# How to Calculate Entropy Changes at Non-Standard Conditions

I am trying to calculate the values of $$\Delta H$$ and $$T\Delta S$$ for the reaction taking place at $$\pu{800 ^\circ C}$$

$$\ce{CO2 + H2O + 2CH4 -> 3CO + 5H2}$$

I calculated $$\Delta H$$ to be $$\pu{\approx 480 kJ mol^{-1}}$$ using Hess’ Law, i.e., by taking the reactants down to $$\pu{ 25 ^\circ C}$$ and calculating the enthalpy change, performing the reaction at standard conditions, and then taking the products back up to $$\pu{800 ^\circ C}$$.

However, I am having trouble with $$T\Delta S$$. Can I take the same approach as I did for the enthalpy change, taking the reactants/products between $$\pu{25 ^\circ C}$$-$$\pu{800 ^\circ C}$$ and calculating the entropy change at $$\pu{25 ^\circ C}$$? Then multiplying by the temperature in Kelvin?

I tried using the Shomate equation expression below for the reactants/products (taken from NIST), where A-G are Shomate equation coefficients; however, I’m not getting the correct answer of $$\pu{\approx -80 kJ mol^{-1}}$$, so unsure if this is the correct approach:

\begin{align} \Delta S = &A \ln{(T_2-T_1)} + B(T_2-T_1) + \dfrac{C}{2}(T_2^2-T_1^2)\\ &+ \dfrac{D}{3}(T_2^3-T_1^3) - \dfrac{E}{2(T_2^2-T_1^2)} + G \end{align}

• Chem+Math Expression formatting reference: MathJax Basics / Chem+Math expressions/formulas/equations / Upright vs italic / Math SE Mathjax tutorial // MathJax is preferred not to be used in CH SE Q titles. Commented Jun 3, 2023 at 11:52
• As you can evaluate integral Cp.dT for enthalpy, you can do the same for Cp.dT/T for entropy. Commented Jun 3, 2023 at 11:56
• Are you using absolute temperatures in your entropy calculation? Commented Jun 3, 2023 at 13:11

You maybe applying the Shomate equation incorrectly:

$$\ln{T_2} - \ln{T_1} = \ln{\dfrac{T_2}{T_1}}$$

$$\dfrac{1}{T_2^2} - \dfrac{1}{T_1^2} = \dfrac{T_1^2-T_2^2}{T_1^2T_2^2}$$

The correct application of the Shomate equation would be:

\begin{align} S ^\circ &= A \ln{(T)} + B(T) + \dfrac{C}{2}(T^2)\\ &+ \dfrac{D}{3}(T^3) - \dfrac{E}{2(T^2)} + G\\ \implies \Delta S ^\circ &= A \ln{\dfrac{T_2}{T_1}} + B(T_2-T_1) + \dfrac{C}{2}(T_2^2-T_1^2)\\ &+ \dfrac{D}{3}(T_2^3-T_1^3) - \dfrac{E}{2}\left( \dfrac{T_1^2-T_2^2}{T_1^2T_2^2}\right) + G \end{align}

How to Calculate Entropy Changes at Non-Standard Conditions

Do note that conditions are still standard, and not non-standard, as indicated by the $$\circ$$ superscript in the above equations.

• Thank you very much for your help ananta! I am still slightly confused however. If the Shomate equation is only applicable for standard conditions, how is it applicable to calculate the entropy change between two temperatures? Thanks again. Commented Jun 3, 2023 at 12:18
• @KarlNow yes! Temperature can change; however, pressure must be kept constant at $\pu{1 bar}$. Commented Jun 3, 2023 at 12:21
• @KarlNow another factor to consider is the phase change of water. I am not sure if the Shomate equation is valid for liquid phase of water. I could help you out if you share your calculations as well. Commented Jun 3, 2023 at 12:37
• After working through the problem I've figured out the issue now! Thanks very much for all your help ananta!! Commented Jun 4, 2023 at 15:17
• @KarlNow happy to help :) Commented Jun 4, 2023 at 15:18