All parameters, including temperature, depend on each other. This dependence is expressed by equations like
F(X 1 ,X 2 ,...,x 1 ,x 2 ,...,T) = 0,
where X 1, X 2,... are generalized forces, x 1, x 2,... are generalized coordinates, and T is temperature. Equations that establish the relationship between parameters are called equations of state.
Equations of state are given for simple systems, mainly for gases. For liquids and solids, assumed, as a rule, to be incompressible, practically no equations of state were proposed.
By the middle of the twentieth century. a significant number of equations of state for gases were known. However, the development of science has taken such a path that almost all of them have not found application. The only equation of state that continues to be widely used in thermodynamics is the equation of state of an ideal gas.
Ideal gas is a gas whose properties are similar to that of a low-molecular-weight substance at very low pressure and a relatively high temperature (quite far from the condensation temperature).
For an ideal gas:
Boyle's law - Mariotte(at a constant temperature, the product of gas pressure and its volume remains constant for given quantity substances)
Gay-Lussac's law(at constant pressure the ratio of gas volume to temperature remains constant)
Charles's law(at constant volume, the ratio of gas pressure to temperature remains constant)
.
S. Carnot combined the above relations into a single equation of the type
.
B. Clapeyron gave this equation a form close to the modern one:
The volume V included in the equation of state of an ideal gas refers to one mole of the substance. It is also called molar volume.
The generally accepted name for the constant R is the universal gas constant (very rarely you can find the name “Clapeyron’s constant” ). Its value is
R=8.31431J/mol TO.
Approaching a real gas to an ideal one means achieving such large distances between molecules that their own volume and the possibility of interaction can be completely neglected, i.e. the existence of forces of attraction or repulsion between them.
Van der Waals proposed an equation that takes these factors into account in the following form:
,
where a and b are constants determined for each gas separately. The remaining quantities included in the van der Waals equation have the same meaning as in the Clapeyron equation.
The possibility of the existence of an equation of state means that to describe the state of the system, not all parameters can be specified, but their number is less by one, since one of them can be determined (at least hypothetically) from the equation of state. For example, to describe the state of an ideal gas, it is enough to indicate only one of the following pairs: pressure and temperature, pressure and volume, volume and temperature.
Volume, pressure and temperature are sometimes called external parameters of the system.
If simultaneous changes in volume, pressure and temperature are allowed, then the system has two independent external parameters.
The system, located in a thermostat (a device that ensures constant temperature) or a manostat (a device that ensures constant pressure), has one independent external parameter.
Since the equation of state pV = nRT has a simple form and reflects with reasonable accuracy the behavior of many gases in wide range external conditions, it is very useful. But, of course, it is not universal. It is obvious that this equation does not obey any substance in the liquid or solid state. There are no condensed substances whose volume would decrease by half when the pressure doubles. Even gases under severe compression or near the condensation point exhibit noticeable deviations from this behavior. Many other more complex equations of state have been proposed. Some of them are different high accuracy in a limited area of change in external conditions. Some apply to special classes of substances. There are equations that apply to a wider class of substances at more widely varying external conditions, but they are not very accurate. We won't spend time looking at these equations of state in detail here, but we'll still give some insight into them.
Let us assume that the gas molecules are perfectly elastic solid balls, so small that their total volume can be neglected in comparison with the volume occupied by the gas. Let us also assume that there are no attractive or repulsive forces between the molecules and that they move completely chaotically, colliding randomly with each other and with the walls of the container. If we apply the elementary classical mechanics, then we obtain the relation pV = RT without resorting to any generalizations of experimental data such as the Boyle-Mariotte and Charles-Gay-Luss laws. In other words, the gas that we called “ideal” behaves as a gas consisting of very small solid balls that interact with each other only at the moment of collisions should behave. The pressure exerted by such a gas on any surface is simply equal to the average amount of momentum transferred per unit time by the molecules to a unit of surface upon collision with it. When a molecule of mass m hits a surface, having a velocity component perpendicular to the surface, and is reflected with a velocity component, then the resulting momentum transferred to the surface, according to the laws of mechanics, is equal to These velocities are quite high (several hundred meters per second for air under normal conditions), therefore The collision time is very short and the transfer of momentum occurs almost instantly. But the collisions are so numerous (about 1023 per 1 cm2 per 1 s in air at atmospheric pressure) that when measured by any instrument, the pressure turns out to be absolutely constant in time and continuous.
Indeed, most direct measurements and observations show that gases are a continuous medium. The conclusion that they must consist of a large number of individual molecules is purely speculative.
We know from experience that real gases do not obey the rules of behavior predicted by the ideal model just described. At sufficiently low temperatures and sufficiently high pressures, any gas condenses into a liquid or solid state, which, compared to a gas, can be considered incompressible. Thus, the total volume of molecules cannot always be neglected compared to the volume of the container. It is also clear that there are attractive forces between molecules, which at sufficiently low temperatures can bind molecules, leading to the formation of a condensed form of the substance. These considerations suggest that one way to obtain an equation of state that is more general than that of an ideal gas is to take into account the finite volume of real molecules and the attractive forces between them.
Taking into account molecular volume is not difficult, at least at a qualitative level. Let us simply assume that the free volume available for the movement of molecules is less than the total volume of the gas V by an amount of 6, which is related to the size of the molecules and is sometimes called the bound volume. Thus, we must replace V in the ideal gas equation of state with (V - b); then we get
This relationship is sometimes called the Clausius equation of state in honor of the German physicist Rudolf Clausius, who played a major role in the development of thermodynamics. We will learn more about his work in the next chapter. Note that equation (5) is written for 1 mole of gas. For n moles you need to write p(V-nb) = nRT.
Taking into account the forces of attraction between molecules is somewhat more difficult. A molecule located in the center of the gas volume, i.e., far from the walls of the vessel, will “see” same number molecules in all directions. Consequently, the attractive forces are equal in all directions and cancel each other out, so that no net force arises. When a molecule approaches the wall of a container, it “sees” more molecules behind itself than in front of it. As a result, an attractive force appears directed towards the center of the vessel. The movement of the molecule is somewhat restrained, and it hits the wall of the vessel less forcefully than in the absence of attractive forces.
Since the pressure of a gas is due to the transfer of momentum by molecules colliding with the walls of the container (or with any other surface located inside the gas), the pressure created by attracting molecules is somewhat less than the pressure created by the same molecules in the absence of attraction. It turns out that the decrease in pressure is proportional to the square of the gas density. Therefore we can write
where p is the density in moles per unit volume, is the pressure created by an ideal gas of non-attracting molecules, and a is the proportionality coefficient characterizing the magnitude of the attractive forces between molecules of a given type. Recall that , where n is the number of moles. Then relation (b) can be rewritten for 1 mole of gas in a slightly different form:
where a has a characteristic value for a given type of gas. The right side of equation (7) represents the “corrected” ideal gas pressure, which needs to be used to replace p in the equation. If we take into account both corrections, one due to volume in accordance with (b) and the other due to attractive forces according to (7), we obtain for 1 mole of gas
This equation was first proposed by the Dutch physicist D. van der Waals in 1873. For n moles it takes the form
The van der Waals equation takes into account in a simple and visual form two effects that cause deviations in the behavior of real gases from the ideal. It is obvious that the surface representing the van der Waals equation of state in p, V, Ty space cannot be as simple as the surface corresponding to an ideal gas. Part of such a surface for specific values of a and b is shown in Fig. 3.7. Isotherms are shown as solid lines. Isotherms corresponding to temperatures above the temperature at which the so-called critical isotherm corresponds do not have minima or inflections and look similar to the ideal gas isotherms shown in Fig. 3.6. At temperatures below isotherms have maxima and minima. At sufficiently low temperatures, there is a region in which the pressure becomes negative, as shown by the portions of the isotherms depicted by dashed lines. These humps and dips, as well as the region of negative pressures, do not correspond to physical effects, but simply reflect the shortcomings of the van der Waals equation, its inability to describe the true equilibrium behavior of real substances.
Rice. 3.7. Surface p - V - T for a gas obeying the van der Waals equation.
In fact, in real gases at lower temperatures and high enough pressures, the forces of attraction between molecules lead to the condensation of the gas into a liquid or solid state. Thus, the anomalous region of peaks and dips in isotherms in the negative pressure region, which is predicted by the van der Waals equation, in real substances corresponds to the region of the mixed phase, in which vapor and a liquid or solid state coexist. Rice. 3.8 illustrates this situation. Such “discontinuous” behavior cannot be described at all by any relatively simple and “continuous” equation.
Despite its shortcomings, the van der Waals equation is useful for describing corrections to the ideal gas equation. The values of a and b for various gases are determined from experimental data, some typical examples are given in table. 3.2. Unfortunately, for any given gas there are no single values of a and b that will provide an accurate description of the relationship between p, V and T over a wide range using the van der Waals equation.
Table 3.2. Characteristic values van der Waals constants
However, the values indicated in the table give us some quality information about the expected magnitude of deviation from the behavior of an ideal gas.
It is instructive to consider specific example and compare the results obtained using the ideal gas equation, the Clausius equation, and the van der Waals equation with measured data. Consider 1 mole of water vapor in a volume of 1384 cm3 at a temperature of 500 K. Remembering that (mol K), and using the values from table. 3.2, we get
a) from the equation of state of an ideal gas:
b) from the Clausius equation of state: atm;
c) from the van der Waals equation of state:
d) from experimental data:
For these specific conditions, the ideal gas law overestimates the pressure by about 14%, Eq.
Rice. 3.8. A surface for a substance that contracts when cooled. A surface like this cannot be described by a single equation of state and must be constructed based on experimental data.
The Clausius equation gives an even larger error of about 16%, and the van der Waals equation overestimates the pressure by about 5%. Interestingly, the Clausius equation gives a larger error than the ideal gas equation. The reason is that the correction for the finite volume of molecules increases the pressure, while the term for attraction decreases it. Thus, these amendments partially compensate each other. The ideal gas law, which does not take into account either one or the other correction, gives a closer approximation actual meaning pressure than the Clausius equation, which takes into account only its increase due to a decrease in free volume. At very high densities, the correction for the volume of molecules becomes much more significant and the Clausius equation turns out to be more accurate than the ideal gas equation.
Generally speaking, for real substances we do not know the explicit relationship between p, V, T and n. For most solids and liquids there are not even rough approximations. Nevertheless, we are firmly convinced that such a relationship exists for every substance and that the substance obeys it.
A piece of aluminum will occupy a certain volume, always exactly the same, if the temperature and pressure are at the given values. We write this general statement in mathematical form:
This entry asserts the existence of some functional relationship between p, V, T and n, which can be expressed by an equation. (If all terms of such an equation are moved to the left, the right-hand side will obviously be equal to zero.) Such an expression is called an implicit equation of state. It means the existence of some relationship between variables. It also says that we do not know what this ratio is, but the substance “knows” it! Rice. 3.8 allows us to imagine how complex an equation must be that would describe real matter in a wide range of variables. This figure shows the surface of a real substance that contracts when it freezes (almost all substances behave this way except water). We are not skilled enough to predict by calculation what volume a substance will occupy given arbitrarily given values of p, T, and n, but we are absolutely sure that the substance “knows” what volume it will occupy. This confidence is always confirmed by experimental testing. Matter always behaves in an unambiguous way.
Equation of state is called an equation that establishes the relationship between thermal parameters, i.e. ¦(P,V,T) = 0. The form of this function depends on the nature of the working fluid. There are ideal and real gases.
Ideal is a gas for which the intrinsic volume of molecules and the interaction forces between them can be neglected. The simplest equation of state for an ideal gas is the Mendeleev–Clapeyron equation = R = const, where R is a constant, depending on the chemical nature of the gas, and which is called the characteristic gas constant. From this equation it follows:
Pu = RT (1 kg)
PV = mRT (m kg)
The simplest equation of state real gas is the van der Waals equation
(P + ) × (u - b) = RT
where is internal pressure
where a, b are constants depending on the nature of the substance.
In the limiting case (for an ideal gas)
u >> b Pu = RT
To determine the characteristic gas constant R, we write the Mendeleev-Clapeyron equation (hereinafter M.-K.) for P 0 = 760 mmHg, t 0 = 0.0 C
multiply both sides of the equation by the value m, which is equal to the mass of a kilomol of gas mP 0 u 0 = mRT 0 mu 0 = V m = 22.4 [m 3 /kmol]
mR = R m = P 0 V m / T 0 = 101.325*22.4/273.15 = 8314 J/kmol×K
R m - does not depend on the nature of the gas and is therefore called the universal gas constant. Then the characteristic constant is equal to:
R= R m /m=8314/m;[J/kg×K].
Let us find out the meaning of the characteristic gas constant. To do this, we write the M.-K. equation. for two states of an ideal gas participating in an isobaric process:
P(V 2 -V 1)=mR(T 2 -T 1)
R= = ; where L is the work of the isobaric process.
m(T 2 -T 1) m(T 2 -T 1)
Thus, the characteristic gas constant is mechanical work(work of volume change) performed by 1 kg of gas in an isobaric process when its temperature changes by 1 K.
Lecture No. 2
Caloric state parameters
The internal energy of a substance is the sum of its kinetic energy thermal movement atoms and molecules potential interaction energy, energy chemical bonds, intranuclear energy, etc.
U = U KIN + U SWEAT + U CHEM + U POISON. +…
In other processes, only the first 2 quantities change, the rest do not change, since in these processes the chemical nature of the substance and the structure of the atom do not change.
In calculations, it is not the absolute value of internal energy that is determined, but its change, and therefore in thermodynamics it is accepted that internal energy consists only of the 1st and 2nd terms, because in calculations the rest are reduced:
∆U = U 2 +U 1 = U KIN + U SOT ... For an ideal gas U SOT = 0. In the general case
U KIN = f(T); U POT = f(p, V)
U = f(p, T); U POT = f(p, V); U = f(V,T)
For an ideal gas we can write the following relation:
Those. internal energy depends only on
temperature and does not depend on pressure and volume
u = U/m; [J/kg] - specific internal energy
Let us consider the change in the internal energy of the working fluid performing a circular process or cycle
∆u 1m2 = u 2 - u 1 ; ∆U 1n2 = u 1 – u 2 ; ∆u ∑ = ∆u 1m2 – ∆u 2n1 = 0 du = 0
It is known from higher mathematics that if a given integral is equal to zero, then the value du represents the total differential of the function
u = u(T, u) and is equal to
For an equilibrium thermodynamic system, there is a functional relationship between the state parameters, which is called equation withstanding. Experience shows that the specific volume, temperature and pressure of the simplest systems, which are gases, vapors or liquids, are related termic equation state of the species.
The equation of state can be given another form: 
These equations show that of the three main parameters that determine the state of the system, any two are independent.
To solve problems using thermodynamic methods, it is absolutely necessary to know the equation of state. However, it cannot be obtained within the framework of thermodynamics and must be found either experimentally or by methods of statistical physics. Specific view equation of state depends on the individual properties of the substance.
Equation of state of ideal hectarescall
From equations (1.1) and (1.2) it follows that 
.
Consider 1 kg of gas. Considering what it contains N molecules and therefore 
, we get: 
.
Constant value Nk, per 1 kg of gas is denoted by the letter R and call gas constantNoah. That's why
, or 
.
(1.3)
The resulting relationship is the Clapeyron equation.
Multiplying (1.3) by M, we obtain the equation of state for an arbitrary gas mass M: 
.
(1.4)
The Clapeyron equation can be given a universal form if we relate the gas constant to 1 kmol of gas, i.e., to the amount of gas whose mass in kilograms is numerically equal to the molecular mass μ. Putting in (1.4) M=μ and V= V μ , We obtain the Clapeyron-Mendeleev equation for one mole:
.
Here 
is the volume of kilomoles of gas, and 
-universal gas constant.
In accordance with Avogadro's law (1811), the volume of 1 kmol, the same under the same conditions for all ideal gases, under normal physical conditions is equal to 22.4136 m 3, therefore
The gas constant of 1 kg of gas is 
.
Equation of state of real hectarescall
In real gases V The difference from ideal ones is that the forces of intermolecular interactions are significant (attractive forces when the molecules are at a considerable distance, and repulsive forces when they are sufficiently close to each other) and the own volume of the molecules cannot be neglected.
The presence of intermolecular repulsive forces leads to the fact that molecules can approach each other only up to a certain minimum distance. Therefore, we can assume that the volume free for the movement of molecules will be equal to 
,
Where b
- the smallest volume to which a gas can be compressed. In accordance with this, the free path of molecules decreases and the number of impacts on the wall per unit time, and therefore the pressure increases compared to an ideal gas in the ratio
,
i.e.
.
Attractive forces act in the same direction as external pressure and result in molecular (or internal) pressure. The force of molecular attraction of any two small parts of a gas is proportional to the product of the number of molecules in each of these parts, i.e., the square of the density, therefore the molecular pressure is inversely proportional to the square of the specific volume of the gas: Rthey say= a/ v 2 where A - proportionality coefficient depending on the nature of the gas.
From this we obtain the van der Waals equation (1873):
,
At large specific volumes and relatively low pressures of a real gas, the van der Waals equation practically degenerates into the Clapeyron equation of state for an ideal gas, because the quantity a/v 2
(compared with p) And b (compared with v) become negligibly small.
The van der Waals equation qualitatively describes the properties of a real gas quite well, but the results of numerical calculations do not always agree with experimental data. In a number of cases, these deviations are explained by the tendency of real gas molecules to associate into separate groups consisting of two, three or more molecules. The association occurs due to the asymmetry of the external electric field of the molecules. The resulting complexes behave like independent unstable particles. During collisions, they break up, then unite again with other molecules, etc. As the temperature increases, the concentration of complexes with a large number of molecules quickly decreases, and the proportion of single molecules increases. Polar molecules of water vapor exhibit a greater tendency to associate.
EQUATIONS OF STATE, equations expressing the relationship between the parameters of the state of a physically homogeneous system under thermodynamic conditions. equilibrium. The thermal equation of state connects pressure p with volume V and temperature T, and for multicomponent systems - also with composition (molar fractions of components). The caloric equation of state expresses the internalenergy of the system as a function of V, T and composition. Usually, the equation of state, unless specifically stated, means thermal. equation of state. From it you can directly obtain the coefficient. thermal expansion, coefficient isothermal compression, thermal coefficient pressure (elasticity). The equation of state is a necessary addition to thermodynamics. laws
Using the equations of state, it is possible to reveal the dependence of the thermodynamic. functions from V and p, integrate the differential. thermodynamic relationships, calculate the volatility (fugacity) of the components of the system, through which the conditions of phase equilibrium are usually written. Thermodynamics establishes a connection between the equations of state and any of the thermodynamic potentials of the system, expressed in the form of a function of its natural variables. For example, if the Helmholtz energy (free energy) F is known as the function T and V, then
the equation of state cannot be obtained using the laws of thermodynamics alone; it is determined from experience or derived by statistical methods. physics. The last task is very difficult and may. solved only for simplified models of the system, for example, for an ideal gas. The equations of state used for real systems are empirical. or semi-empirical. character. Below we consider some of the most well-known and promising equations of state.U The equation of state of an ideal gas has the form pV=RT, where V is the molar volume, R is the universal gas constant. Real gases at high rarefaction obey this equation (see Clapeyron-Mendeleev equation).. Usually limited to the B 2 /V term (rarely B 3 /V 2). In lit. give an experiment.
values of virial coefficients, developed and theoretical. methods for their determination. Equation of state with second virial coefficient. B 2 is widely used for modeling the gas phase when calculating phase equilibria in the case of not too high pressures (up to 10 atm). It is also used to describe the properties of diluted solutions of high molecular weight. in-in (see Polymer solutions). For practical purposes calculations of phase equilibria in a wide range
t-r range
and pressure, equations of state are important, capable of simultaneously describing the properties of the liquid and gas phases. Such an equation was first proposed by I. Van der Waals in 1873:
p = RT(V-b)-a/V 2,
If detailed experiments are known. data on p-V-T dependencies; multiparameters are used to generalize them.
empirical equations of state. One of the most common equations of state of this type is the Benedict-Webb Rubin equation (BVR equation), developed in 1940 on the basis of the virial equation of state. In this equation, the pressure p is presented as a polynomial of the density of the substance with coefficients depending on the temperature. The terms of a number of higher orders are neglected, and to compensate, an exponential term is included in the equation. This leads to the appearance of S-shaped isotherms and makes it possible to describe the liquid phase and liquid-gas equilibria. For non-polar and weakly polar substances, the BVR equation gives very accurate results. For an individual substance it contains eight adjustable parameters; for a mixture, the parameters of mixed (“binary”) interaction are additionally introduced. Estimating a large number of fitting parameters is a very complex task, requiring numerous and varied experiments. data. The parameters of the BVR equation are known only for several. dozens of , ch. arr. hydrocarbons and inorganic gases Modifications of the level, aimed, in particular, at increasing the accuracy of the description of the holy specific items , also contain larger number
fitting parameters. Despite this, it is not always possible to achieve satisfactory results for polar substances. The complexity of the form makes it difficult to use equations of state of this type in calculations of distillation processes, when it is necessary to perform multiple assessments of the volatilities of the components, the volume and enthalpy of the system. When describing mixtures in-in
empirical the constant equations of state are considered to depend on the composition. For cubic equations of state of the van der Waals type, quadratic mixing rules are generally accepted, according to which the constants a and b for a mixture are determined from the relations: where x i, x j are the molar fractions of the components, the values a ij and b ij are associated with constants for individual items
a ii, a jj and b ii, b jj according to combination rules:
a ij = (a ii a jj) 1/2 (1-k ij);
M. Huron and J. Vidal in 1979 formulated mixing rules of a new type, based on local composition models, which successfully convey the asymmetry of concentrations. dependences of the excess Gibbs potential G E for liquid mixtures and can significantly improve the description of phase equilibria. The essence of the approach is that they equate the values of G E liquid solution, obtained from the equations of state and calculated according to the selected model of local composition [Wilson equation, NRTL (Non-Random Two Liquids equation), UNIQAC (UNIversal QUAsi-Chemical equation), UNIFAC (UNIque Functional group Activity Coefficients model); CM. Non-electrolyte solutions].
This direction is developing intensively.
Many two-parameter
equations of state (van der Waals, virial with a third virial coefficient, etc.) can be represented in the form of the reduced equation of state: f(p pr, T pr, V pr)= 0, where p pr = p/r crit, T pr =T/T crit, V pr = V/V crit - the given state parameters. Materials with the same values of p pr and T pr have the same reduced volume V np; The compressibility factors also coincide Z = pV/RT, coefficient.
volatility and certain other thermodynamic. functions (see Law of Corresponding States). More
general approach
, which allows you to expand the range of considered parameters, is associated with the introduction of additional parameters into the given equation of state. Naib, the simple ones among them are the critical factor. compressibility Z crit = p crit V crit / RT crit.
and acentric factor w = -Ig p pr -1 (at T pr = 0.7). Acentric factor is an indicator of the nonsphericity of the intermolecular field. strength of a given substance (for noble gases it is close to zero).established when choosing n-octane as a “reference” liquid. It is accepted that Z"(T crit, p crit) = /w *, where w * is the acentricity factor of n-octane, Z* is its compressibility factor according to the BVR equation. A methodology for applying the Lee-Kessler equation has been developed for liquid mixtures. This equation of state most accurately describes the thermodynamic properties and phase equilibria for non-polar substances and mixtures.
Along with the above-mentioned empirical equations of state have acquired important equations that have the ability to take into account the structural features of molecules and intermolecular structures. interaction They rely on the provisions of statistics. theories and results of numerical experiments for model systems.
According to molecular statistics. Interpretation, the van der Waals equation describes a fluid of solid attractive spheres, considered in the mean field approximation. In the new equations, first of all, the member of the van der Waals equation, determined by the forces of interparticle repulsion, is clarified. Much more accurate is the Cariahan-Starling approximation, which is based on the results of numerical modeling of solid sphere fluid in a wide range of densities. It is used in many equations of state, but the equations of state of model systems of solid particles have great potential, in which the asymmetry of the mol. is taken into account. forms. For example, in the BACK (Boublik-Alder-Chen-Kre-glewski) equation, the equation of state of the fluid of solid particles shaped like dumbbells is used to estimate the contribution of repulsive forces. To take into account the contribution of attractive forces, an expression is used that approximates the results obtained by the mol. dynamics for a fluid with interparticle potentials such as a rectangular well (see Molecular dynamics). The BACK equation and its analogs make it possible to describe mixtures that do not contain high-boiling components with sufficient accuracy. taking into account additional rotational vibrations.
degrees of freedom associated with displacements of segments of chain molecules (for example, C 8 alkenes). For these systems, the PHCT (Perturbed Hard Chain Theory) equation, proposed by J. Prausnitz and M. Donahue in 1978, has become widespread. parameters in the PHCT equation. Combination rules for a mixture contain one mixture interaction parameter. Further improvement of the PHCT equation is based on replacing the rectangular well potential, which describes the attraction of molecules, with the Lennard-Jones potential [PSCT equation (Perturbed Soft Chain Theory)] and taking into account the anisotropy of intermolecular molecules. forces [level PACT (Perturbed Anisotropic Chain Theory)]. The last equation well describes phase equilibria in systems with polar components even without the use of adjustable pair interaction parameters.
component molecules. The ever-increasing interest in equations of state is primarily due to practical reasons. development needs of many modern technologies related to absorption
separation
, exploitation of oil and gas fields, etc., since in these cases quantities, description and prediction of phase equilibria are required in a wide range of temperatures and pressures.
However, there is not yet enough universal. equations of state. All the mentioned equations of state turn out to be inaccurate when describing states near the critical point. points and are not intended to address critical phenomena. For these purposes, special equations of state are being developed, but they are still poorly adapted for specific practical applications. applications.
