Chemical potential of a component of ideal solutions. Chemical potential of a gas phase component Examples of problem solving

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

The chemical potential of a component of a mixture of ideal gases is equal to:

Where p i– gas partial pressure.

Expressing the partial pressure of a gas p i through total pressure and mole fraction of gas x i, obtain an 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 the equation applies

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 considered dependencies are not applicable. 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, a fictitious quantity is introduced into them f i, 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 fugacity coefficient:

;

Likewise, for liquid real solutions the actual concentration is replaced by the corresponding fictitious value - activity and 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, and i And f i→ c i And p i 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 a standard solution into a given real solution.

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

Symmetrical method. The same standard state is chosen for the solvent and 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's consider a one-component system. In this case:

Hence

suppose that 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 this function, chemical potential is!

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

Let us assume that an equilibrium gas mixture contains k individual substances, and all of them are in an ideal gas state. In a mixture of ideal gases, both the internal energy and entropy of the system are additive functions of the composition. Let us first consider 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 in the form

where is the molar heat capacity at a constant volume of the i-th gas. As a first approximation, let us assume 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 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 it in the form: .

Let's look at 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 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 quantity and is called the chemical potential, and the quantity is the standard chemical potential, that is, the chemical potential of 1 mole of an ideal gas at standard pressure and temperature.

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

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

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

Lecture No. 6

Lecture outline:

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

2. Law of mass action. Standard equilibrium constant.

3. Practical equilibrium constants.

4. Chemical equilibrium in heterogeneous systems.

The chemical potential of a component of a mixture of ideal gases is equal to:

Where p i– gas partial pressure.

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 the equation applies

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.

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 fugacity coefficient:

Likewise, for liquid real solutions the actual concentration is replaced by the corresponding fictitious value - activity and 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, and i And f i→ c i And p i 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 a standard solution into a given real solution.

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

Asymmetrical method. A different standard state is selected for the solvent and solute. For the solvent - as in the symmetrical method: x i → 1, a i → 1and γ i → 1. For a solute, the standard state is the state of the substance in an infinitely dilute solution: x i → 0, a i → x i and γ i → 1. The method is most often used in the thermodynamics of electrolyte solutions.

Energy transformations occurring in systems during processes under various conditions are described using the corresponding thermodynamic functions U,H,G,A. It should be noted that these functions were introduced for an ideal process in which the amount of each substance was considered constant and equal to one mole. However, their values must depend on the amount of a given substance in the system, which can change during the process. For example, in a closed system, when a chemical reaction occurs, the amount of starting substances decreases and the amount of products increases while maintaining the total mass of the substance (the qualitative and quantitative composition of the system changes). To take into account the influence of this circumstance on the values of thermodynamic functions, the concept of chemical potential was introduced.

The increase in the internal energy of a system with an increase in the amount of a given substance under conditions of constant entropy of the system and its volume, with constant amounts of other substances, is called chemical potential of the i-th substance:

It can be shown that the value of the chemical potential i of the th substance is determined by the change in the thermodynamic function when the amount of this substance changes by one mole in processes that occur with constant corresponding parameters and constant amounts of other substances:

In isobaric-isothermal processes, the change in Gibbs energy with a change in the amount i-substance will be determined by the expression dG=m i×dn i. When a chemical reaction occurs, the amounts of all substances participating in the reaction change, therefore dG=Sm i × dn i .

The equilibrium condition for a chemical reaction occurring under isobaric-isothermal conditions, D r G=0, therefore , Sm i×dn i= 0. For reaction n A A+n b B=n With C+n d The equilibrium condition will be Sm i × n i= 0,

(m c× n C+ m d× n D)–(m a × n A+ m b × n B)=0.

It is obvious that the chemical potential i-th substance will depend on its quantity per unit volume - on the concentration of the substance. This dependence can be obtained by considering the change in the Gibbs energy during isobaric-isothermal mixing of two ideal gases.

Let two ideal gases, under standard conditions, be separated by a partition and occupy volumes V 1 and V 2 respectively (Fig. 5.5).

Rice. 5‑5 Mixing of two ideal gases under isobaric-isothermal conditions as a result of mutual diffusion

The amount of the first gas is equal to one mole (n 1 = 1), and the second is n 2. If you remove the partition, mixing of gases occurs as a result of mutual diffusion. Each gas will occupy the entire volume of the system, and the volume of each will be V 1 + V 2. In this case, the concentration of each gas (the amount of substance per unit volume) will decrease. Each gas will perform work of expansion at constant pressure and temperature. Obviously, as a result of this process, the Gibbs energy of the system will decrease by the amount of perfect expansion work.

The change in Gibbs energy as a result of a decrease in the concentration of the first gas will be equal to its work of expansion. The work of expansion of the first gas is determined as follows:

dA=p 0 × dV, considering that p× V=n× R× T and n 1 =1,

® A=–R× T×ln.

Since equal volumes of ideal gases contain the same number of moles of substance,

Where X 1 – mole fraction of the 1st gas; p 1 – partial pressure of the 1st gas; R 0 = 1.013×10 5 Pa – standard pressure; WITH 1 – molar concentration of the 1st gas; WITH 0 =1 mol/l standard concentration.

Thus, the Gibbs energy of the 1st gas will change by the amount D G 1 =R× T×ln X 1 . Since n 1 =1 mol, then, obviously, D f G i=D f G 0 i+R× T×ln Xi.

Thus, the chemical potential of a substance depends on its concentration in the mixture:

m i=m i 0 + R× T×ln X i, m i=m i 0 + R× T×ln , m i=m i 0 + R× T×ln .

It should be noted that these concentration dependences of the chemical potential characterize ideal gases and solutions. Intermolecular interactions in real gases and solutions lead to a deviation of the calculated chemical potentials from the values obtained for ideal systems. To take this into account, the concepts of fugacity and activity are introduced.

Fugacity f(volatility) is a thermodynamic quantity used to describe the properties of real gas mixtures. It allows the use of equations expressing the dependence of the chemical potential of an ideal gas on temperature, pressure and composition of the system. In this case, the partial pressure of a component of the gas mixture p i replaced by its fugacity f i. Intermolecular interaction leads to a decrease in the effective partial pressure of a component of the gas mixture. To take this into account, the partial pressure value is multiplied by the fugacity coefficient (g i<1).Очевидно, что при p i®0 g i®1 and f i® p i.

Unlike ideal ones, in real solutions there are intermolecular interactions and interactions between ions formed as a result of electrolytic dissociation. This leads to the fact that the effective concentration of molecules and ions in real solutions decreases. Therefore, when calculating the chemical potential, they use instead of concentration WITH size activity a. Activity and molar concentration i-components are related by the relation and i=g i× With i, where g i– molar activity coefficient (g i<1). Очевидно, что при With i®0 g i®1 and and i® With i.

Control questions.

1. Thermodynamic system, parameters and state functions. Thermodynamic process.

2. The first law of thermodynamics. Internal energy and enthalpy.

3. Thermal effect of a chemical reaction. Enthalpy of formation of a substance.

4. Temperature dependence of enthalpy.

5. Entropy. Second law of thermodynamics.

7. Temperature dependence of the Gibbs energy value.

8. Concentration dependence of the Gibbs energy. Activity and fugacity.

9. Thermodynamic calculations of the thermal effect of a chemical reaction.

10 Assessment of the thermodynamic possibility of a chemical reaction.

The main feature of a 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 during various processes depends not only on thermodynamic parameters (P, V, T, S, etc.), but also on the amount of substance participating in the process. Let's look at the Gibbs energy as an example.

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

When P, T = const

G = f (n 1, n 2, n 3)

Total change in Gibbs energy:

Magnitude - called chemical potential.

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

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 pressure.

- for individual gas.

- for gas in a mixture,

where are 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 standard pressure, and therefore dimensionless.

If 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 mmHg. In the case of a real gas, instead of pressure, we substitute relative fugacity:

- for individual gas

- for gas in a mixture

Examples of problem solving

V(N 2) = 200 m 3; V(He) = 500 m 3 ;

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

Solution : Mixing DS is calculated using the equation

DS= - 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 mixing gases, so it is necessary to find the temperature of the mixture Tx. When mixed, the temperature of nitrogen decreases, that is, nitrogen transfers some amount of heat to helium, and helium accepts 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 create a heat balance equation, one of the heats should be taken with the opposite sign, that is, Q (N 2) = - Q (He)

Let's take C p = const and calculate according to classical theory. Molar heat capacity for diatomic gases Ср = 7/2 R, for monatomic gases С р = 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, the entropy of nitrogen and helium changed

= -62,5 . 10 3 J/C

Now we calculate the change in entropy during mixing

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

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