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2.10 Solution

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

exp⁡(c)=kA=exp⁡(T⁢S-HR⁢T).\exp(c)=\frac{k}{A}=\exp\left(\frac{TS-H}{RT}\right).

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

Since SS, cc and RR are all kept constant, we have:

∂⁡H∂⁡T=∂∂⁡T⁢(T⁢(S-c⁢R))=S-c⁢R=HT.\frac{\partial H}{\partial T}=\frac{\partial}{\partial T}\left(T(S-cR)\right)% \,{=S-cR}\,{=\frac{H}{T}.}