Chapter 9 Chemical Equilibrium

Chemical reactions move toward a dynamic equilibrium in which both reactants and products are present but don’t undergo net change. The direction of 'spontaneous' change is toward lower values of Gibbs energy, G.

9.1 The Gibbs energy minimum

We locate the equilibrium composition of a reaction mixture by calculating the Gibbs energy of the reaction mixture and identifying the composition that corresponds to minimum G.

(a) The reaction Gibbs energy

Let's consider . Suppose an infinitesimal amount, dx, of A turns into B, Then the change in the amount of A present is dn_A = -dx and the change in the amount of B present is dn_B = +dx

The quantity x is called the extent of reaction; it has the dimensions of amount of substance and is reported in moles. When the extent of reaction changes by a finite amount Dx, the amount of A present changes from n_A to n_A,0 - Dx and the amount of B changes from n_B to n_B,0 + Dx.

The reaction Gibbs energy, D_rG, is defined as the slope of the graph of Gibbs energy plotted against the extent of reaction:

Usually, D signifies a difference, but in this case D_r signifies a derivative, the slope of G with respect to x.

Thus D_rG can be interpreted as the difference between chemical potentials of the reactants and products at the composition of the reaction mixture.

Because the chemical potentials vary with composition, the slope of G against extent of reaction changes also. For A Û B, the reaction is spontaneous when m_A > m_B.

The slope is zero and the reaction is spontaneous in neither direction when D_rG = 0.

(b) Exergonic and endergonic reactions

If D_rG < 0, the forward reaction is spontaneous. exergonic
If D_rG > 0, the reverse reaction is spontaneous. endergonic
If D_rG = 0, the reaction is at equilibrium. (neither exer- or endergonic)

(c) Perfect gas equilibria

When A and B are perfect gases we can write,

If we denote the ratio of partial pressures by Q, we obtain

The ratio Q is defined as the reaction quotient. It ranges from 0 (pure A) to infinity (pure B). The standard reaction Gibbs energy, D_rG° is defined as the difference in the standard molar Gibbs energies of the products and reactants.

We know that the difference in standard molar Gibbs energies of the products and reactants is equal to the difference in their standard Gibbs energies of formation, thus

At equilibrium, D_rG = 0 and the ratio of partial pressures is denoted as K

Molecular interpretation 9.1

In molecular terms, the minimum in the Gibbs energy, which corresponds to D_rG = 0, stems from the energy of mixing of the two gases. An important contribution to the position of chemical equilibrium is the mixing of the products with reactants as the products are formed.

We see that when D_rG^° > 0, K < 1. Then at equilibrium, the partial pressure of A exceeds that of B, which means that the reactant A is favored in the equilibrium. When D_rG^° < 0, K > 1 and B is favored at equilibrium.

(d) A general reaction

For the general case the change in the amount of any species J is n_JDx, where n_J is the stoichiometric number of J in the chemical equation.

The P symbol to denote product multiplication. Each species is raised to a power given by its stoichiometric coefficient, n.

Recall that for pure solids and liquids, the activity is 1, so there is no contribution to Q even though they may appear in the chemical equation. For a gas, a_J = f_J/p⁰, where f_J is its fugacity. For a perfect gas f_J = p_J, the partial pressure of J.

Consider the reaction 2A + 3B ® C + 2D

At equilibrium, the slope of G is zero: D_rG = 0, and the activities have their equilibrium values, so

These expressions have the same form as Q but are evaluated using equilibrium activities. An equilibrium constant K expressed in terms of activities (or fugacities) is called a thermodynamic equilibrium constant. Note that, because activities are dimensionless numbers, the thermodynamic equilibrium constant is also dimensionless. We often can approximate activities with concentrations and fugacities with partial pressures.

(e) The relation between equilibrium constants

To express the equilibrium constants in terms of mole fractions, x_J, or molalities b_J, of the species we need to know the activity coefficients. Recall

For example

The activity coefficients must be evaluated at the equilibrium composition of the mixture, which may involve a complicated calculation, because the latter is known only if the equilibrium composition is known. In elementary applications we usually assume K_g = 1. Thus K is approximately = K_b. In chapter 10 we will show how electrochemistry can be used to estimate activity coefficients.

Molecular interpretation of equilibrum constants

Consider molecules of reactant A in the energy levels available to the molecule. In a large group (such as a mole of 10²³ molecules) the statistical Boltzmann distribution of molecules over the available states shows a greater population (number) of molecules at lower energies, decreasing as the energy of the level increases.

At a given temperature, their will be a specific distribution of populations, and hence a specific composition of reaction mixture.

In the next figure, although B energy levels still lie higher than A energy levels, in this instance they are much more closely spaced. As a result, their total population may be considerable and B could even dominate in the reaction mixture at equilibrium. Closely spaced energy levels correlate with high entropy, so in this case we see that entropy effects can override a positive enthalpy. A positive enthalpy results in lowering the equilibrium constant

however a large, positive entropy may favor an equilibrium shifted to the reaction products, despite the endothermic character of the reaction.

Examples of closely spaced energy levels include non-rigid molecules. Several conformations are available with very little energy required to interconvert. This is the source of the closely spaced energy levels. For example, consider ethylene, H₂C=CH₂ has one configuration while ethane, H₃C-CH₃ has three rotational isomers with equivalent energy. Higher entropy and closer spaced energy levels are expected for ethane.

The equilibrium constant for a catalyzed reaction changes the reaction rate at which equilibrium is attained, but does not affect the equilibrium position.

9.2 How equilibria respond to pressure

The equilibrium constant depends upon the value of D_rG°, which is defined at a single standard pressure. Therefore D_rG°, and K are independent of the pressure at which equilibrium is actually established. Formally

Even though K is independent of pressure, the equilibrium composition can change. There are two ways to increase the pressure of a reaction:

1. Add an inert gas If the gas behaves ideally, the partial pressure of the products and reactants do not change, hence the equilibrium composition stays the same.

2. Compression The pressure of a system can be increased by confining the gases to a smaller volume. In this case, partial pressures will change. Consider the case A Û 2B,

To remain constant, an increase in p_A must offset an increase in the square of p_B. The number of A molecules will increase.

The increase in the number of A molecules and the corresponding decrease in the number of B molecules brought about by compression is a special case of LeChaelier's principle.

A system at equilibrium, when subjected to a disturbance, responds in a way that tends to minimize the effect of the disturbance.

For instance compression of the previous case: the system will try to minimize particles in the gas phase, A ¬ 2B.

The extent of dissociation a, of A into B is inversely related to pressure. As p is increased a decreases.

In general, the composition will shift towards the side of the reaction with fewer moles in the gas phase.

Illustration

To predict the effect of an increase in pressure on the composition of ammonia synthesis at equilibrium, we note that the number of gas molecules decreases (from 4 to 2). So LeChatelier's principle predicts an increase in pressure will favor product.

9.3 The response of equilibrium to temperature

LeChatelier's principle predicts that a system at equilibrium will tend to shift in the endothermic direction if the temperature is raised, for then energy is absorbed as heat. Conversely, an equilibrium can be expected to shift in the exothermic direction if the temperature is lowered, for the reduction in temperature is opposed.

Exothermic reactions:

increased temperature favors the reactants.

Endothermic reactions:

increased temperature favors the products.

(a) The van't Hoff equation

An expression for the slope of a plot of equilibrium constant as a function of temperature.

Justification 9.3

The differentials are complete because K and D_rG⁰ depend only on T, not p. From the Gibbs-Helmholtz equation

The b form is obtained by noting

When reaction is exothermic under standard conditions

(D_rH° < 0) .

A negative slope means that lnK, and therefore K itself, decreases as temperature rises.

Some insight into the thermodynamic behavior can be found in the expression

When the reaction is exothermic, -DH/T corresponds to a positive change in entropy of surroundings, and favors formation of products. When T is raised, -DH/T decreases, and the increasing entropy of surroundings has a less potent role. As a result the equilibrium lies less to the right.

When the reaction is endothermic, the principle factor is the increasing entropy of the reaction system. The importance of the unfavorable change of the entropy of the surroundings is reduced if the temperature is raised (because DH/T is smaller), and the reaction is able to shift towards products.

(b) The value of K at different temperatures.

We integrate to obtain

If we assume reaction enthalpy is constant over the temperature range of interest, then we have

Illustration

To estimate the equilibrium constant for the synthesis of ammonia at 500 K from its value at 298 K (6.0 x 10⁵⁾ we use the standard reaction enthalpy (Table 2.6), (D_fH°=0 for H₂ and N₂.) Then with D_rH^° = -92.2 kJ/mol, we find

Acid-base equilibria in water

The only change is to use activity instead of concentration in the equilibrium expression.

For dilute solution the activity of water is close to 1 (the value of pure water), and the equilibrium can be expressed in terms of the acidity constant, K_a :