# Thermodynamic relations from Gibbs-Duhem

Given the Gibbs-Duhem relation $V dp = S dT - N d \mu$, I am having trouble deriving the following identity:

$\ (\frac{\partial N}{\partial \mu})_{V,T} = N (\frac{\partial \rho}{\partial p})_T$

The problem is that the variables $N$ and $\rho$ don't appear as infinitisemals in the equation. How can I proceed?

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Start by noting that the density is inversely proportional to volume: ρ = m / V, and also that the mass m equals the substance's molar mass M multiplied by the number of moles N: m = M N. Together we have, ρ = M (N / V) (M will be carried through the rest of the derivation as an arbitrary constant with dimensions kg / mol.) –  David H Nov 27 '12 at 21:16