Chemical Equilibrium

The goal of this section is to derive the law of mass action and to obtain an expresion of the equilibrium constant for a chemical reaction in the gas phase in terms of the canonical partition function of the molecular constituents. In order to achieve these goals, we first obtain an expression of the chemical potential for the constituent molecules in terms of their canonical partition functions and then we derive the law of mass action by using the minimum energy principle. Finally, we combine both results and we obtain an expression of the equilibrium constant in terms of the molecular canonical partition functions.

The grand canonical ensemble of a multicomponent system is described by the density operator, introduced by Eq. (72) but where

$$p_j=\frac{e^{-\beta E_j + \beta \sum_k \mu_k N_j(k)}}{\sum_j e^{-......a \sum_k \mu_k N_j(k)}} = \Xi^{-1} e^{-\beta E_j + \beta \sum_k \mu_k N_j(k) },$$ (238)

with $\mu_k$ the chemical potential of species $k$ and $N_j(k)$ the number of particles of species $k$ in quantum state $j$. Eq. (238) is obtained by maximizing the entropy of the system, introduced by Eq. (23), subject to the constraints of constant volume, average internal energy $E_j$ and average number of particles $N_j(k)$ for all the different species $k$ in the system. Substituting Eq. (238) into Eq. (74), we obtain

$$S = \frac{E}{T} - \frac{1}{T} \sum_k \mu_k \overline{N(k)} + k \Xi.$$ (239)

Therefore,

$$G \equiv \sum_k \mu_k \overline{N(k)} = E - T S + T k \Xi,$$ (240)

and since $G = H - T S = E + P V - T S$,

$$P V = k T \Xi.$$ (241)

Eqs. (239)--(241) provide an expression for the change in internal energy$dE$ due to changes in the extensive properties of the system such as changes of volume $dV$, number of particles $d\overline{N(k)}$ and entropy $dS$,

$$dE = T dS - P dV + \sum_k \mu_k d\overline{N(k)}.$$ (242)

According to Eq. (242), $\mu_k$ can be computed as follows,

$$\mu_k = \frac{\partial E}{\partial \overline{N(k)}} \Bigg)_{S,V},$$ (243)

and since $A=E-TS$ and $dA = dE - T dS - S dT$,

$$\mu_k = \frac{\partial A}{\partial \overline{N(k)}} \Bigg)_{T,V}.$$ (244)

Furthermore, according to Eqs. (244) and (47),

$$\beta \mu_k = - \frac{\partial \text{ln} Z}{\partial \overline{N(k)}} \Bigg)_{T,V}.$$ (245)

The canonical partition function,

$$Z = \prod_k \frac{(q_t(k)*q_{\text{int}}(k))^{\overline{N(k)}}}{\overline{N(k)}!},$$ (246)

is computed according to Eq. (172), where $q_t(j)= V (2\pi m_jkT)^{3/2}/h^3$ and $q_{\text{int}}$ are the translational and internal canonical partition functions of species $j$, respectively. The underlying assumption, when computing $Z$ according to Eq. (246), is that the constituents of the systems in the gas phase do not interact with each other except when they undergo reactive collisions. Substituting Eq. (246) into Eq. (245) and using the Stirling Formula, introduced by Eq. (54), we obtain

$$\boxed{\beta \mu_k = - \text{ln} \Bigg(\frac{q_t(k)*q_{\text{int}}(k)}{\overline{N(k)}}\Bigg)}.$$ (247)

In order to derive the law of mass action, we consider the following chemical reaction,

$$a A + b B \rightleftharpoons c C + d D,$$ (248)

where the stoichiometric coefficients ($c_k=$a, b, c and d) determine the relative changes in the number of moles of the molecular constituents due to the chemical reaction, as follows:

$$\frac{d N(A)}{a}=\frac{d N(B)}{b}=-\frac{d N(C)}{c}=-\frac{d N(D)}{d}.$$ (249)

Substituting Eq. (249) into Eq. (242) we obtain an expression for the change in internal energy at constant entropy S and volume V,

$$dE \Bigg)_{S,V} = \sum_j \mu_j d\overline{N}(j) = d \overline{N(A)} \sum_j \mu_j \nu_j,$$ (250)

where $\nu_j=c_j/a$. The minimum energy principle establishes that

$$dE \Bigg)_{S,V} \geq 0,$$ (251)

for all arbitrary changes $d\overline{N(A)}$ in a system that was initially at equilibrium. Therefore, according to Eqs. (250) and (251),

$$\boxed{\sum_j \beta \mu_j \nu_j = 0.}$$ (252)

Substituting Eq. (247) into Eq. (252), we obtain

$$\sum_j \Bigg( \Bigg[\frac{q_{\text{int}}*(2\pi m_jkT)^{3/2}}{h^3} \Bigg)^{\nu_j} \Bigg[\frac{V}{\overline{N(j)}}\Bigg]^{\nu_j} \Bigg) = 0,$$ (253)

and

$$\prod_j \Bigg[\frac{q_{\text{int}}*(2*\pi*m_j*k*T)^{3/2}}{h^3} \Bigg)^{\nu_j} \Bigg[\frac{V}{\overline{N}(j)}\Bigg]^{\nu_j} = 0.$$ (254)

Therefore,

$$K(T) \equiv \prod_j \Bigg(\frac{q_{\text{int}}*(2*\pi*m_j*k*T)^{3/2}}{h^3} \Bigg)^{-\nu_j} = \prod_j \Bigg(\frac{V}{\overline{N(j)}}\Bigg)^{\nu_j},$$ (255)

which is the law of mass action. Such law establishes that the concentrations of the constituent molecules in chemical equilibrium define an equilibrium constant $K(T)$ that depends only on the temperature of the system and on the nature of the chemical species. The first equality, in Eq. (255), provides a molecular expression of such equilibrium constant in terms of the canonical partition functions of the molecular constituents and T.