# How to calculate the Gibbs free energy from the equilibrium constant for the decomposition of sodium bicarbonate?

Solid $$\ce{NaHCO3}$$ is heated to $$90~^\circ\mathrm{C}$$. At equilibrium the total pressure of the gases produced is $$0.545~\mathrm{atm}$$. Calculate $$\Delta G^\circ$$ at $$90~^\circ\mathrm{C}$$ for the reaction.

$$\ce{2 NaHCO3(s) <=> Na2CO3(s) + H2O(g) + CO2(g)}$$

Using the $$\Delta G^\circ = -\mathcal{R}T\cdot\ln(K)$$ formula, I did the work below:

$$\Delta G^\circ = -(8.31)(273.15 + 90)\cdot\ln(0.545)$$

This gave me $$1.83~\mathrm{kJ/mol}$$ and therefore $$3.66~\mathrm{kJ}$$ for the entire reaction, $$2~\mathrm{mol}$$. Unfortunately, the answer is marked as $$7.85~\mathrm{kJ}$$, suggesting that $$K=1$$, why is this?

Shouldn't $$K=(0.545)/1$$ because of the gases present?

$$\ce{2 NaHCO3(s) ⇌ Na2CO3(s) + H2O(g) + CO2(g)}$$

$$K=P_{\ce{CO2}}\cdot P_{\ce{H2O}}$$

However, $K\ne0.545\text{ atm}$ because the sum of the pressures is $0.545\text{ atm}$ and the equilibrium constant is the product of the pressures.

$$P_{\ce{CO2}}+P_{\ce{H2O}}=0.545\text{ atm}$$

Since you started with no gas, and the stoichiometric coefficients of both gases are equal, then

$$P_{\ce{CO2}}=P_{\ce{H2O}}=\dfrac{0.545\text{ atm}}{2}$$

What is the value of $K$ now?

• $K=\frac{{\left(\frac{0.545}{2}\right)}^2}{1}$ which means ∆G° is 7846J or 7.85kJ. Thank you for pointing out my mistake! May 18 '14 at 19:23