# Calculating ideal chemical work using extent of reaction and chemical potential [closed]

I have a electrochemical reactor assumed isothermal and isobaric with 4 reactions and I am trying to calculate the "ideal chemical work' exerted by each reaction using the extent of reactions for each reaction (which I have already calculated).

I am calculating the 'chemical work' $$w$$ by multiplying the extent of reaction by the Gibbs energy of the respective reaction $$k$$:

$$w_k = \mathrm{d}G_k = \Delta_\mathrm{r} G_k \cdot \mathrm{d}\xi _k$$

I'm calculating the reaction Gibbs energy $$\Delta_\mathrm{r}G$$ as basically the stoichiometric sum of the chemical potentials

$$\Delta_\mathrm{r} G = \sum_i(\nu_i \cdot \mu_i)$$

However, I am not sure which conditions to use for the calculation for the reaction Gibbs energy as the reactions are reversible and for 2 of the reactions I have taken the forward reaction as the opposite of the expected (i.e. the forward reaction is the fuel cell reaction but I am operating in electrolyser mode so the extent of reaction for these reactions is negative). Do I use the conditions at the reactor outlet? Average of the inlet and outlet or am I going about this completely wrong?

• If the concentrations of reactants and products are changing over the course of the reaction, you have to take that into account and integrate over the Gibbs free energy of reaction with the extent of reaction as integration variable (or over the electrochemical potential with the charge through the wire as integration variable). See also chemistry.stackexchange.com/questions/6076/… – Karsten Theis Jul 9 '19 at 17:50
• Thanks for responding! Okay so if I integrate over the course of the reaction, with respect to the extent of reaction, $\xi$, where at the beginning of the reaction $\xi = 0$ and at the end of reaction, the extent is known, then the solution of the integral would simply mean the change in Gibbs energy (work) is the product of Gibbs energy of reaction (second reaction in OP) using the final conditions and the final/overall extent of reaction. When I use these conditions I get similar results to when I did an overall Gibbs energy change but not the exact value which is why I'm so confused. – Farough Jul 10 '19 at 10:59