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I have a system whose entropy is specified by three variables: Salinity $S$, pressure$p$, and density$\rho$.

I have the following partial derivative:

\begin{equation} (\frac{\partial\eta}{\partial p})_{\rho,S}= -(\frac{\partial\eta}{\partial \rho})_{p,S}(\frac{\partial\rho}{\partial p})_{n,S} \end{equation}

Where $\eta$ is the specific entropy:

My question is where does the minus sign come from? I though it just was a chain rule but I can not explain this sign.

Thanks

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  • $\begingroup$ Should the last derivative be holding eta or n constant? $\endgroup$
    – Eddy
    Commented Sep 11, 2020 at 21:59
  • $\begingroup$ This is the triple product rule; see here. $\endgroup$
    – Chemomechanics
    Commented Sep 11, 2020 at 22:23

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Start with $$d\eta=\left(\frac{\partial \eta}{\partial p}\right)_{\rho}dp+\left(\frac{\partial \eta}{\partial \rho}\right)_pd\rho$$Then set $d\eta$ equal to zero and solve for $d\rho/dp$ (at constant $\eta$).

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