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:


$$
\begin{align}
\Delta S = &A \ln{(T\_2-T\_1)} + B(T\_2-T\_1) + \dfrac{C}{2}(T\_2^2-T\_1^2)\\ &+ \dfrac{D}{3}(T\_2^3-T\_1^3) - \dfrac{E}{2(T\_2^2-T\_1^2)} + G
\end{align}
$$




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.


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