How to compute backward reaction rate from forward one and an equilibrium data

Question

Suppose we have have an ideal gas mixture and a reversible elementary reaction: $$ \ce{O + H_2 <=> H + OH} $$ Its forward reaction rate may be computed as follows: $$ \frac{d[\ce{H}]}{dt} = \frac{d[\ce{OH}]}{dt} = -\frac{d[\ce{O}]}{dt} = -\frac{d[\ce{H2}]}{dt} = k^{(f)}\left[\ce{O}\right]\left[\ce{H2}\right] = A T^n e^{\left(\frac{E_a}{RT}\right)}\left[\ce{O}\right]\left[\ce{H2}\right] $$ where $A, n$ and $E_a$ are known values. The exponents of concentrations $\left[\ce{O}\right]$ and $\left[\ce{H2}\right]$ are both equal to one due since the reaction is elementary.

Also, there is thermodynamic data for the species - heat of formation, entropy, heat capacity - for all species. For example, for H these values are: $$ \Delta H^{298K}_f\left(\ce{H}\right) = 52.1~\text{kcal/mol}, \quad S^{300K}\left(\ce{H}\right) = 27.4~\text{cal/mol K},\quad C_p(\ce{H}) = 4.97~\text{cal/mol K}. $$ For simplicity, heat capacity is assumed constant here.

Parameters $A, n$ and $E_a$ for backward reaction rate $\left(k^{(b)}\right)$ are unknown. According to Warnatz's et al "Combustion" book backward reaction may instead be computed from equilibrium constant, which, in turn, is determined by thermodynamic data: $$ K_c = \frac{k^{(f)}}{k^{(b)}}= exp\left(-\Delta_R\bar{A}^0/RT\right) $$

But how to compute $\Delta_R\bar{A}^0$?

I assume that it equals to an increment of free energy (Helmholtz function) $A = U - TS$ corresponding to consumption of one mole of each reactant, but not sure how to properly calculate it from thermo data above.

Answer 1

You are right that you can determine the backwards rate if you know the forward rate and the equilibrium constant. In fact, you have the right formula

$$ K_c = \frac{k^f}{k^b} $$

In general, the equilibrium constant can be calculated from the change of the Gibbs free energy during the reaction

$$ \Delta G^r = - R T \ln K_c$$

Therefore in principle you need to calculate the total Gibbs free energy of the reactants and the products and take their difference to get the equilibrium constant. How to do this depends on the nature of the data you have; it might involve quite complicated thermodynamic calculations.

Edit: I was mistaken previously, the equilibrium coefficient is expressed with the rate coefficients.