Chapter 14: Fundamental Equilibrium Concepts

# 14.5 Equilibrium and Thermodynamics

### Learning Outcomes

- Relate standard free energy changes to equilibrium constants

## Free Energy and Equilibrium

The free energy change for a process may be viewed as a measure of its driving force. A negative value for [latex]\Delta[/latex]*G* represents a driving force for the process in the forward direction, while a positive value represents a driving force for the process in the reverse direction. When [latex]\Delta[/latex]*G* is zero, the forward and reverse driving forces are equal, and so the process occurs in both directions at the same rate (the system is at equilibrium).

In the section on equilibrium the *reaction quotient*, *Q*, was introduced as a convenient measure of the status of an equilibrium system. Recall that *Q* is the numerical value of the mass action expression for the system and that you may use its value to identify the direction in which a reaction will proceed in order to achieve equilibrium. When *Q* is lesser than the equilibrium constant, *K*, the reaction will proceed in the forward direction until equilibrium is reached and *Q* = *K*. Conversely, if *Q* < *K*, the process will proceed in the reverse direction until equilibrium is achieved.

The free energy change for a process taking place with reactants and products present under nonstandard conditions, [latex]\Delta[/latex]*G*, is related to the standard free energy change, [latex]\Delta[/latex]*G*°, according to this equation:

[latex]\Delta G=\Delta G^{\circ }+RT\text{ln}Q[/latex]

*R* is the gas constant (8.314 J/K mol), *T* is the kelvin or absolute temperature, and *Q* is the reaction quotient. We may use this equation to predict the spontaneity for a process under any given set of conditions as illustrated in Example 6.

### Example 14.5.1: **Calculating [latex]\Delta[/latex]***G* under Nonstandard Conditions

*G*under Nonstandard Conditions

What is the free energy change for the process shown here under the specified conditions?

*T* = 25 °C, [latex]{P}_{{\ce{N}}_{2}}=\text{0.870 atm}[/latex], [latex]{P}_{{\ce{H}}_{2}}=\text{0.250 atm}[/latex], and [latex]{P}_{{\ce{NH}}_{3}}=\text{12.9 atm}[/latex]

[latex]{\ce{2NH}}_{3}\left(g\right)\longrightarrow {\ce{3H}}_{2}\left(g\right)+{\ce{N}}_{2}\left(g\right)\qquad\Delta G^{\circ }=\text{33.0 kJ/mol}[/latex]

## Show Solution

The equation relating free energy change to standard free energy change and reaction quotient may be used directly:

[latex]\Delta G=\Delta G^{\circ }+RT\text{ln}Q=33.0\frac{\text{kJ}}{\text{mol}}+\left(8.314\frac{\text{J}}{\text{mol K}}\times \text{298 K}\times \text{ln}\dfrac{\left({0.250}^{3}\right)\times 0.870}{{12.9}^{2}}\right)=9680\frac{\text{J}}{\text{mol}}\text{or 9.68 kJ/mol}[/latex]

Since the computed value for [latex]\Delta[/latex]*G* is positive, the reaction is nonspontaneous under these conditions.

#### Check Your Learning

For a system at equilibrium, *Q* = *K* and [latex]\Delta[/latex]*G* = 0, and the previous equation may be written as

[latex]0=\Delta G^{\circ }+RT\text{ln}K\qquad\left(\text{at equilibrium}\right)[/latex]

[latex]\Delta G^{\circ }=-RT\text{ln}K\qquad\text{ or }\qquad{K}={e}^{-\frac{\Delta G^{\circ }}{RT}}[/latex]

This form of the equation provides a useful link between these two essential thermodynamic properties, and it can be used to derive equilibrium constants from standard free energy changes and vice versa. The relations between standard free energy changes and equilibrium constants are summarized in Table 14.5.1.

### Example 14.5.2: **Calculating an Equilibrium Constant using Standard Free Energy Change**

Given that the standard free energies of formation of [latex]\ce{Ag+}[/latex](*aq*), [latex]\ce{Cl-}[/latex](*aq*), and [latex]\ce{AgCl}[/latex](*s*) are 77.1 kJ/mol, −131.2 kJ/mol, and −109.8 kJ/mol, respectively, calculate the solubility product, *K*_{sp}, for [latex]\ce{AgCl}[/latex].

## Show Solution

The reaction of interest is the following:

[latex]\ce{AgCl}\left(s\right)\rightleftharpoons {\ce{Ag}}^{\text{+}}\left(aq\right)+{ce{Cl}}^{-}\left(aq\right){K}_{\text{sp}}=\left[{\ce{Ag}}^{\text{+}}\right]\left[{\ce{Cl}}^{-}\right][/latex]

The standard free energy change for this reaction is first computed using standard free energies of formation for its reactants and products:

[latex]\begin{array}{rll}\Delta G^{\circ }&=&\Delta {G}_{298}^{\circ }=\left[\Delta {G}_{\text{f}}^{\circ }\left({\text{Ag}}^{\text{+}}\left(aq\right)\right)+\Delta {G}_{\text{f}}^{\circ }\left({\ce{Cl}}^{-}\left(aq\right)\right)\right]-\left[\Delta {G}_{\text{f}}^{\circ }\left(\ce{AgCl}\left(s\right)\right)\right]\\{}&=&\left[\text{77.1 kJ/mol}-\text{131.2 kJ/mol}\right]-\left[-\text{109.8 kJ/mol}\right]=\text{55.7 kJ/mol}\end{array}[/latex]

The equilibrium constant for the reaction may then be derived from its standard free energy change:

[latex]{K}_{\text{sp}}={e}^{-\dfrac{\Delta G^{\circ }}{RT}}=\text{exp}\left(-\frac{\Delta G^{\circ }}{RT}\right)=\text{exp}\left(-\dfrac{55.7\times {10}^{3}\text{J/mol}}{8.314\text{J/mol}\cdot\text{K}\times 298.15\text{K}}\right)=\text{exp}\left(-22.470\right)={e}^{-22.470}=1.74\times {10}^{-\text{10}}[/latex]

This result is in reasonable agreement with the value provided in Solubility Products.

**Check Your Learning**

To further illustrate the relation between these two essential thermodynamic concepts, consider the observation that reactions spontaneously proceed in a direction that ultimately establishes equilibrium. As may be shown by plotting the free energy change versus the extent of the reaction (for example, as reflected in the value of *Q*), equilibrium is established when the system’s free energy is minimized (Figure 14.5.1). If a system is present with reactants and products present in nonequilibrium amounts (*Q* ≠ *K*), the reaction will proceed spontaneously in the direction necessary to establish equilibrium.

### Key Concepts and Summary

Reactions proceed in the direction that establishes equilibrium, where *Q *= *K* and free energy is minimized

#### Key Equations

- [latex]\Delta{G}=\Delta G^{\circ }+RT\text{ln}K[/latex]
- [latex]\Delta G^{\circ }=-RT\text{ln}K[/latex]

### Try It

- Calculate [latex]\Delta[/latex]
*G*° for each of the following reactions from the equilibrium constant at the temperature given.- [latex]{\ce{Cl}}_{2}\left(g\right)+{\ce{Br}}_{2}\left(g\right)\longrightarrow \ce{2BrCl}\left(g\right)\,\,\,{;}\,\,\,\text{T}=25^{\circ}\text{C}\,\,\,{;}\,\,\,{K}_{p}=4.7\times {10}^{-\text{2}}[/latex]
- [latex]{\ce{2SO}}_{2}\left(g\right)+{\ce{O}}_{2}\left(g\right)\rightleftharpoons {\ce{2SO}}_{3}\left(g\right)\,\,\,{;}\,\,\,\text{T}=500^{\circ}\text{C}\,\,\,{;}\,\,\,{K}_{p}=48.2[/latex]

- Calculate the equilibrium constant at 25 °C for each of the following reactions from the value of [latex]\Delta[/latex]
*G*° given.- [latex]{\text{O}}_{2}\left(g\right)+{\text{2F}}_{2}\left(g\right)\longrightarrow {\ce{2OF}}_{2}\left(g\right)\,\,\,\,\,\,\,{;}\,\,\,\,\,\,\,\Delta G^{\circ }=-\text{9.2 kJ}[/latex]

## Show Selected Solutions

2. Equilibrium constants are calculated from [latex]\text{ln}K=\frac{-\Delta G^{\circ }}{RT}[/latex]. Note that *K* is a function of *T* and thus changes as *T* changes.

- [latex]\text{ln}K=-\left[\frac{-9.2\times {10}^{3}\text{J}}{\left({\text{8.314 J mol}}^{-\text{1}}{\text{K}}^{-\text{1}}\times \text{298.15 K}\right)}\right]=3.71[/latex],
*K*= 41