## Change of free energy in bimolecular reaction

Some conceptual notes regarding the change of free energy in reactive systems in thermodynamic equilibrium.

System undergoing

$$A + B \rightleftharpoons C$$

in equilibrium in an isothermal and isobaric system (can be translated into isochoric system with Helholtz free energy $F$). The Gibbs free energy is the appropriate thermodynamic potential

$$\mathrm{d}G = V\mathrm{d}p - S\mathrm{d}T + \sum_{i\in\{A,B,C\}}\mu_i\mathrm{d}N_i$$

Define $\xi$, the extent/amount of a species undergoing a reaction. Interprete the change of particle numbers $\mathrm{d}N$, as multiples of the extent $\xi$ via the stoichiometric coefficients $\nu$

$$\mathrm{d}N_i=\nu_i\mathrm{d}\xi$$

The change of Gibb’s energy with respect to one reaction instance is

$$\left(\frac{\partial G}{\partial \xi}\right)_{T,p} = \sum_i\mu_i\nu_i =\Delta_r G_{T,p}$$

For our system

$$\Delta_r G_{T,p} = \mu_C - \mu_A - \mu_B$$

The chemical potential is rewritten in a standard chemical potential of each species (denoted by $^{\ominus}$) and the activity ${i}$ of each species for this particular system.

$$\mu_i = \mu_i^\ominus + RT \ln \{i\}$$

Hence,

\begin{aligned} \Delta_r G_{T,p} &= \mu_C^\ominus - \mu_A^\ominus - \mu_B^\ominus + RT \ln \frac{\{C\}}{\{A\}\{B\}}\\ &= \Delta_r G_{T,p}^\ominus + RT \ln \frac{\{C\}}{\{A\}\{B\}} \end{aligned}

with the standard Gibbs free energy change $\Delta_r G_{T,p}^\ominus$. In thermodynamic equilibrium it holds

$$\left(\frac{\partial G}{\partial \xi}\right)_{T,p}=0$$

and the standard Gibbs free energy change can be measured given the activities

$$\rightarrow \Delta_r G_{T,p}^\ominus =- RT \ln \frac{\{C\}_\mathrm{eq}}{\{A\}_\mathrm{eq}\{B\}_\mathrm{eq}}$$

### Activities, equilibrium constants and standard concentrations

In a reaction with as many educts as products (because of units) and with low concentrations, e.g.

$$A \rightleftharpoons B$$

the equilibrium constant

$$K=\frac{\rho_{B,\mathrm{eq}}}{\rho_{A,\mathrm{eq}}}$$

is a unitless number. However in general the activity at a given time is measured in terms of a standard concentration ($\rho$ can also be another observable, as long as its normalized w.r.t. a standard value)

$$\{i\}=\frac{\rho_i}{\rho_i^\ominus}$$

Thus for the equilibrium constant to be a unitless number, one writes it in terms of the activities and not the concentrations

$$K = \frac{\{A\}_\mathrm{eq}}{\{B\}_\mathrm{eq}}$$

In turn also the equilibrium constant for the bimolecular reaction can be written as a unitless number, given the appropriate standard concetrations

$$A + B \rightleftharpoons C \quad K=\frac{\{A\}_\mathrm{eq}\{B\}_\mathrm{eq}}{\{C\}_\mathrm{eq}}$$

The choice of standard concentrations influences the value of the equilibrium constant and the value of the equilibrium Gibbs free energy change

For example: One may choose for all species the standard concentration

$$\rho^\ominus = 1/V$$

Given the dissociation constant $K_d=k_\mathrm{off}/k_\mathrm{on}$ the equilibrium constant is

$$K_\mathrm{eq} = \frac{\{A\}_\mathrm{eq}\{B\}_\mathrm{eq}}{\{C\}_\mathrm{eq}} = \frac{[A]_\mathrm{eq}[B]_\mathrm{eq}}{[C]_\mathrm{eq}} V = K_dV$$

Hence the change of standard Helmholtz free energy during a reaction in this system at equilibrium is

$$\Delta_r F_{T,V}^\ominus =- RT \ln (K_\mathrm{eq}) =- RT \ln (K_dV)$$