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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 |
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