Chemical potential of the gas. Chemical potentials of ideal gases. With pure solvent

A change in the composition of the system cannot but affect the nature of the process, for example, the position of chemical equilibrium. The question arises of taking into account the influence of a variable composition on the main reaction parameters, the main of which is the Gibbs energy.

Because G is an extensive quantity, then its partial derivative with respect to the number of moles is a partial molar quantity:

.

The Gibbs partial molar energy is called "chemical potential".

The chemical potential of a component is equal to the change in the Gibbs energy when 1 mol of this component is added to a large volume of the system in an isobaric-isothermal process, provided that the composition of the system remains constant:

The course of the process depends on the value of the chemical potential.

In order for the process to go in the right direction, the condition for reducing the chemical potential must be observed:

What is equivalent or

Equilibrium condition: .

According to the PMA property (I Gibb-Duhem equation):

and one can calculate the change in the Gibbs energy for any composition.

According to Hess' law:

The chemical potential of an ideal gas is equal to its Gibbs energy or in differential form at T = const and the amount of gas substance is one mole

Where ,

where μ * is the integration constant.

To determine μ * use the concept of the so-called. standard state (P = 1 atm., T = 298 K), taking into account which

,

where is the standard chemical potential, is the relative pressure; is the ratio of the current partial pressure of an ideal gas to its pressure under standard conditions. Relative pressure is a dimensionless quantity, but numerically the relative pressure is equal to the partial pressure of the gas, expressed in atmospheres.

Change in chemical potential

If there is a mixture of ideal gases, then for a component of the mixture:

And

In the situation of a real gas, it is necessary to take into account the interaction of gas molecules with each other, which is taken into account using the concept fugacity in the chemical potential equation of an ideal gas:

For a one-component system

for a mixture of non-ideal gases

In both equations, the standard chemical potential. A hypothetical gas at 298 K and 1 atm is taken as the standard state of the gas. with the properties of an ideal gas ( =1, and f 0 = R 0); is the relative fugacity of the gas. Similar to the relative partial pressure, , where f 0 - standard fugacity - the fugacity of the gas in the standard state, the relative fugacity is numerically equal to the fugacity expressed in atmospheres.

Change in the chemical potential of a non-ideal gas:

For a one-component system

For a mixture of non-ideal gases.

The chemical potential of an ideal solution component is described by the differential equation

where x i is the molar fraction of the component in the solution.

After solving the differential equation, the integral form is obtained:

,

where is the chemical potential of the pure component of the solution; = f(T, the nature of matter)

If the total pressure of the gas mixture is small, then each gas will exert its own pressure, moreover, such as if it alone occupied the entire volume. This pressure is called partial. Total observed pressure R is equal to the sum of the partial pressures of each gas (Dalton's law):

The chemical potential of a component of an ideal gas mixture is:

,

where p i is the partial pressure of the gas.

Expressing the partial pressure of a gas p i through the total pressure and the mole fraction of the gas x i, get the expression for the dependence of the chemical potential i-th component from the mole fraction:

where is the chemical potential of an ideal gas at x i= 1 (i.e. in the individual state) at pressure R and temperature T; depends on both temperature and pressure.

For ideal liquid solutions applicable equation

,

where is the standard chemical potential of an individual component in the liquid state () depends on temperature and pressure; x i is the mole fraction of the component.

Chemical potential of a component of real solutions.

For real solutions, all the dependences considered are inapplicable. Chemical potential of the component real gas solution calculated using the Lewis method. In this case, to preserve the form of thermodynamic equations, instead of partial pressure, they introduce a fictitious quantity fi, which is called partial fugacity, or volatility. Then

,

where is the chemical potential of a component of a real gas mixture in the standard state.

The ratio of volatility to the partial pressure of a real gas solution is called the volatility coefficient:

;

Similarly, for liquid real solutions the actual concentration is replaced by the corresponding fictitious value - activity a i:

,

where is the chemical potential of a component of a real liquid solution in the standard state.

Activity is related to concentration through the activity coefficient:

where γ i is the activity coefficient.

Depending on the method of expressing the concentration of a solution, rational, molar and molal activity coefficients are distinguished:

The activity coefficient depends on the concentration of the solution. In infinitely dilute solutions γ → 1, a i And fi→ c i And pi respectively.

Let us rewrite the equation for the chemical potential in the form

,

therefore, thermodynamic activity is the work of transfer of 1 mol i th component from the standard solution to the given real solution.

There are two main ways to choose standard condition- symmetrical and asymmetrical.

symmetrical way. The same standard state is chosen for the solvent and the solute - the state of the pure component at the temperature of the solution. Then in the standard state x i = 1, a i = 1and γ i = 1. This method is more often used for non-electrolyte solutions.

Let us assume that the equilibrium gas mixture contains i individual substances.

From the definition of the Gibbs energy (III, 17) follows:

G=U+PV-TS

Consider each term in this expression.

According to equation (I, 25a), the dependence of the internal energy of 1 mol of individual i th substance on temperature is represented as follows:

where is the molar heat capacity at constant volume i th gas. Since for an ideal gas the heat capacity does not depend on temperature, integrating the equation () from zero to T, we get:

Or (III, 38)

where is the internal energy of 1 mol i th substance at 0 TO. If a mixture of this substance contains a mole, then multiplying both parts of equation (III, 38) by and summing over all individual substances of the system, we will have:

The second term in the expression for the Gibbs energy, based on the Mendeleev-Clapeyron equation, can be written as:

Let's consider the third term. From equation (II, 21) for 1 mol of an ideal gas follows:

Let's put 1 atm and then

Or (III, 41),

where - standard (because it refers to = 1 atm.) entropy of 1 mole of an ideal gas at 1 TO, which is also called the entropy constant of an ideal gas. Index " 2 » can now be discarded and the expression will be written in the form:

where is the relative pressure. The quantities and under the sign of the logarithm are dimensionless. It should be noted that the values ​​and can be expressed in any, but always the same units - atmospheres, pascals, millimeters of mercury, etc. However, expressing pressure in atmospheres has an obvious advantage, as in this case, pressure and relative pressure are numerically the same.

Thus, for 1 mole i th component of the gas mixture, we can write:

where is the relative partial pressure i th component.

Multiplying both parts of expression (III, 42) by and summing over all individual substances in the system, we get:

Substituting the values U, PV And S from equations (III, 39), (III, 40) and (III, 43) into the expression for the Gibbs energy, we find the following expression:

The first five terms in this equation depend on the nature of the individual i th substance and temperature, but do not depend on the composition of the mixture and pressure. The algebraic sum of these five terms in brackets will be denoted by . Then

or, if we introduce the notation

then the expression (III, 45) can finally be represented in the following form:

The value is called the chemical potential of the individual i th substance, and the value is the standard chemical potential (at =1).

Since for an ideal gas mixture and , the equation

(III, 46) can be reduced to the form:

If there is a function of only temperature, then it depends not only on temperature, but also on pressure.

Substituting the values ​​from equations (III, 46) and (III, 48) into (III, 47), we respectively obtain:

To clarify the meaning of the concept of "chemical potential", we differentiate the expression (III, 49) as a product at constants R And T:

It is easy to show that for constant R And T second term (depends only on temperature).

Then for a system of variable composition

Let us take constant the number of moles of all components of the mixture, except i th component, then

The following follows from the definition of the chemical potential as a partial derivative. If at constant temperature T and pressure R add one mole of some component to an infinitely large amount of a mixture (solution) of a certain composition, then the chemical potential will be equal to the increase in the Gibbs energy.

The foregoing allows us to define the chemical potential as the Gibbs energy per mole of a component in a mixture, or, in other words, the partial molar Gibbs energy.

The total differential of the Gibbs energy in accordance with (III, 26) and (III, 53) will be written as follows:

Based on this expression, it can be shown that for systems with variable composition, the fundamental thermodynamic equations will have the following form:

Equations (III, 55) - (III, 58) imply:

Thus, the chemical potential is a partial derivative with respect to the number i-th component of any characteristic function G,F,U And H with a constant amount of other individual substances in the system and the constancy of the corresponding independent variables.

It should be noted that the chemical potential is an intensive property of the system.

Chemical potential for one mole of a pure substance in the state of an ideal gas at any temperature T and pressure R can be calculated according to the equation:

In general, the chemical potential of a pure substance is its Gibbs molar energy: , where is the Gibbs energy of 1 mole of a pure substance.

For practical purposes, the Gibbs molar energy in the standard state (at and T = 298TO).

In this case

The value we have denoted is also defined as the standard molar Gibbs energy of formation of one mole of a substance from simple substances in their standard states.

It is assumed that the Gibbs energy of formation () of all elements at all temperatures is equal to zero.

The standard Gibbs energies of formation of many compounds are tabulated. Using the values ​​taken from the tables, the change in the Gibbs energy of a chemical reaction can be calculated in the same way as the heat effect of a chemical reaction is calculated from the values ​​of the standard heats of formation of substances involved in a chemical reaction.

The use of statistical methods in thermodynamics makes it possible to calculate the entropy of an ideal gas as a function of temperature and pressure (see Chap. V)

To accurately calculate the temperature part of the entropy, it is necessary to have the most complete spectroscopic information about the molecular quantities that characterize the rotation of molecules and the vibration of atoms in them, as well as information about the energy levels of excitation of electron shells. A certain minimum of such information is also necessary for the use of semiclassical formulas, the accuracy of which in most cases turns out to be quite sufficient and which, due to their simplicity, are indispensable for solving many applied problems of chemical thermodynamics. For example, in the temperature range, when the rotational degrees of freedom are fully excited, according to equation (5.83)

Here the entropy constant Zvrashch), the electronic part of the entropy.

The temperature dependence of the internal energy of an ideal gas is determined by the same molecular constants used to calculate the temperature part of the entropy. Thus, having the necessary spectroscopic information, it is possible to establish how the total thermodynamic potential of an ideal gas changes with temperature, which is equal to the chemical potential of the gas for a pure phase:

or, what is the same (because

For a one-component (pure) phase of an ideal gas, formulas (7.110) and (7.111) are equally valid. But for a mixture of ideal gases, the situation is different. For the total thermodynamic potential of a mixture of gases, only formula (7.111) is valid, while (7.110), if we understand the total pressure of the mixture in it, turns out to be incomplete: it reveals the absence of an important term that corresponds to the entropy of mixing gases. The foregoing follows from the Gibbs theorems, one of which will now be explained, and the second considered at the end of the section.

First of all, it must be emphasized that in this paragraph only mixtures of gases that do not chemically react with each other are meant. If gases are prone to chemical transformation, then only such temperature and pressure conditions (in the absence of catalysts) should be considered, when chemical transformation is practically excluded or, in any case, extremely “inhibited”.

Rice. 25. On the proof of the Gibbs theorem

According to the Gibbs theorem, the energy, entropy, and potentials of a mixture of ideal and chemically non-reacting gases are additive quantities, i.e., each of these quantities is the sum in the same thermodynamic state in which it is in the mixture, i.e. at the same temperature and, in addition, at the same density (or, equivalently, at the same partial pressure as it has in the mixture, or, finally, with the same volume as the mixture as a whole, and therefore, in particular, and its given component). Here, apparently, it is appropriate to note right away that there will be no additivity if the components are taken at least at the same temperature, but in the volume or pressure that were characteristic of them before the mixing process.

The Gibbs theorem on additivity follows from the concept of an ideal gas as a system of particles that do not interact with each other. Let a mixture of ideal gases fill a cylindrical vessel B, which is inserted inside another similar vessel (Fig. 25, position 1). Imagine that all the walls of the outer vessel are impermeable to all molecules of the mixture, except for the lid-diaphragm of this vessel a, which is permeable to the molecules of the component of the mixture. All walls of the inner vessel B, including its lid (which at the beginning of the experiment is adjacent to a), are impermeable only to the molecules of the component and are completely permeable to all other components of the mixture. It is obvious that since the gas particles are not bound by interaction forces and the chemical transformation of the components is also excluded, then, using the described device (permitted by the principle of thermodynamic admissibility, p. 201), it is possible to move vessel B out of vessel A (Fig. 25, position 2), without spending neither work nor heat on such isolation of a component and without changing the thermodynamic state of both this component and the rest of the mixture, In such a mental experiment, the division of a mixture of gases into components occurs without a change in energy and without a change in entropy. Therefore, the energy and entropy of a mixture of ideal gases (each of these quantities) is equal to the sum of the same quantities taken for the components of the mixture, considered in the same thermodynamic state and in the same quantities in which they enter the mixture. More specifically, the components of the mixture must be taken at the same temperature and density such that each of them, in the same amount in which it enters the mixture, occupies the entire volume of the mixture (i.e., has the same pressure as its partial pressure in mixtures).

If means the number of moles of a component in the mixture, then under the conditions just indicated (and, accordingly, correctly chosen arguments), the following additivity relations are valid for a mixture of gases:

But for any system, therefore (for the indicated

conditions and for the indicated choice of independent variables), the chemical potential of a gas in a mixture of ideal gases is equal to the total thermodynamic potential of a mole of the mixture component

We see, therefore, that from the two formulas (7.110) and (7.111) for the -potential (and chemical potential) of a pure gas, formula (7.111) remains valid for the -potential of one mole of a mixture of gases, while (7.110) turns out to be suitable for calculating the chemical potential of a component of a mixture of gases (of course, if we understand the partial pressure in it

This widely used expression for the chemical potential of a gas is often written in a slightly different form; namely, instead of partial pressure, the molar volume concentration c is used as the main independent variable. Substituting into (7.114)

we get

It often turns out that it is most convenient to use mole fractions. Since

where is the number of moles of the solvent), then according to (7.1. 4)

what does it have to do with

For several decades, formula (7.115) served as the basis for the thermodynamic theory of ideal solutions.

According to Van't Hoff, Planck, Nernst, an ideal solution is an infinitely dilute solution in which the interaction between the molecules of solutes can be completely neglected (due to the large average distance between these molecules), while the interaction between the molecules of solutes and solvent can be very strong.

If, in the reasoning illustrated above, we assumed that the thought experiment with vessels is carried out in a solvent environment surrounding these vessels, and that the walls of both vessels are completely permeable to the solvent, then the conclusion about additivity would turn out to be justified for the "ideal gas mixture of solutes" .

In essence, it is this way of deriving formulas for solutes

(7.114) and (7.116) and was adopted by Planck, who in the last years of the last century most rigorously substantiated the classical theory of dilute solutions.

From what has been said, it is clear that, as applied to ideal solutions, formulas (7.114) - (7.116) in the given outline are valid, in fact, only for dissolved substances, and not for a solvent (for which we will take the designation However, it turns out that with other expressions for the first term of these formulas, the same formulas with a slightly worse approximation can be used for the solvent.

That the above formulas are to a certain extent also suitable for the solvent follows from completely different considerations. The fact is that formula (7.114) can be considered correct for a solvent if: 1) it means not the partial pressure of the solvent in the solution, but the partial pressure of the saturated vapor of the solvent over the solution, and 2) this saturated vapor can be approximately considered ideal gas. Then the right side of (7.114) at can be considered as the chemical potential of the solvent in the gaseous phase in equilibrium with the solution, and this potential, due to thermodynamic equilibrium, is naturally equal to the chemical potential of the solvent in the solution. (Such an interpretation of formula (7.114), of course, is also permissible for dissolved substances, but for them such an interpretation and the indicated restriction on the ideality of the vapor are not necessary.)

According to Raoult's law (which can be obtained from (7.114) and which is discussed in more detail below), the saturated vapor pressure of the solvent over the solution is proportional to the mole fraction of the solvent, and therefore, to the molar volume concentration. This justifies the use of formulas (7.115) and (7.116) for the solvent , in which, however, the values ​​obtained for the solvent are different expressions than those indicated above for solutes.

For a correct understanding of the further development of the thermodynamics of solutions, it is important to pay attention to the fact that the conclusion about additivity remains valid in one more very important case.? Namely, the reasoning reproduced above, that a component can be isolated from other components without the expenditure of heat and work, is valid for a solution not only when the interaction between the molecules of the dissolved substances is negligible. All this reasoning will also be valid when the interaction between the molecules of the solute is intense (due, for example, to a high concentration of the solute), but when this interaction does not differ quantitatively from the interaction of the molecules of the solute with the solvent medium. In this case, the extension of vessel B from vessel A would again not require heat or work, since the break in the bonding forces between the molecules of the component would be exactly compensated by replacing these bonding forces with identical, by condition, bonding forces between the molecules of the component and those molecules of the solvent, which when moving, they occupy places that previously belonged to the molecules of the component.

The studies of the Lewis school, as well as E. V. Biron, Guggenheim and other authors over the years have shown that, based on what was said in the previous paragraph, the idea of ​​ideal solutions can be extended so that some solutions of significant concentration turn out to be ideal. To this end, ideal solutions began to be understood as mixtures of substances that are very similar in their molecular physical properties (at a noticeable concentration; or any mixture at infinite dilution).

The extent to which the forces of intermolecular interaction of substances coincide qualitatively and quantitatively can be judged by the change

volume and thermal effect during mixing. Experience shows that these effects are indeed very small for non-polar liquids, which have more or less the same type of chemical structure and similar physical properties. The closer the solution is to ideal, the smaller the change in volume during mixing and the closer to zero the heat of mixing and dilution. At the same time, Raoult's law is more precisely justified: for ideal solutions, the partial pressure of the saturated vapor of the solvent is equal to the saturation vapor pressure of the pure phase of this substance at the same temperature, multiplied by the mole fraction of the solvent in the solution:

Similar to Raoult's law, according to Henry's law, the saturation vapor pressure of a solute at a given temperature is proportional to its mole fraction in solution

Here, the coefficient of proportionality depends on the nature of the solvent and the temperature and is determined by the relation

Raoult's and Henry's laws were experimentally established for dilute solutions and for them they were also thermodynamically substantiated by Planck. Subsequently, it was found that these laws are also valid for some doubly concentrated solutions formed by mixing substances that have molecules related in properties.

As rightly noted, this could be “foreseen from the kinetic theory, since if the molecules of two components are so similar to each other that the forces acting between different molecules are the same as between the molecules of the same component, then in As a result, due to the laws of probability theory, the number of molecules of each component passing into the gas phase will be proportional to the relative number of molecules in the liquid” [A - 16, p. 163].

Along with what has been said, the classical theory of dilute solutions by van't Hoff, Planck, and others began to raise objections from some authors who pointed out that, in their opinion, the idea of ​​the partial pressures of the components of the solution and the osmotic pressure is not sufficiently strictly substantiated. For example, in Guggenheim's book [A - 5, pp. 82-85] his rather lengthy arguments are given, based on the fact that semi-permeable partitions supposedly cannot be thought of as completely ideal partitions. Of course, if such partitions are not ideally permeable selectively for certain substances, then with their help, even in an imaginary experiment, it is impossible to accurately measure either the osmotic pressure or the partial pressures of the components of the solution.

It suffices, however, to turn to the principle of thermodynamic admissibility (see p. 201) to recognize Guggenheim's reasoning as unfounded. Nevertheless, considerations of a similar nature (about insufficient ideality, etc.) have found many followers. In his fundamental textbook on chemical thermodynamics, Lewis wrote: “At the beginning of the development of the theory of solutions, the concept of osmotic pressure was widely used by van't Hoff and led to valuable results. However, with the exception of its historical value, osmotic pressure is no longer of paramount importance” [A-16, p. 158].

Many rebelled against this assessment. For example, Ginshelwood wrote: “In presenting the theory of solutions, we will adhere to the van't Hoff method, despite the objection sometimes raised against him that osmotic pressure is allegedly not the main property of solutions.

This objection seems to us completely unfounded. The second law of thermodynamics can be considered as a direct consequence of the molecular-kinetic nature of matter. In the tendency of a solute to diffuse in solution, this molecular kinetic nature is found in its simplest form. Osmotic pressure is a direct measure of this trend. Thus, from a theoretical point of view, osmotic pressure is the most characteristic property of a solution, ”Ginshelwood said, one could only add that for those who refuse to use the visual representation of partial pressures and osmotic pressure as their sum (or in any way limit these representations), the laws of Henry, Raoult and van't Hoff must look like unexpected, almost mysterious relationships.

Without dwelling on a more detailed coverage of the issues raised, it must be said that at present, an ideal solution is usually understood as a system for which Raoult's law is valid and for which, in connection with this, the chemical potential of all components (including, i.e., the solvent) can be expressed, as for ideal gases, by the equation

where is the partial vapor pressure of the component in the equilibrium gas phase, or by the equation

Lewis proposed - and this gradually became generally accepted - to apply equations (7.117) and (7.118) for any real phases (pure or mixtures), but replacing the actual values ​​in them with some effective values ​​of pressure and concentration. Effective pressure determined similarly to (7.117) by the formula

called volatility (or fugacity). And the effective concentration a, determined similarly to (7.115) and (7.118) by the formula

called activity.

All of the above formulas define the chemical potentials for a mole of a component. But in some cases it is necessary to apply potentials not for one mole, but specific ones, i.e., for a unit mass of a component. So, for example, the equilibrium distribution of a substance in two phases is characterized by the equality of chemical potentials, but, generally speaking, specific, and not molar. If the molecular weight of a substance is not the same in the compared phases (due to association in them or dissociation of molecules), then from (7.48) and (7.115) it follows that

This is the Nernst distribution law. It is a generalization of the Henry's law mentioned above, which is valid when there is no association or dissociation of molecules or when it is the same in the compared phases, i.e. when If one of the phases is saturated vapor and for the component under consideration it is so rarefied,

that the Clapeyron equation is valid for it, then the partial vapor pressure of this component in the gas phase and the concentration of the component in the solution are proportional to each other.

We have considered the applicable generalizations of the formulas for the chemical potential of an ideal gas, without touching upon the theorems on the entropy of mixing gases. But one could, on the contrary, rely mainly on the mixing theorem; so did, for example, Planck in his theory of dilute solutions [A - 18, p. 250].

Hence, in confirmation of (7.123), we obtain

Actually, in any equilibrium state of a gas, self-diffusion constantly occurs - continuous mixing of parts of the gaseous phase. But according to the thermodynamic interpretation, such processes correspond to the idea of ​​a thermodynamic state, and the entropy increase theorem has nothing to do with such processes. In order for Theorem (7.123) to be applicable, there must be a qualitative difference between the gases being mixed: they must differ chemically, or in the mass of molecules, like isotopes, or in some other objectively ascertainable feature. But although the specified qualitative difference between the mixing gases is required, however, quantitatively the entropy of mixing according to (7.123) does not depend in any way on the physicochemical properties of the mixture components. Entropy

mixing is completely determined by the numbers that characterize the deviation from the homogeneity of the composition. To many scientists and philosophers, everything said was paradoxical (the Gibbs paradox). A number of articles are devoted to the analysis of the issues raised. In the comments to the Russian translation of Gibbs' works, V. K. Semenchenko rightly writes [A - 4, p. 476] that Gibbs's paradox was resolved by Gibbs himself as early as 1902 in his Fundamental Principles of Statistical Mechanics.

A more detailed exposition of some of the issues raised in this chapter can be found in the monograph by A. V. Storonkin "Thermodynamics of Heterogeneous Systems" (parts I and II. L., Publishing House of Leningrad State University, 1967; part III, 1967). In its first part, the principle and conditions for the equilibrium of heterogeneous systems, the stability criteria, the general theory of critical phases, and the principles of equilibrium shift are stated. In its second part, based on the van der Waals method, regularities characterizing the relationship between and the concentration of two coexisting phases are discussed. The third part contains the main results obtained by the author and his collaborators on the thermodynamics of multicomponent multiphase systems. You can also refer to the monographs of V. B. Kogan "Heterogeneous equilibria" (L., publishing house "Chemistry", 1968), D. S. Tsiklis "The stratification of gas mixtures" (M., publishing house "Chemistry", 1969 ) and V. V. Sventoslavsky "Azeotropy and polyazeotropy" (M., publishing house "Chemistry", 1968). (Ed. note)

We express the Gibbs energy as a function of T and p. Let us substitute into the equation G = H -TS instead of the values ​​H and S their values ​​expressed in terms of T and p. From equation (16) we find

We determine dS from equation (88):

dS = 1fT (dU + pdV) (140)

Here dU = μc V dT, a pdV we find from the characteristic equation, having previously differentiated it:

pdV μ + V μ dp=R dT,

pdV μ = RdT – V μ = RdT – V μ dp = RdT – RTfp dp.

Substituting the resulting expressions into equation (140), we have

After integration

where 5° is the entropy of a substance in a state taken as standard. Finally we get

For a separate component of the gas mixture

G i 0 = G i (T) + RTlnp (142)

and since (135) G i 0 = μ i, then the chemical potential of the i-th component of a mixture of ideal gases can be represented as

μ i = μ i (T) + RTlnp i , (143)

where p i - partial pressure of the i-th gas; μ i (T) - part of the chemical potential, depending on the temperature (and on the nature of the gas).

The chemical potential can also be expressed in terms of the concentration of the i-th component; to do this, replace p i with its value expressed in terms of concentration. From the characteristic

equations p i V = n i RT find p i = n i f V i RT = C i RT,

where C i = n i f V i- kilomol-volume concentration; n i V i - respectively, the number of kilomoles and the volume of the mixture component. Then

μ i \u003d μ i (T) + RT In C + RT In (RT)

or by including the value RT In (RT), which is a function of temperature, into the quantity μ i (T) and denoting μ i (T) + RT In (RT) through (T), we get

(144)

If in equation (143) we substitute the value of partial pressure, expressed through the volume fraction p i =r i p, then the equation will look like

μ i = μ i * (T) + RT In r i + RT In p . (145)

Combining the terms depending on temperature and pressure in equation (145) and denoting the sum as μr i (T, p), we obtain the dependence of the chemical potential of the i-th gas, expressed in terms of volume fractions in the form

μ i = μ r (Т,р) + RTlnr i. (146)

Similarly, replacing pi on the n i pfn(because V i fV = n i fn), we have

Combining in this expression terms that depend on temperature and on volume

μ i (T) + RT ln RTfV= (T, F), we get

μ i = (T,V) + RT ln n i. (147)

FUGITIVITY AND ACTIVITY

The above equations, as was stated above, are valid for ideal gases. They are not suitable for real gases.

However, at the suggestion of Lewis (1901), the method of using these equations for real gases with the introduction of thermodynamic quantities into them: fugacity (or volatility) f and activities but, the first of which characterizes the pressure, and the second - the concentration. Under fugacity (or volatility) is meant the thermodynamic value of a given gas, taken separately or as part of a gas mixture, which is a function of temperature, pressure and gas mixture composition and has the property that the ratio of its values ​​for various states of this gas at T = const is associated with corresponding values ​​of its chemical potential by the relation

(148)

and its absolute value is determined by the equality

where p is the total pressure; p i - partial pressure of gas i, determined through its kilomol fraction:

p i = r i p.

The dimension of fugacity coincides with the dimension of pressure. The fugacity of an ideal gas is equal to its pressure ( f = p).

Thus, for real gases and their mixtures, the thermodynamic relations of an ideal gas for isothermal processes will be valid when the pressures in them are replaced by the corresponding fugacity values.

For example, the chemical potential of the i-th component of a mixture of real gases can be represented as

μ i = μ i (T) + RT ln f i

or the maximum useful work of expansion of a real gas in the form

(149)

Consequently,

ΔG = G i – G i 0 = (150)

The use of fugacity proved to be convenient due to the fact that ways were found to determine them in terms of parameters that are relatively easy to measure. For example, for the case when it can be considered that the real gas differs slightly from the ideal one, the fugacity can be determined by the relation

f/p = p/p id

where p is the gas pressure; p id - gas pressure taken as ideal, r id \u003d RT / V μ for given T and V μ.

The ratio of fugacity to the pressure of an ideal gas is called the fugacity coefficient γ, which shows the degree of difference between the gas and the ideal:

γ = f/p id.

When studying the thermodynamic properties of real gases with the help of fugacity, it is customary to use their values ​​for gases in the standard state. The standard state is considered to be such a state of an ideal gas, the fugacity of which at a given temperature is equal to unity, and the enthalpy is equal to the enthalpy of a real gas at the same temperature and pressure equal to zero. The change in the Gibbs energy can be represented as

ΔG = G – G 0 = =RT ln f , (151)

since for the standard state f 0 = 1.

Another thermodynamic quantity, which is usually substituted into the equations relating the properties of a real gas, is activity. There are absolute and relative thermodynamic activities (the latter is usually called simply activity).

Absolute activity is a dimensionless quantity determined for substance i in a given phase (solution) by the equality

In λ i = μ i /RT, (152)

where μ i is the chemical potential of the i-th substance at a given temperature.

Relative thermodynamic activity is a dimensionless quantity determined for the i-th substance in a given phase by the equality

a i = λi /λi 0(153)

In a i = (μ i - μ i 0)/RT. (154)

The relative activity also depends on the concentration of each of the substances contained in the phase, temperature and pressure. For standard state and i 0 = 1. By substituting activity values ​​instead of concentration values ​​into equations expressing the relationship between various thermodynamic properties of an ideal gas and its concentration, it is possible to apply these equations to a real gas.

From equation (148), activity can be expressed in terms of fugacity:

a i = f i / f i 0. (155)

Since for gases fi 0 = 1, then, therefore,

a i = f i.

Substituting the activity value instead of the concentration value into equation (144), we obtain the relation for the chemical potential in the form

μ i = μ i a i (T) + RT ln a i(156)

or, substituting the activity value instead of the fugacity value into equation (150), we obtain the Gibbs energy equation

G i – G i 0 = RT ln a i .(157)

The ratio of the activity value of the component to the value characterizing the concentration of the component in a given gas mixture is called the activity coefficient

φ i = a i / r i ,(158)

where r i- volume or mole fraction.

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