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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. |
Maxwell justified the rule based on the fact that the area on a "pV" diagram corresponds to mechanical work, saying that work done on the system in going from "c" to "b" should equal work released on going from "a" to "b". This is because the change in free energy "A"("T","V") equals the work done during a reversible process, and, as a state variable, the free energy must be path-independent. In particular, the value of "A" at point "b" should be the same regardless of whether the path taken is from left or right across the horizontal isobar, or follows the original Van der Waals isotherm. |
This derivation is not entirely rigorous, since it requires a reversible path through a region of thermodynamic instability, while "b" is unstable. Nevertheless, modern derivations from chemical potential reach the same conclusion, and it remains a necessary modification to the Van der Waals and to any other analytic equation of state. |
The Maxwell equal area rule can also be derived from an assumption of equal chemical potential "μ" of coexisting liquid and vapour phases. On the isotherm shown in the above plot, points "a" and "c" are the only pair of points which fulfill the equilibrium condition of having equal pressure, temperature and chemical potential. It follows that systems with volumes intermediate between these two points will consist of a mixture of the pure liquid and gas with specific volumes equal to the pure liquid and gas phases at points "a" and "c". |
The Van der Waals equation may be solved for "VG" and "VL" as functions of the temperature and the vapor pressure "pV". Since: |
where "A" is the Helmholtz free energy, it follows that the equal area rule can be expressed as: |
Since the gas and liquid volumes are functions of "pV" and "T" only, this equation is then solved numerically to obtain "pV" as a function of temperature (and number of particles "N"), which may then be used to determine the gas and liquid volumes. |
A pseudo-3D plot of the locus of liquid and vapor volumes versus temperature and pressure is shown |
in the accompanying figure. One sees that the two locii meet at the critical point (1,1,1) smoothly. An isotherm of the Van der Waals fluid taken at "T r" = 0.90 is also shown where the intersections of the isotherm with the loci illustrate the construct's requirement that the two areas (red and blue, shown) are equal. |
We reiterate that the extensive volume "V" is related to the volume per particle "v=V/N" where "N = nN"A is the number of particles in the system. |
The equation of state does not give us all the thermodynamic parameters of the system. We can take the equation for the Helmholtz energy "A" |
From the equation derived above for ln"Q", we find |
Where Φ is an undetermined constant, which may be taken from the Sackur–Tetrode equation for an ideal gas to be: |
This equation expresses "A" in terms of its natural variables "V" and "T" , and therefore gives us all thermodynamic information about the system. The mechanical equation of state was already derived above |
The entropy equation of state yields the entropy ("S" ) |
from which we can calculate the internal energy |
Similar equations can be written for the other thermodynamic potential and the chemical potential, but expressing any potential as a function of pressure "p" will require the solution of a third-order polynomial, which yields a complicated expression. Therefore, expressing the enthalpy and the Gibbs energy as functions of their natural variables will be complicated. |
Although the material constant "a" and "b" in the usual form of the Van der Waals equation differs for every single fluid considered, the equation can be recast into an invariant form applicable to "all" fluids. |
Defining the following reduced variables ("fR", "fC" are the reduced and critical variable versions of "f", respectively), |
The first form of the Van der Waals equation of state given above can be recast in the following reduced form: |
This equation is "invariant" for all fluids; that is, the same reduced form equation of state applies, no matter what "a" and "b" may be for the particular fluid. |
This invariance may also be understood in terms of the principle of corresponding states. If two fluids have the same reduced pressure, reduced volume, and reduced temperature, we say that their states are corresponding. The states of two fluids may be corresponding even if their measured pressure, volume, and temperature are very different. If the two fluids' states are corresponding, they exist in the same regime of the reduced form equation of state. Therefore, they will respond to changes in roughly the same way, even though their measurable physical characteristics may differ significantly. |
The Van der Waals equation is a cubic equation of state; in the reduced formulation the cubic equation is: |
At the critical temperature, where formula_66 we get as expected |
For "TR" < 1, there are 3 values for "vR". |
For "TR" > 1, there is 1 real value for "vR". |
The solution of this equation for the case where there are three separate roots may be found at Maxwell construction |
The equation is also usable as a PVT equation for compressible fluids (e.g. polymers). In this case specific volume changes are small and it can be written in a simplified form: |
where "p" is the pressure, "V" is specific volume, "T" is the temperature and "A, B, C" are parameters. |
The Murnaghan equation of state is a relationship between the volume of a body and the pressure to which it is subjected. This is one of many state equations that have been used in earth sciences and shock physics to model the behavior of matter under conditions of high pressure. It owes its name to Francis D. Murnaghan who proposed it in 1944 to reflect material behavior under a pressure range as wide as possible to reflect an experimentally established fact: the more a solid is compressed, the more difficult it is to compress further. |
The Murnaghan equation is derived, under certain assumptions, from the equations of continuum mechanics. It involves two adjustable parameters: the modulus of incompressibility "K"0 and its first derivative with respect to the pressure, "K"'0, both measured at ambient pressure. In general, these coefficients are determined by a regression on experimentally obtained values of volume "V" as a function of the pressure "P". These experimental data can be obtained by X-ray diffraction or by shock tests. Regression can also be performed on the values of the energy as a function of the volume obtained from ab-initio and molecular dynamics calculations. |
The Murnaghan equation of state is typically expressed as: |
If the reduction in volume under compression is low, i.e., for "V"/"V"0 greater than about 90%, the Murnaghan equation can model experimental data with satisfactory accuracy. Moreover, unlike many proposed equations of state, it gives an explicit expression of the volume as a function of pressure "V"("P"). But its range of validity is limited and physical interpretation inadequate. However, this equation of state continues to be widely used in models of solid explosives. Of more elaborate equations of state, the most used in earth physics is the Birch–Murnaghan equation of state. In shock physics of metals and alloys, another widely used equation of state is the Mie–Grüneisen equation of state. |
The study of the internal structure of the earth through the knowledge of the mechanical properties of the constituents of the inner layers of the planet involves extreme conditions; the pressure can be counted in hundreds of gigapascal and temperatures in thousands of degrees. The study of the properties of matter under these conditions can be done experimentally through devices such as diamond anvil cell for static pressures, or by subjecting the material to shock waves. It also gave rise to theoretical work to determine the equation of state, that is to say the relations among the different parameters that define in this case the state of matter: the volume (or density), temperature and pressure. |
Dozens of equations have been proposed by various authors. These are empirical relationships, the quality and relevance depend on the use made of it and can be judged by different criteria: the number of independent parameters that are involved, the physical meaning that can be assigned to these parameters, the quality of the experimental data, and the consistency of theoretical assumptions that underlie their ability to extrapolate the behavior of solids at high compression. |
Generally, at constant temperature, the bulk modulus is defined by: |
The easiest way to get an equation of state linking "P" and "V" is to assume that "K" is constant, that is to say, independent of pressure and deformation of the solid, then we simply find the Hooke's law. In this case, the volume decreases exponentially with pressure. This is not a satisfactory result because it is experimentally established that as a solid is compressed, it becomes more difficult to compress. To go further, we must take into account the variations of the elastic properties of the solid with compression. |
The assumption Murnaghan is to assume that the bulk modulus is a linear function of pressure : |
Murnaghan equation is the result of the integration of the differential equation: |
We can also express the volume depending on the pressure: |
This simplified presentation is however criticized by Poirier as lacking rigor. The same relationship can be shown in a different way from the fact that the incompressibility of the product of the modulus and the thermal expansion coefficient is not dependent on the pressure for a given material. This equation of state is also a general case of the older Polytrope relation which also has a constant power relation. |
In some circumstances, particularly in connection with ab initio calculations, the expression of the energy as a function of the volume will be preferred, which can be obtained by integrating the above equation according to the relationship "P" = −"dE"/"dV" . It can be written to "K"'0 different from 3, |
Despite its simplicity, the Murnaghan equation is able to reproduce the experimental data for a range of pressures that can be quite large, on the order of "K"0/2. It also remains satisfactory as the ratio "V"/"V"0 remains above about 90%. In this range, the Murnaghan equation has an advantage compared to other equations of state if one wants to express the volume as a function of pressure. |
Nevertheless, other equations may provide better results and several theoretical and experimental studies show that the Murnaghan equation is unsatisfactory for many problems. Thus, to the extent that the ratio "V"/"V"0 becomes very low, the theory predicts that "K"' goes to 5/3, which is the Thomas–Fermi limit. However, in the Murnaghan equation, "K"' is constant and set to its initial value. In particular, the value "K"'0 = 5/3 becomes inconsistent with the theory under some situations. In fact, when extrapolated, the behavior predicted by the Murnaghan equation becomes quite quickly unlikely. |
Regardless of this theoretical argument, experience clearly shows that "K"' decreases with pressure, or in other words that the second derivative of the incompressibility modulus "K"" is strictly negative. A second order theory based on the same principle (see next section) can account for this observation, but this approach is still unsatisfactory. Indeed, it leads to a negative bulk modulus in the limit where the pressure tends to infinity. In fact, this is an inevitable contradiction whatever polynomial expansion is chosen because there will always be a dominant term that diverges to infinity. |
These important limitations have led to the abandonment of the Murnaghan equation, which W. Holzapfel calls "a useful mathematical form without any physical justification". In practice, the analysis of compression data is done by using more sophisticated equations of state. The most commonly used within the science community is the Birch–Murnaghan equation, second or third order in the quality of data collected. |
Finally, a very general limitation of this type of equation of state is their inability to take into account the phase transitions induced by the pressure and temperature of melting, but also multiple solid-solid transitions that can cause abrupt changes in the density and bulk modulus based on the pressure. |
In practice, the Murnaghan equation is used to perform a regression on a data set, where one gets the values of the coefficients "K"0 and "K"'0. These coefficients obtained, and knowing the value of the volume to ambient conditions, then we are in principle able to calculate the volume, density and bulk modulus for any pressure. |
The data set is mostly a series of volume measurements for different values of applied pressure, obtained mostly by X-ray diffraction. It is also possible to work on theoretical data, calculating the energy for different values of volume by ab initio methods, and then regressing these results. This gives a theoretical value of the modulus of elasticity which can be compared to experimental results. |
The following table lists some of the results of different materials, with the sole purpose of illustrating some numerical analyses that have been made using the Murnaghan equation, without prejudice to the quality of the models obtained. Given the criticisms that have been made in the previous section on the physical meaning of the Murnaghan equation, these results should be considered with caution. |
To improve the models or avoid criticism outlined above, several generalizations of the Murnaghan equation have been proposed. They usually consist in dropping a simplifying assumption and adding another adjustable parameter. This can improve the qualities of refinement, but also lead to complicated expressions. The question of the physical meaning of these additional parameters is also raised. |
A possible strategy is to include an additional term "P"2 in the previous development, requiring that formula_7. Solving this differential equation gives the equation of the second-order Murnaghan: |
where formula_9. Found naturally in the first order equation taking formula_10. Developments to an order greater than 2 are possible in principle, but at the cost of adding an adjustable parameter for each term. |
In physics and thermodynamics, an equation of state is a thermodynamic equation relating state variables which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature (PVT), or internal energy. Equations of state are useful in describing the properties of fluids, mixtures of fluids, solids, and the interior of stars. |
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