Beginning with the expression $C_{p} = \left( \frac{\partial H}{\partial T}\right)_{N,p}$, I use the Leibniz rule to get: $$ C_{p} = \left( \frac{\partial H}{\partial S}\right)_{N,p} \left( \frac{\partial S}{\partial T}\right)_{N,p} $$

Why is it true that $\left( \frac{\partial H}{\partial S}\right)_{N,p} = T$? I don't see how this is the case; I know that the definition of temperature is $\left( \frac{\partial U}{\partial S}\right)_{N,V} = T$, but I'm not sure how it can possibly relate to this.

EDIT: I meant the derivative of enthalpy $H = U + PV$, not just the energy.

  • 1
    $\begingroup$ hint: what is the internal energy change at constant pressure $\endgroup$
    – hyportnex
    Mar 11, 2017 at 14:23

1 Answer 1


$\text{d}H=T\text{d}S+V\text{d}P$, then $\left(\frac{\partial H}{\partial S}\right)_P=T$, doesn't it?


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