MIXTURES OF GASES AND EQUILIBRIUM IN GASEOUS MIXTURES

CHAPTER 10 MIXTURES OF GASES AND EQUILIBRIUM IN GASEOUS MIXTURES

In this chapter we will apply the concepts developed in Chapter 11 to gaseous systems, first to mixtures of ideal gases, then to pure real gases, and finally to mixtures of real gases.

10.1 MIXTURES OF IDEAL GASES In Chapter 5, we defined an ideal gas on the basis of two properties 1 :

1 We showed in Exercise 3 in Chapter 6 that Equation (5.2) can be derived from Equation (5.1) on the basis of the second law of thermodynamics.

Chemical Thermodynamics: Basic Concepts and Methods, Seventh Edition . By Irving M. Klotz and Robert M. Rosenberg Copyright # 2008 John Wiley & Sons, Inc.

MIXTURES OF GASES AND EQUILIBRIUM IN GASEOUS MIXTURES

We define an ideal gas mixture as one that follows Dalton’s law:

It also follows from the second law of thermodynamics that the partial molar energy U mi of each component is dependent only on the temperature and is independent of the pressure. We make part of the definition of an ideal gas mixture that U mi is independent of the composition.

We will see that the relationships that are derived for mixtures of ideal gases will form convenient bases for the treatment of nonideal gases and solutions.

The Entropy and Gibbs Function for Mixing Ideal Gases The change in entropy and the change in the Gibbs function for mixing ideal gases

can be calculated on the basis of a thought experiment with a van’t Hoff equilibrium box (Fig. 10.1). Consider a cylinder in equilibrium with a thermal reservoir at temperature T so that the experiment is isothermal. Initially, both A and B are present in the separate compartments at the same pressure P. The two gases in the cylinder are separated by two semipermeable pistons; the one on the right is permeable only to A, and the one on the left is permeable only to B. To carry out

the mixing process in a reversible manner, the external pressure P 0 on the right piston is kept infinitesimally less than the pressure of B in the mixture; and the exter- nal pressure P 00 on the left piston is kept infinitesimally less than the pressure of A in the mixture. The work performed by the gases in the mixing process is the sum of the work performed by A in expanding against the left piston and the work performed by B

Figure 10.1. van’t Hoff equilibrium box.

229 in expanding against the right piston. That is,

10.1 MIXTURES OF IDEAL GASES

As A and B initially were at the same temperature and pressure, and as the final mixture is at the original temperature, and the total pressure of the mixture is equal

to the initial pressures of the individual gases, the volumes V A ,V B , and V A þV B are in the same proportion as the respective numbers of moles of gas. Thus, Equation (10.2) can be written as

in which X A and X B are the mole fractions of the two gases. As the mixing process is isothermal and the gases form an ideal mixture, DU ¼ 0, and

A RT ln X A B RT ln X B (10 :4) Then from the entropy change in a reversible, isothermal process [Equation (6.72)]

Q rev

rev

DS ¼ mixing rev

A R ln X A B R ln X B (10 :5)

T As both X A and X B are less than 1, DS mixing is a positive quantity.

For the reversible mixing, the entropy change in the surroundings is equal, but opposite in sign, and the total entropy change is zero. If the mixing process were allowed to proceed irreversibly by puncturing the two pistons, DS for the system would be the same, but DS for the surroundings would be zero because no work would be performed and no heat would be exchanged. Thus, the total change in entropy for the irreversible process would be positive.

For the isothermal process involving ideal gases, DH is zero, and from Equation (7.26)

DG mixing

mixing

MIXTURES OF GASES AND EQUILIBRIUM IN GASEOUS MIXTURES

so that DG mixing ¼n A RT ln X A þn B RT ln X B (10 :7)

The Chemical Potential of a Component of an Ideal Gas Mixture DG mixing also is equal to the difference between the Gibbs function for the mixture

and the Gibbs function for the unmixed gases. That is, DG mixing ¼G mixture

puregases

A G † þn B G m,A † m,B ] unmixed (10 :8) where G † m is the molar free energy of the pure gas. From Equation (7.43) and the ideal

¼ [n A m A þn B m B ] mixed

gas law, we can obtain for the change in Gibbs function in the isothermal expansion of an ideal gas:

DG ¼ nRT ln

If the change in state is the expansion of one mole of ideal gas from a standard pressure P8 ¼ 0.1 MPa to a pressure P, Equation (10.9) can be written as

(10 :10) Substituting for G m from Equation (10.10) into Equation (10.8), we obtain DG mixing ¼n A m A þn B m B A (G8 mA þ RT ln P=P8)

DG ¼ G m

m 8 ¼ RT ln P=P8

(10 :11) From Equations (10.7) and (10.11), we have

B (G8 mB þ RT ln P=P8)

n A RT ln X A þn B RT ln X B ¼n A m A þn B m B A G mA 8 A RT ln P =P8

B G mB 8 B RT ln P =P8 (10 :12)

The coefficients of n A and n B on the two sides of the equation must be equal: Thus,

(10 :13) Therefore

RT ln X A ¼m A mA

m A ¼ G8 mA þ RT ln X A þ RT ln P=P8

¼G mA 8 þ RT ln PX A =P8

231 and similarly for m B . We shall define the partial pressure of an ideal gas as the

10.1 MIXTURES OF IDEAL GASES

product of its mole fraction and the total pressure. Thus, we can write

(10 :15) The right-hand equality in Equation (10.10), which gives the molar free energy

m A ¼G mA 8 þ RT ln p A =P8

of a pure ideal gas, is of the same form as Equation (10.15), which gives the chemical potential of a component of an ideal gas mixture, except that for the latter, partial pressure is substituted for total pressure. If the standard state of a component of the mixture is defined as one in which the partial pressure of that component is 0.1 MPa, then

(10 :16) and we can write

Chemical Equilibrium in Ideal Gas Mixtures For the reaction [Equation (9.47) applied to a mixture of ideal gases]

R ) þ sS( p s ) (10 :18) we can substitute the expression in Equation (10.17) for the chemical potentials into

a A( p A ) þ bB( p B )

Equation (9.53) that is DG m

B þ RT ln p B =P8) þ r(m8 R þ RT ln p R =P8) þ s(m8 s þ RT ln p s =P8) ¼0

A þ RT ln p A =P8)

(10 :19) or, if we gather together the chemical potential terms, (p R =P8) r (p S =P8) s

(rm8 R þ sm8 S

a b :20)

(p

A =P8) (p B =P8) equil

We can define the left side of Equation 10.20 as DG8 m , where the process described is one mole of reaction at constant chemical potential for reactants and products, that is, for a system large enough so that one mole of reaction can take place in the mixture without any significant change in composition or chemical potential, an infinite-copy model. As DG8 m is a constant at constant temperature, the quantity in brackets is also a constant at constant temperature, and, in particular, independent of the total pressure and the initial composition of the system. We therefore designate the quantity in brackets as K P , which is the equilibrium constant in terms of partial pressures for a

MIXTURES OF GASES AND EQUILIBRIUM IN GASEOUS MIXTURES

mixture of ideal gases. Thus,

(p R =P8) r (p

S =P8)

(p A =P8) a (p B =P8) b equil

because DG8 m ¼ 2TDY8 m . The subscript P distinguishes the ideal gas equilibrium constant in terms of partial pressures from other forms for the constants that will

be derived for real systems. If we use the symbol Q for the quotient of pressures not at equilibrium on the right side of Equation (10.21), then we can write

DG m ¼ DG m 8 þ R ln Q P þ RT ln Q

(10 :23) Thus, if the initial quotient of pressures is greater than K P , DG m is positive and the

¼ RT ln (Q =K P )

reaction will be spontaneous to the left, whereas if the initial quotient of pressures is less than K P , DG m is negative and the reaction will be spontaneous to the right.

The form of the equilibrium constant in Equation (10.21) is different from that pre- sented in introductory courses. It has the advantages that 1) it is explicit that K P is a dimensionless quantity; 2) it is explicit that the numerical value of K P depends on the choice of standard state but not on the units used to describe the standard state pressure; the equilibrium constant has the same value whether P8 is expressed as 750.062 Torr, 0.98692 atm, 0.1 MPa, or 1 bar.

Dependence of K on Temperature From the value of DY 8 m at a single temperature, it is possible to calculate the equili-

brium constant K P at that temperature. It is also desirable to be able to calculate K P as a function of the temperature, so that it is not necessary to have values of DY 8 m at frequent temperature intervals. All that is required is to differentiate the relationship between DY 8 m and ln K P [Equation (10.22)] and to use Equation (7.57) for the derivative of DY 8 m . Then

DH m

@ ln K P

¼R

@T

233 and

10.1 MIXTURES OF IDEAL GASES

When Equation (10.24) is applied to the temperature dependence of ln K P , where K P applies to an isothermal transformation, the DH 8 m that is used is the enthalpy change at zero pressure for gases and at infinite dilution for substances in solution (see Section 7.3).

We have observed in Chapter 4 that a general expression for DH m as a function of temperature can be written in the form [Equation (4.74)]

DH m ¼ DH m0 þ Ð DC P m dT

If the heat capacities of the substances involved in the transformation can be expressed in the form of a simple power series [Equation (4.67)]

¼a þa

0 1 T þa 2 T

in which a 0 ,a 1 , and a 2 are constants, then Equation (4.74) becomes

Da 1 Da

DH m 2 ¼ DH m0 þ R Da 0 T þ T 2 þ T 3 (10 :25)

in which the D’s refer to the sums of the coefficients for the products minus the sums of the coefficients for the reactants. Equation (10.25) can be inserted into Equation (10.24), which then can be integrated at constant pressure. If terms higher than T 3 are neglected, the result is

2 3 RT If the constant of integration is I, the result of the integration can be written as

2 6 RT The value of DH m0 can be found if the enthalpy of reaction is known at one temp-

erature. Similarly, the constant I can be determined if DH m0 and ln K P are known at one temperature.

MIXTURES OF GASES AND EQUILIBRIUM IN GASEOUS MIXTURES

Comparison of Temperature Dependence of DG88888 m and ln K [1]

DY m 8 ¼ R ln K

From Equation (10.22), when applied to a reaction in solution, we use K, not K P ,

In view of the relationship in Equation (10.28), one might expect that changes in DG8 m and ln K with temperature would be congruent. If K changes monotonically with temperature, this expectation is fulfilled. If, however, K goes through a maximum or minimum as temperature is changed, DG8 m may still change monotonically in one direction.

Such behavior is exhibited by the simple chemical reaction HC 2 þ H 3 O 2 (aq) þH 2 O( l) ¼H 3 O (aq) þC 2 H 3 O 2 (aq) Very precise, classic, measurements of the ionization constant of acetic acid were

made many decades ago [2]. The dependence of ln K on temperature is illustrated in Figure 10.2. As the temperature is increased from 273 K, ln K and the degree of ionization increases gradually, reaching a maximum just below 298 K, and then

Figure 10.2. The temperature dependence of ln K for the ionization of acetic acid. Data from Ref. 2.

10.1 MIXTURES OF IDEAL GASES

Figure 10.3. The temperature dependence of DG8 m for the ionization of acetic acid. Data from Ref. 2.

decreases with increasing temperature. In contrast, DG8 m is a monotonically increasing function of the temperature (Fig. 10.3).

When we compare chemical reactions at a fixed temperature, that reaction with the more positive value of DG8 m is less “spontaneous,” less capable of progressing from reactants to products. On the other hand, when we compare a given reaction at different temperatures, a temperature with a more positive DG8 m may show a more

Figure 10.4. The temperature dependence of DY 8 m for the ionization of acetic acid. Data from Ref. 2.

MIXTURES OF GASES AND EQUILIBRIUM IN GASEOUS MIXTURES

spontaneous reaction, as can be observed from Figures 10.2 and 10.3. For example, the value of DG8 m for the ionization of acetic acid is 5976.4 cal mol 21 at 273 K and 6489.6 cal mol 21 at 298 K. Nevertheless, the degree of ionization is greater at 298 K,

25 25 . In contrast to the Gibbs func- tion, the Planck function DY 8 m does vary with temperature congruently with the

extent of reaction, as measured by ln K, as is illustrated in Figure 10.4.

10.2 THE FUGACITY FUNCTION OF A PURE REAL GAS We expressed the molar free energy of a pure ideal gas as [Equation (10.10)]

G mA ¼ G8 mA þ RT ln P=P8

by substituting V ¼ nRT/P in the integral

Similarly, we obtained the chemical potential of a component of an ideal gas mixture as [Equation (10.17)] from an analysis of the van’t Hoff mixing experiment, using the same integral.

m A ¼ m8 A þ RT ln p A =P8

It would be possible to apply Equation (10.22) to real gases by substituting a different empirical expression for V m as a function of P for each gas, but no simple closed form is applicable to all gases. A simple form of the equation for the chemical potential and

a simple form of the equation for an equilibrium constant that is independent of the gases involved is so convenient, however, that G. N. Lewis suggested an alternative procedure. He defined a new function, the fugacity f, with a universal relationship to the chemical potential, and let the dependence of f on P vary for different gases. The fugacity is defined to have the dimensions of pressure.

An advantage of the fugacity over the chemical potential as a measure of escaping tendency is that an absolute value of the fugacity can be calculated, whereas an absolute value of the chemical potential cannot be calculated.

One part of the definition of fugacity can be stated as

in which m8 is a function only of the temperature. The standard chemical potential is characteristic of each gas and the standard state chosen. For a pure gas, the value of f 8 is chosen equal to P8, 0.1 MPa.

237 As all gases approach ideality as their pressure is decreased, and as Equation

10.2 THE FUGACITY FUNCTION OF A PURE REAL GAS

(10.29) is of the same form as Equation (10.14) for an ideal gas, it is convenient to complete the definition of f by stating

That is, as the pressure approaches zero, the fugacity approaches the pressure. Figure 10.5 indicates the relationship between P and f for ideal and real gases. The standard state for a real gas is chosen as the state at which the fugacity is equal to

0.1 MPa, 1 bar, along a line extrapolated from values of f at low pressure, as indicated in Figure 10.5. The standard state for a real gas is then a hypothetical 0.1 MPa stan- dard state.

From Equation (10.29), we can see that the change in the Gibbs function for the isothermal expansion of a real gas is

DG ¼ nRT ln

As the pressure approaches zero, Equation (10.30) applies and DG approaches the value calculated from Equation (10.9).

Change of Fugacity with Pressure The dependence of fugacity on pressure can be derived by differentiating

Figure 10.5. Characteristics of the fugacity for ideal and real gases.

MIXTURES OF GASES AND EQUILIBRIUM IN GASEOUS MIXTURES

Combining Equation (10.31) with Equation (9.25), we have

because f 8 is independent of P. This equation can be integrated to find a fugacity at one pressure from that at another:

RT ln ¼ V m dP

Change of Fugacity with Temperature Let us consider an isothermal process in which a gas is transformed form one state A

at a pressure P to another A at a different pressure P . Such a transformation can be represented as follows:

(10 :35) The change in the Gibbs function for such a transformation is given by the expression DG ¼ m

The partial derivative of the fugacity with respect to temperature is given by @(m =T)

@(m=T)

From Equation (9.57), we have @(m =T)

@(m=T)

2 þ (10 :39) @T

T 2 Therefore, @ ln f

@T P

@ ln f

@T

@T

RT 2

RT 2

239 If the pressure P approaches zero, the ratio of the fugacity to the pressure approaches

10.3 CALCULATION OF THE FUGACITY OF A REAL GAS

one, and we can write

If we substitute from Equation (10.40) into Equation (10.41), we obtain

in which H m is the partial molar enthalpy of the substance in State A , that is, the state of zero pressure. Therefore, the difference (H m 2H m ) is the change in molar enthalpy when the gas goes from State A to its state of zero pressure, that is, at infinite volume.

The pressure dependence of this enthalpy change is given by the expression

because ( @H /@P) T is zero, as H m is the partial molar enthalpy at a fixed (zero) pressure.

From Equation (5.68), we know that the pressure coefficient of the molar enthalpy of a gas is related to the Joule – Thomson coefficient m J.T by the equation

If we combine Equations (10.43) and (10.44), we find that

Because of this relationship between (H m 2H m ) and m J.T. , the former quantity fre- quently is referred to as the “Joule – Thomson enthalpy.” The pressure coefficient of this Joule – Thomson enthalpy change can be calculated from the known values of the Joule – Thomson coefficient and the heat capacity of the gas. Similarly, as

(H m 2H m ) is a derived function of the fugacity, knowledge of the temperature dependence of the latter can be used to calculate the Joule – Thomson coefficient. As the fugacity and the Joule – Thomson coefficient are both measures of the devi- ation of a gas from ideality, it is not surprising that they are related.

10.3 CALCULATION OF THE FUGACITY OF A REAL GAS Several methods have been developed for calculating fugacities from measurements

of pressures and molar volumes of real gases.

MIXTURES OF GASES AND EQUILIBRIUM IN GASEOUS MIXTURES

Graphical or Numerical Methods Using the a Function. A typical molar volume – pressure isotherm for a real gas

is illustrated in Figure 10.6, together with the corresponding isotherm for an ideal gas. From Equation (10.34) we can write

f m 1 dP

RT ln 2 ¼ V

The ratio of the fugacity f 2 at the pressure P 2 to the fugacity f 1 at the pressure P 1 can

be obtained by graphical or numerical integration, as indicated by the area between the two vertical lines under the isotherm for the real gas in Figure 10.6. However, as P 1 approaches zero, the area becomes infinite. Hence, this direct method is not suitable for determining absolute values of the fugacity of a real gas. Equation (10.34) takes cognizance of only one part of the definition of fugacity. The second part of the definition states that although f approaches zero as P approaches zero, the ratio f /P approaches one. Hence, this ratio might be integrable to zero pressure.

If we take the pressure coefficient of the ratio f /P, we obtain @ ln ( f =P)

Figure 10.6. Comparison of molar volume – pressure isotherms for a possible real gas and an ideal gas.

10.3 CALCULATION OF THE FUGACITY OF A REAL GAS

Figure 10.7. The a function for hydrogen gas at 300 K. Data from Ref. 3.

The pressure coefficient of ln f is given by Equation (10.33),

@ ln f

¼ @P T RT

in which V m is the molar volume of the gas. Thus, Equation (10.46) becomes

If we call the quantity within the parentheses 2a, that is, if

a¼ RT

we obtain

Integration of this equation for isothermal conditions from zero pressure to some pressure P gives

ln (f ð =P)

d ln

a dP