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2.10 Solution

Solution. First of all, divide through by $A$ and set $c=\log\frac{k}{A}$ . Since $k$ and $A$ are constants, so is $c$ . Also,

\exp(c)=\frac{k}{A}=\exp\left(\frac{TS-H}{RT}\right).

Now take logarithms, to get: $c=\,{\frac{TS-H}{RT},}$ so $TS-H=cRT$ , whence $H=\,{T(S-cR).}$

Since $S$ , $c$ and $R$ are all kept constant, we have:

\frac{\partial H}{\partial T}=\frac{\partial}{\partial T}\left(T(S-cR)\right)% \,{=S-cR}\,{=\frac{H}{T}.}