Consider a one-component system. In this case:

Hence

suppose the system contains one mole of an ideal gas, then:

P 0 is the beginning of the pressure report, which is most often equated to standard pressure.

Expression for the chemical potential of 1 mole of an ideal gas.

Let's try to figure out what kind of function it is, the chemical potential!

Let's find the relationship internal energy(U), entropy (S) and product PV.

Let us assume that the equilibrium gas mixture contains k individual substances, and all of them are in an ideal gaseous state. In a mixture of ideal gases, both the internal energy and the entropy of the system are additive functions of the composition. Consider first the first term in the expression for the Gibbs energy. According to the equation, the dependence of the internal energy for 1 mole of the i-th individual substance on temperature can be represented as

where is the molar heat capacity at a constant volume of the i-th gas. Let us assume as a first approximation that Cv does not depend on temperature. Integrating this expression under this condition, we obtain: .

- internal energy of 1 mole of the i-th gas at 0 K. If the gas

mixture containsn i mole of the i-th gas, then: .

The second term in the expression for the Gibbs energy, based on the equation

Mendeleev - Clapeyron, we write in the form: .

Let's consider the third term. The dependence of the entropy S of one mole of the i-th gas in a gas mixture on its relative partial pressure and temperature can be written as:

where is the molar heat capacity of the i-th component of the gas mixture. In this case:

Substituting the expressions for the internal energy (U), entropy (S) and the product PV into the equation , we obtain

The first five terms of this equation depend only on the nature of the individual i-th substance and temperature and do not depend on the composition of the mixture and pressure. Their sum is indicated. Then:

or , where the value and is called the chemical potential, and the value is the standard chemical potential, that is, the chemical potential of 1 mole of an ideal gas at standard pressure and temperature.

The chemical potential is the Gibbs energy, the absolute value of which is unknown, therefore the value of the standard chemical potential is also unknown. If the system contains several components, we should talk about the chemical potential of individual components:

Relative partial pressure of components in the system; it is the gas pressure that would produce that amount of gas in the system if there were no other gases.

The partial pressure of a gas in a system is related to the total pressure using Dalton's law:

Lecture #6

Lecture plan:

1. Equation of the isotherm of the system. Relationship between the Gibbs energy and the chemical potential of the reaction components.

2. Law of acting masses. Standard equilibrium constant.

3. Practical equilibrium constants.

4. Chemical equilibrium in heterogeneous systems.

main feature chemical reaction and many processes in solutions is a change in the composition of the system. Therefore, the total change in the energy of the system at various processes depends not only on thermodynamic parameters (P, V, T, S, etc.), but also on the amount of substance involved in the process. Consider the Gibbs energy as an example.

So, G \u003d f (P, T, n 1, n 2, n 3 .....)

At P, T = const

G \u003d f (n 1, n 2, n 3)

Total change in Gibbs energy:

Value is called the chemical potential.

Chemical potential of the ith component- this is the change in the Gibbs energy of the entire system with an infinitesimal change in the amount of a given gas (per 1 mol), at constant P and T and with unchanged amounts of other gases (the sign is “except n i”).

The chemical potential of an individual gas,, is equal to the Gibbs energy of one mole of this gas, at constant P and T. The chemical potential can also be expressed in terms of the Helmholtz energy:

At T = const, the chemical potential depends on the pressure.

- for individual gas.

- for gas in the mixture,

where are the standard chemical potentials (at P i = 1)

It should be noted that the value of P under the logarithm is relative, that is, related to the standard pressure, therefore, dimensionless.

If the pressure is expressed in atmospheres, then it refers to 1 atm. , if in Pascals - to 1.0133 × 10 5 Pa; if in mm Hg. - to 760 mm Hg. In the case of a real gas, instead of pressure, we substitute the relative fugacity:

- for individual gas

- for gas in the mixture

Examples of problem solving

V (N 2) \u003d 200 m 3; V(He) \u003d 500 m 3;

T (N 2) \u003d 700 K; T (He) = 300 K

Solution : The calculation of DS mixing is made according to the equation

D.S. = - R.

This equation can be used if the pressure and temperature of both gases are the same. In this case, the pressures are equal, and the temperature will equalize when the gases are mixed, so it is necessary to find the temperature of the mixture T x. When mixed, the temperature of nitrogen decreases, that is, nitrogen transfers some amount of heat to helium, and helium receives this heat and increases its temperature. In absolute value, the amount of heat is the same, but the signs are different, therefore, in order to draw up the heat balance equation, one of the heats should be taken with the opposite sign, that is, Q (N 2) \u003d - Q (He)

Let's take С р = const and calculate according to the classical theory. Molar heat capacity for diatomic gases C p \u003d 7/2 R, for monatomic gases C p \u003d 5/2 R, J / mol K; R = 8.31 J/mol K;

20,3 10 3 mole

mole

101 . 10 3 (T x -700) = -422 10 3 (T x -300)

When the temperatures equalized, there was a change in the entropy of nitrogen and helium

= -62,5 . 10 3 J/K

Now we calculate the change in entropy when mixing

The total change in the entropy of the system is equal to the sum of the changes in the entropy of all stages of the process

DS = -62.5 10 3 +530 10 3 + 82.3 10 3 = 549 10 3 J/K

For an ideal gas, we write the chemical potential in molar units

H = h-Ts. (12.18)

Take the starting point p0, T0, at which the entropy has the value s(To). The standard value of the chemical potential at this point is /i°(pc To).

For an ideal gas, the entropy for other values ​​of p and T can be calculated by the formula

Z(T) = 5(To) +

Where Cp is the average value of the isobaric heat capacity, kJ / (mol-K); R is the universal gas constant, kJ/(mol-K).

We substitute (12.19) into (12.18):

P = h-Ts(T0)--cpTn(T0)--cpTn(T0)--cpTn(T0) + RTn(t) =/λ(T) + DG(T), (12.20) where it is denoted

Р°(Т) = ft - Тя(Т0) - cpT ln(J). (12.21)

For an ideal gas, enthalpy is a function of temperature only, so the value of p? (T) depends only on temperature. Let's transform (12.20):

/і\u003d p ° (T) - RT lnpo + RTlnp \u003d q * (ro, T) + DPpr. (12.22)

The standard value of p * (po, T) includes a term that takes into account the initial value of the pressure

Consider a mixture of ideal gases. The number of components r \u003d 1 - N, the total number of moles n \u003d] G pg "mole fraction of the ith gas X ( \u003d w / n. Let's introduce the notation: po is the pressure of a mixture of gases; pi is the partial pressure of the ith gas; N

Po \u003d Y ^ Pi - For ideal gases, the partial pressure can be expressed і

Through their mole fractions

Pi - %iPo> (12.23)

In a mixture of ideal gases, each of them remains ideal and the expression (12.20) is valid for it

/іі=/і?(G) + DG1P(^)1 (12.24)

Where the pressure corresponding to the pressure of the mixture ro-

We express the partial pressure in terms of the mole fraction of the gas

Pi = p°i(T) - f RTn(x?). (12.25)

The standard value of the chemical potential is equal to the chemical

What is the potential of a pure ideal gas (хі = 1; р1 - р0).

We express (Fig. 12.9), the dependence (12.25) graphically in the coordinates pi - f (n (xi)). When Xi - 1 1pa? r = 0 and pi = p°i(T).

With a decrease in the molar fraction of r-ro gas, the value of lnx * decreases, and the value of the chemical potential fii decreases proportionally. The slope of the straight line pi corresponds to the value of RT, that is, it depends on the temperature.

Thus, having determined the standard value of the chemical potential of a pure ideal gas Рі(Т) according to the tables of thermophysical properties and drawing a straight line at an angle RT, we obtain the change in the chemical potential of a given gas рі in the required range of changes in the partial pressure of a gas in a mixture of gases in the range of changes in the molar share Xi.

12.2.5. Chemical potentials of substances in aqueous solutions

Aqueous solutions of substances can be divided into two groups: ideal solutions and real (non-ideal) solutions.

For ideal solutions, the dependence of the chemical potential on the mole fraction of a substance is similar to the dependence for an ideal gas:

Pi = p°i(p, T) + RT ln(a?0. (12.26)

The difference lies in the fact that the standard value of the chemical potential /i? depends not only on temperature, but also on pressure. This is due to the fact that the enthalpy of water depends on pressure and temperature, and for an ideal gas - only on temperature.

Graphically, the chemical potential of an ideal solution is shown in fig. 12.10.

For real solutions, the dependence of the chemical potential on In Xi becomes non-linear (Fig. 12.11).

In this case, the chemical potential of the solute is expressed in terms of the activity of its sc

//r = /i?(p, T) + RTln(c4). (12.27)

"5 installations

The activity of a substance a^ and its mole fraction are related by the activity coefficient jf.

Q>i:=: "Y%X%

For strongly dilute solutions Xi and « 1, a^ w

The physical meaning of the concept of the activity of a solute can be traced from Fig. 12.11. Let us assume that the mole fraction of the substance is хі.

This value corresponds to the value of the chemical potential at the point a(ra) on the curve for a real solution. If the solution were ideal, then this value p, a would correspond to point b with the mole fraction x "a \u003d (c. Therefore, the activity of the dissolved substance o, i in a real solution is equal to the mole fraction of this substance that should have been in ideal solution x -1D to get the same value of the chemical potential.

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Application statistical methods in thermodynamics allows one to calculate the entropy of an ideal gas as a function of temperature and pressure (see Ch. 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). Let us 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 is adjacent to a at the beginning of the experiment), are impermeable only to the molecules of the component and are completely permeable to all other components of the mixture. Obviously, 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 an 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 separation 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 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 last years 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 formulas indicated above 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 correct understanding further development 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 solutes is negligible. All this reasoning will be valid even 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 the expenditure of 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. For this purpose, 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 saturated vapor pressure of a solute at a given temperature is proportional to its mole fraction in solution

Here, the proportionality coefficient 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 classical theory dilute solutions of 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 supposedly 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 solution” to what 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 should look like as unexpected, almost mysterious correlations.

Without dwelling on a more detailed coverage of the issues raised, it must be said that at present, by an ideal solution, they usually began to understand 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 molecular weight substances 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 indicated qualitative difference between the mixing gases is required, the entropy of mixing according to (7.123) does not depend quantitatively on physical and chemical properties 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's works, V.K. Semenchenko rightly writes [A - 4, p. 476] that Gibbs's paradox was resolved by Gibbs himself back in 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 A. V. Storonkin's monograph "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 criteria for stability, general theory critical phases and principles of equilibrium displacement. 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)

chemical potential.

It is important to note that for a system consisting of one substance, it is true:

Any extensive state function is a function of the amount of matter in the system.

For this reason, if the system consists of several components, then

where n i is the number of moles i-th component. Differentiate (62) with respect to n i

p, Т, n j≠i =const

Gibbs named the quantity chemical potential and denoted μ i It is also called Gibbs partial molar energy(partial thermodynamic Gibbs function)

We can give the following definition of chemical potential:

This change in the Gibbs energy of a homogeneous multicomponent system when 1 mole of this component is added to it at constant pressure, temperature and composition of the system (ᴛ.ᴇ. addition should occur at infinitely large quantities all components so that the composition of the system does not change).

The chemical potential, unlike, for example, the Gibbs energy G, is an intense quantity, ᴛ.ᴇ. it does not depend on the mass of the system, but depends on the nature of the system and its composition, temperature, and pressure. Generally speaking, m i depends on the strength of the chemical interaction of a given component with other components: the stronger this interaction, the smaller m i . The strength of the interaction depends on the concentration of the component, and the lower the concentration of the i-th component, the stronger the interaction, and the less m i . The substance tends to move from a state where its m is greater to a state where its m is less (ᴛ.ᴇ. to where the interaction of this component with other components is stronger).

Any energy characteristic is the product of an intensive factor by an extensive one. In our case, μ i is an intensive parameter, and n i is an extensive one. Then:

For T, p = const. (64)

The introduction of a certain amount of dn i moles of the i-th component with a constant amount of other components and constant Т and р will increase the value of the Gibbs energy by . Similar changes will be caused by the addition of other components. The total change in the Gibbs energy of the system when several components are added to it is:

or, in general,

This equation is called fundamental Gibbs equation.

We integrate relation (64) at a constant composition of the system (ᴛ.ᴇ. when m i = const):

Relation (67) is sometimes called Gibbs-Duhem equation(more often this equation is written as follows:

Where x i is the mole fraction of the i-th component.)

When p, T \u003d const for a chemical reaction, it is true:

Calculation of the chemical potential of an ideal gas:

If we have one pure component, then its chemical potential m is equal to the Gibbs molar energy:

(Here and - molar volume and the molar entropy of the substance), then we get:

Let the ideal gas be at T = const, then

Let us integrate expression (71) from p 0 = 1 atm to any p and, accordingly, from m 0 to m; we get:

But for an ideal gas, the Mendeleev–Clapeyron law is fulfilled, which for 1 mole of gas has the form:

p= RT, hence = . (73)

Then we get:

If p 0 = 1 A tm, then

In equation (75), p is not the pressure itself, but dimensionless a value numerically equal to pressure expressed in atmospheres ().

m 0 - standard chemical potential, ᴛ.ᴇ. chemical potential at standard pressure p 0 = 1 atm;

If there is a mixture of gases, then for any i-th component of the mixture.

m i = m 0 i + RT ln (76)

Here, is a dimensionless quantity numerically equal to the partial pressure of the i-th component of the mixture (ᴛ.ᴇ. that part of the total pressure that falls on the i-th component), expressed in atmospheres ().

Since , where is the mole fraction of the i-th gas in the mixture, p is the total pressure in the system, then

chemical potential. - concept and types. Classification and features of the category "Chemical potential." 2017, 2018.

- Chemical potential

Thermodynamics of phase transitions. Definitions Consider the thermodynamics of systems in which phase transitions can take place. A thermodynamic system that can exchange matter with environment, is called open. - Thermodynamic ... .

- Chemical potential is the Gibbs energy per 1 mole of a substance in a given system

; Chemical potential is not given in reference tables. It serves as evidence. The chemical potential of a substance in a solution depends on the concentration: m(X) = m°(X) + RTlnc(X) This is a heuristic equation proposed in a logical way for ideal solutions. With him... .

- Chemical potential of a component of ideal solutions.

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 pressure. The total observed pressure p is equal to the sum of the partial pressures of each gas....

- Chemical potential of an individual ideal gas.

Consider a thermodynamic system that is an ideal gas. The chemical potential of an ideal gas is: , where is the Gibbs molar energy (isobaric potential of 1 mole of an ideal gas). Since, then, where is the molar volume of an ideal gas (the volume of 1 mol of gas). ... .

Enthalpy, the Gibbs thermodynamic function, If, when a certain amount of heat is transferred to a gas, it expands isobarically, then the first law of thermodynamics for an elementary process in this case can be written as: . The value under the sign ... .

- I.e. the chemical potential of the i-th component of the system is equal to the derivative of the Gibbs energy of the system with respect to the mass of the i-th component at constant P. T, V, n and the masses of the remaining components.

Partial derivatives of extensive properties with respect to n at constant P. T, V, n are called partial quantities. Depending on the units in which the mass of the component is expressed, molar and specific partial quantities are distinguished. Thus, &... .

- Electrochemical systems. Electrode. Electrochemical potential. Absolute electrode potentials and electromotive force.

Electrode processes. The concept of potential jumps and electromotive force (EMF). electrochemical circuits, galvanic cells. Standard hydrogen electrode, standard electrode potential. Classification of electrochemical circuits and electrodes. LECTURE... .