws that could have been discovered with this kind of setup are:


where "P" stands for pressure, "V" for volume, "N" for number of particles in the gas and "T" for temperature; where formula_48 are not actual constants but are in this context because of each equation requiring only the parameters explicitly noted in it changing.


To derive the ideal gas law one does not need to know all 6 formulas, one can just know 3 and with those derive the rest or just one more to be able to get the ideal gas law, which needs 4.


Since each formula only holds when only the state variables involved in said formula change while the others remain constant, we cannot simply use algebra and directly combine them all. I.e. Boyle did his experiments while keeping "N" and "T" constant and this must be taken into account.


Keeping this in mind, to carry the derivation on correctly, one must imagine the gas being altered by one process at a time. The derivation using 4 formulas can look like this:


at first the gas has parameters formula_49


Say, starting to change only pressure and volume, according to Boyle's law, then:


Using then Eq. (5) to change the number of particles in the gas and the temperature,


Using then Eq. (6) to change the pressure and the number of particles,


Using then Charles's law to change the volume and temperature of the gas,


Using simple algebra on equations (7), (8), (9) and (10) yields the result:


Another equivalent result, using the fact that formula_61, where "n" is the number of moles in the gas and "R" is the universal gas constant, is:


where the numbers represent the gas laws numbered above.


If you were to use the same method used above on 2 of the 3 laws on the vertices of one triangle that has a "O" inside it, you would get the third.


Change only pressure and volume first: formula_37 (1´)


then only volume and temperature: formula_64 (2´)


then as we can choose any value for formula_65, if we set formula_66, Eq. (2´) becomes: formula_67(3´)


combining equations (1´) and (3´) yields formula_43 , which is Eq. (4), of which we had no prior knowledge until this derivation.


The ideal gas law can also be derived from first principles using the kinetic theory of gases, in which several simplifying assumptions are made, chief among which are that the molecules, or atoms, of the gas are point masses, possessing mass but no significant volume, and undergo only elastic collisions with each other and the sides of the container in which both linear momentum and kinetic energy are conserved.


The fundamental assumptions of the kinetic theory of gases imply that


Using the Maxwell–Boltzmann distribution, the fraction of molecules that have a speed in the range formula_70 to formula_71 is formula_72, where


and formula_33 denotes the Boltzmann constant. The root-mean-square speed can be calculated by


from which we get the ideal gas law:


Let q = ("q"x, "q"y, "q"z) and p = ("p"x, "p"y, "p"z) denote the position vector and momentum vector of a particle of an ideal gas, respectively. Let F denote the net force on that particle. Then the time-averaged kinetic energy of the particle is:<br>


where the first equality is Newton's second law, and the second line uses Hamilton's equations and the equipartition theorem. Summing over a system of "N" particles yields


By Newton's third law and the ideal gas assumption, the net force of the system is the force applied by the walls of the container, and this force is given by the pressure "P" of the gas. Hence


where dS is the infinitesimal area element along the walls of the container. Since the divergence of the position vector q is


where "dV" is an infinitesimal volume within the container and "V" is the total volume of the container.


which immediately implies the ideal gas law for "N" particles:


where "n" = "N"/"N"A is the number of moles of gas and "R" = "N"A"k"B is the gas constant.


For a "d"-dimensional system, the ideal gas pressure is:


where formula_87 is the volume of the "d"-dimensional domain in which the gas exists. Note that the dimensions of the pressure changes with dimensionality.


In cosmology, the equation of state of a perfect fluid is characterized by a dimensionless number formula_1, equal to the ratio of its pressure formula_2 to its energy density formula_3 :


It is closely related to the thermodynamic equation of state and ideal gas law.


The perfect gas equation of state may be written as


where formula_6 is the mass density, formula_7 is the particular gas constant, formula_8 is the temperature and formula_9 is a characteristic thermal speed of the molecules. Thus


where formula_11 is the speed of light, formula_12 and formula_13 for a "cold" gas.


FLRW equations and the equation of state.


The equation of state may be used in Friedmann–Lemaître–Robertson–Walker (FLRW) equations to describe the evolution of an isotropic universe filled with a perfect fluid. If formula_14 is the scale factor then


If the fluid is the dominant form of matter in a flat universe, then


In general the Friedmann acceleration equation is


where formula_19 is the cosmological constant and formula_20 is Newton's constant, and formula_21 is the second proper time derivative of the scale factor.


If we define (what might be called "effective") energy density and pressure as


the acceleration equation may be written as


The equation of state for ordinary non-relativistic 'matter' (e.g. cold dust) is formula_26, which means that its energy density decreases as formula_27, where formula_28 is a volume. In an expanding universe, the total energy of non-relativistic matter remains constant, with its density decreasing as the volume increases.


The equation of state for ultra-relativistic 'radiation' (including neutrinos, and in the very early universe other particles that later became non-relativistic) is formula_29 which means that its energy density decreases as formula_30. In an expanding universe, the energy density of radiation decreases more quickly than the volume expansion, because its wavelength is red-shifted.


Cosmic inflation and the accelerated expansion of the universe can be characterized by the equation of state of dark energy. In the simplest case, the equation of state of the cosmological constant is formula_31. In this case, the above expression for the scale factor is not valid and formula_32, where the constant "H" is the Hubble parameter. More generally, the expansion of the universe is accelerating for any equation of state formula_33. The accelerated expansion of the Universe was indeed observed. According to observations, the value of equation of state of cosmological constant is near -1.


Hypothetical phantom energy would have an equation of state formula_34, and would cause a Big Rip. Using the existing data, it is still impossible to distinguish between phantom formula_35 and non-phantom formula_36.


In an expanding universe, fluids with larger equations of state disappear more quickly than those with smaller equations of state. This is the origin of the flatness and monopole problems of the Big Bang: curvature has formula_37 and monopoles have formula_26, so if they were around at the time of the early Big Bang, they should still be visible today. These problems are solved by cosmic inflation which has formula_39. Measuring the equation of state of dark energy is one of the largest efforts of observational cosmology. By accurately measuring formula_1, it is hoped that the cosmological constant could be distinguished from quintessence which has formula_41.


A scalar field formula_42 can be viewed as a sort of perfect fluid with equation of state


where formula_44 is the time-derivative of formula_42 and formula_46 is the potential energy. A free formula_47 scalar field has formula_48, and one with vanishing kinetic energy is equivalent to a cosmological constant: formula_31. Any equation of state in between, but not crossing the formula_31 barrier known as the Phantom Divide Line (PDL), is achievable, which makes scalar fields useful models for many phenomena in cosmology.


In fluid mechanics, the Tait equation is an equation of state, used to relate liquid density to pressure. The equation was originally published by Peter Guthrie Tait in 1888 in the form


where formula_2 is the reference pressure (taken to be 1 atmosphere), formula_3 is the current pressure, formula_4 is the volume of fresh water at the reference pressure, formula_5 is the volume at the current pressure, and formula_6 are experimentally determined parameters.


Around 1895, the original isothermal Tait equation was replaced by Tammann with an equation of the form


The temperature-dependent version of the above equation is popularly known as the Tait equation and is commonly written as


The expression for the pressure in terms of the specific volume is


The tangent bulk modulus at pressure formula_3 is given by


Another popular isothermal equation of state that goes by the name "Tait equation" is the Murnaghan model which is sometimes expressed as


where formula_20 is the specific volume at pressure formula_3, formula_22 is the specific volume at pressure formula_2, formula_24 is the bulk modulus at formula_2, and formula_26 is a material parameter.


This equation, in pressure form, can be written as


where formula_28 are mass densities at formula_29, respectively.


For pure water, typical parameters are formula_2 = 101,325 Pa, formula_31 = 1000 kg/cu.m, formula_24 = 2.15 GPa, and formula_26 = 7.15.


Note that this form of the Tate equation of state is identical to that of the Murnaghan equation of state.


The tangent bulk modulus predicted by the MacDonald-Tait model is


A related equation of state that can be used to model liquids is the Tumlirz equation (sometimes called the Tammann equation and originally proposed by Tumlirz in 1909 and Tammann in 1911 for pure water). This relation has the form


where formula_36 is the specific volume, formula_3 is the pressure, formula_38 is the salinity, formula_39 is the temperature, and formula_40 is the specific volume when formula_41, and formula_42 are parameters that can be fit to experimental data.


The Tumlirz-Tammann version of the Tait equation for fresh water, i.e., when formula_43, is


For pure water, the temperature-dependence of formula_45 are:


In the above fits, the temperature formula_39 is in degrees Celsius, formula_2 is in bars, formula_40 is in cc/gm, and formula_50 is in bars-cc/gm.


The inverse Tumlirz-Tammann-Tait relation for the pressure as a function of specific volume is


The Tumlirz-Tammann-Tait formula for the instantaneous tangent bulk modulus of pure water is a quadratic function of formula_3 (for an alternative see )


Like Wilson (1964), Renon & Prausnitz (1968) began with local composition theory, but instead of using the Flory–Huggins volumetric expression as Wilson did, they assumed local compositions followed


with a new "non-randomness" parameter α. The excess Gibbs free energy was then determined to be


Unlike Wilson's equation, this can predict partially miscible mixtures. However the cross term, like Wohl's expansion, is more suitable for formula_7 than formula_8, and experimental data is not always sufficiently plentiful to yield three meaningful values, so later attempts to extend Wilson's equation to partial miscibility (or to extend Guggenheim's quasichemical theory for nonrandom mixtures to Wilson's different-sized molecules) eventually yielded variants like UNIQUAC.


For a binary mixture the following function are used:


Here, formula_11 and formula_12 are the dimensionless interaction parameters, which are related to the interaction energy parameters formula_13 and formula_14 by:


Here "R" is the gas constant and "T" the absolute temperature, and "Uij" is the energy between molecular surface "i" and "j". "Uii" is the energy of evaporation. Here "Uij" has to be equal to "Uji", but formula_16 is not necessary equal to formula_17.


The parameters formula_18 and formula_19 are the so-called non-randomness parameter, for which usually formula_18 is set equal to formula_19. For a liquid, in which the local distribution is random around the center molecule, the parameter formula_22. In that case the equations reduce to the one-parameter Margules activity model:


In practice, formula_18 is set to 0.2, 0.3 or 0.48. The latter value is frequently used for aqueous systems. The high value reflects the ordered structure caused by hydrogen bonds. However, in the description of liquid-liquid equilibria the non-randomness parameter is set to 0.2 to avoid wrong liquid-liquid description. In some cases a better phase equilibria description is obtained by setting formula_25. However this mathematical solution is impossible from a physical point of view, since no system can be more random than random (formula_18 =0). In general NRTL offers more flexibility in the description of phase equilibria than other activity models due to the extra non-randomness parameters. However, in practice this flexibility is reduced in order to avoid wrong equilibrium description outside the range of regressed data.


The limiting activity coefficients, also known as the activity coefficients at infinite dilution, are calculated by:


The expressions show that at formula_22 the limiting activity coefficients are equal. This situation that occurs for molecules of equal size, but of different polarities.<br> It also shows, since three parameters are available, that multiple sets of solutions are possible.


The general equation for formula_29 for species formula_30 in a mixture of formula_31 components is:


There are several different equation forms for formula_36 and formula_37, the most general of which are shown above.


To describe phase equilibria over a large temperature regime, i.e. larger than 50 K, the interaction parameter has to be made temperature dependent.


Two formats are frequently used. The extended Antoine equation format:


Here the logarithmic and linear terms are mainly used in the description of liquid-liquid equilibria (miscibility gap).


The other format is a second-order polynomial format:


The NRTL parameters are fitted to activity coefficients that have been derived from experimentally determined phase equilibrium data (vapor–liquid, liquid–liquid, solid–liquid) as well as from heats of mixing. The source of the experimental data are often factual data banks like the Dortmund Data Bank. Other options are direct experimental work and predicted activity coefficients with UNIFAC and similar models.


Determination of NRTL parameters from LLE data is more complicated than parameter regression from VLE data as it involves solving isoactivity equations which are highly non-linear. In addition, parameters obtained from LLE may not always represent the real activity of components due to lack of knowledge on the activity values of components in the data regression. For this reason it is necessary to confirm the consistency of the obtained parameters in the whole range of compositions (including binary subsystems, experimental and calculated lie-lines, Hessian matrix, etc.).


An equation of state introduced by R. H. Cole


where formula_2 is a reference density, formula_3 is the adiabatic index, and formula_4 is a parameter with pressure units.


In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state; named after Dutch physicist Johannes Diderik van der Waals) is an equation of state that generalizes the ideal gas law based on plausible reasons that real gases do not act ideally. The ideal gas law treats gas molecules as point particles that interact with their containers but not each other, meaning they neither take up space nor change kinetic energy during collisions (i.e. all collisions are perfectly elastic). The ideal gas law states that volume ("V") occupied by "n" moles of any gas has a pressure ("P") at temperature ("T") in kelvins given by the following relationship, where "R" is the gas constant:


To account for the volume that a real gas molecule takes up, the Van der Waals equation replaces "V" in the ideal gas law with formula_2, where "Vm" is the molar volume of the gas and "b" is the volume that is occupied by one mole of the molecules. This leads to:


The second modification made to the ideal gas law accounts for the fact that gas molecules do in fact interact with each other (they usually experience attraction at low pressures and repulsion at high pressures) and that real gases therefore show different compressibility than ideal gases. Van der Waals provided for intermolecular interaction by adding to the observed pressure "P" in the equation of state a term formula_4, where "a" is a constant whose value depends on the gas. The Van der Waals equation is therefore written as:


and can also be written as the equation below:


where "Vm" is the molar volume of the gas, "R" is the universal gas constant, "T" is temperature, "P" is pressure, and "V" is volume. When the molar volume "Vm" is large, "b" becomes negligible in comparison with "Vm", "a/Vm2" becomes negligible with respect to "P", and the Van der Waals equation reduces to the ideal gas law, "PVm=RT".


The Van der Waals equation is a thermodynamic equation of state based on the theory that fluids are composed of particles with non-zero volumes, and subject to a (not necessarily pairwise) inter-particle attractive force. It was based on work in theoretical physical chemistry performed in the late 19th century by Johannes Diderik van der Waals, who did related work on the attractive force that also bears his name. The equation is known to be based on a traditional set of derivations deriving from Van der Waals' and related efforts, as well as a set of derivation based in statistical thermodynamics, see below.


The equation relates four state variables: the pressure of the fluid "p", the total volume of the fluid's container "V", the number of particles "N", and the absolute temperature of the system "T".


The intensive, microscopic form of the equation is:


is the volume of the container occupied by each particle (not the velocity of a particle), and "k"B is the Boltzmann constant. It introduces two new parameters: "a"′, a measure of the average attraction between particles, and "b"′, the volume excluded from "v" by one particle.


The equation can be also written in extensive, molar form:


is a measure of the average attraction between particles,


is the volume excluded by a mole of particles,


is the universal gas constant, "k"B is the Boltzmann constant, and "N"A is the Avogadro constant.


A careful distinction must be drawn between the volume "available to" a particle and the volume "of" a particle. In the intensive equation, "v" equals the total space available to each particle, while the parameter "b"′ is proportional to the proper volume of a single particle – the volume bounded by the atomic radius. This is subtracted from "v" because of the space taken up by one particle. In Van der Waals' original derivation, given below, "b"' is four times the proper volume of the particle. Observe further that the pressure "p" goes to infinity when the container is completely filled with particles so that there is no void space left for the particles to move; this occurs when "V" = "nb".


If a mixture of formula_14 gases is being considered, and each gas has its own formula_15 (attraction between molecules) and formula_16 (volume occupied by molecules) values, then formula_15 and formula_16 for the mixture can be calculated as


and the rule of adding partial pressures becomes invalid if the numerical result of the equation formula_26 is significantly different from the ideal gas equation formula_27 .


The Van der Waals equation can also be expressed in terms of reduced properties:


This yields a critical compressibility factor of 3/8. Reasons for modification of ideal gas equation: The equation state for ideal gas is PV=RT. In the derivation of ideal gas laws on the basis of kinetic theory of gases some assumption have been made.


The "Van der Waals equation" is mathematically simple, but it nevertheless predicts the experimentally observed transition between vapor and liquid, and predicts critical behaviour. It also adequately predicts and explains the Joule–Thomson effect (temperature change during adiabatic expansion), which is not possible in ideal gas.


However, the values of physical quantities as predicted with the Van der Waals equation of state "are in very poor agreement with experiment", so the model's utility is limited to qualitative rather than quantitative purposes. Empirically-based corrections can easily be inserted into the Van der Waals model (see Maxwell's correction, below), but in so doing, the modified expression is no longer as simple an analytical model; in this regard, other models, such as those based on the principle of corresponding states, achieve a better fit with roughly the same work.


Even with its acknowledged shortcomings, the pervasive use of the "Van der Waals equation" in standard university physical chemistry textbooks makes clear its importance as a pedagogic tool to aid understanding fundamental physical chemistry ideas involved in developing theories of vapour–liquid behavior and equations of state. In addition, other (more accurate) equations of state such as the Redlich–Kwong and Peng–Robinson equation of state are essentially modifications of the Van der Waals equation of state.


Textbooks in physical chemistry generally give two derivations of the title equation. One is the conventional derivation that goes back to Van der Waals, a mechanical equation of state that cannot be used to specify all thermodynamic functions; the other is a statistical mechanics derivation that makes explicit the intermolecular potential neglected in the first derivation. A particular advantage of the statistical mechanical derivation is that it yields the partition function for the system, and allows all thermodynamic functions to be specified (including the mechanical equation of state).


Consider one mole of gas composed of non-interacting point particles that satisfy the ideal gas law:(see any standard Physical Chemistry text, op. cit.)


Next, assume that all particles are hard spheres of the same finite radius "r" (the Van der Waals radius). The effect of the finite volume of the particles is to decrease the available void space in which the particles are free to move. We must replace "V" by "V" − "b", where "b" is called the "excluded volume" (per mole) or "co-volume". The corrected equation becomes


The excluded volume formula_16 is not just equal to the volume occupied by the solid, finite-sized particles, but actually four times the total molecular volume for one mole of a Van der waals' gas. To see this, we must realize that a particle is surrounded by a sphere of radius 2"r" (two times the original radius) that is forbidden for the centers of the other particles. If the distance between two particle centers were to be smaller than 2"r", it would mean that the two particles penetrate each other, which, by definition, hard spheres are unable to do.


The excluded volume for the two particles (of average diameter "d" or radius "r") is


which, divided by two (the number of colliding particles), gives the excluded volume per particle:


So "b′" is four times the proper volume of the particle. It was a point of concern to Van der Waals that the factor four yields an upper bound; empirical values for "b′" are usually lower. Of course, molecules are not infinitely hard, as Van der Waals thought, and are often fairly soft. To obtain the excluded volume per mole we just need to multiply by the number of molecules in a mole, i.e. by the avogadro number:


The number of particles in the surface layers is, again by assuming homogeneity, also proportional to the density. In total, the force on the walls is decreased by a factor proportional to the square of the density, and the pressure (force per unit surface) is decreased by


Upon writing "n" for the number of moles and "nV"m = "V", the equation obtains the second form given above,


It is of some historical interest to point out that Van der Waals, in his Nobel prize lecture, gave credit to Laplace for the argument that pressure is reduced proportional to the square of the density.


The canonical partition function "Z" of an ideal gas consisting of "N = nN"A identical (non-interacting) particles, is:


where formula_40 is the thermal de Broglie wavelength,


with the usual definitions: "h" is Planck's constant, "m" the mass of a particle, "k" Boltzmann's constant and "T" the absolute temperature. In an ideal gas "z" is the partition function of a single particle in a container of volume "V". In order to derive the Van der Waals equation we assume now that each particle moves independently in an average potential field offered by the other particles. The averaging over the particles is easy because we will assume that the particle density of the Van der Waals fluid is homogeneous.


The interaction between a pair of particles, which are hard spheres, is taken to be


"r" is the distance between the centers of the spheres and "d" is the distance where the hard spheres touch each other (twice the Van der Waals radius). The depth of the Van der Waals well is formula_43.


Because the particles are not coupled under the mean field Hamiltonian, the mean field approximation of the total partition function still factorizes,


but the intermolecular potential necessitates two modifications to "z". First, because of the finite size of the particles, not all of "V" is available, but only "V − Nb"', where (just as in the conventional derivation above)


exp[" - ϕ/2kT"] to take care of the average intermolecular potential. We divide here the potential by two because this interaction energy is shared between two particles. Thus


The total attraction felt by a single particle is


where we assumed that in a shell of thickness d"r" there are "N/V" 4"π" "r"2"dr" particles. This is a mean field approximation; the position of the particles is averaged. In reality the density close to the particle is different than far away as can be described by a pair correlation function. Furthermore, it is neglected that the fluid is enclosed


between walls. Performing the integral we get


so that we only have to differentiate the terms containing V. We get


Below the critical temperature, the Van der Waals equation seems to predict qualitatively incorrect relationships. Unlike for ideal gases, the p-V isotherms oscillate with a relative minimum ("d") and a relative maximum ("e"). Any pressure between "pd" and "pe" appears to have 3 stable volumes, contradicting the experimental observation that two state variables completely determine a one-component system's state. Moreover, the isothermal compressibility is negative between "d" and "e" (equivalently formula_52), which cannot describe a system at equilibrium.


To address these problems, James Clerk Maxwell replaced the isotherm between points "a" and "c" with a horizontal line positioned so that the areas of the two shaded regions would be equal (replacing the "a"-"d"-"b"-"e"-"c" curve with a straight line from "a" to "c"); this portion of the isotherm corresponds to the liquid-vapor equilibrium. The regions of the isotherm from "a"–"d" and from "c"–"e" are interpreted as metastable states of super-heated liquid and super-cooled vapor, respectively. The equal area rule can be expressed as:


where "pV" is the vapor pressure (flat portion of the curve), "VL" is the volume of the pure liquid phase at point "a" on the diagram, and "VG" is the volume of the pure gas phase at point "c" on the diagram. A two-phase mixture at "pV" will occupy a total volume between "VL" and "VG", as determined by Maxwell's lever rule.