Bernard Boyer

2022-07-20

Is there a clear analogy here between volume and entropy? Looking at: $\frac{dQ}{T}=dS,\frac{dW}{P}=dV$

minotaurafe

Beginner2022-07-21Added 22 answers

The general expression for the first law (using the OP sign convention) is: $dE=\delta q+\delta w$. When the process is reversible it is possible to write: $dE=TdS+PdV$. This expression is the diferential of a function E(S,V):

$dE=\frac{\mathrm{\partial}E}{\mathrm{\partial}S}dS+\frac{\mathrm{\partial}E}{\mathrm{\partial}V}dV$

The similarity between V and S in the expression is that both are independent functions of the internal energy, while temperature and pressure are partial derivatives:

$T=\frac{\mathrm{\partial}E}{\mathrm{\partial}S}$

$P=\frac{\mathrm{\partial}E}{\mathrm{\partial}V}$

$dE=\frac{\mathrm{\partial}E}{\mathrm{\partial}S}dS+\frac{\mathrm{\partial}E}{\mathrm{\partial}V}dV$

The similarity between V and S in the expression is that both are independent functions of the internal energy, while temperature and pressure are partial derivatives:

$T=\frac{\mathrm{\partial}E}{\mathrm{\partial}S}$

$P=\frac{\mathrm{\partial}E}{\mathrm{\partial}V}$

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