# What is the relation between pressure and chemical potential?

I know that in a two-component gas-liquid system (for simplicity) the chemical potential $\mu_i$ ($i=1,2$) and total pressure $p$ can be defined by

$$\mu_i=\left(\frac{\partial U}{\partial N_i}\right)_{S,V,N_j}$$ and $$p=-\left(\frac{\partial U}{\partial V}\right)_{S,N_1,N_2}.$$ where $U$ and $V$ are the internal energy and volume of the system, $N_i$ is the mole number of each component, and $p=p_1+p_2$ with $p_i$ being the partial pressure. But how the partial pressure and chemical potential are related? Intuitively, I think $\mu_i\propto p_i$. Thank you!

• Are you looking for an explicit equation relating the two quantities, $p = p(\mu)$ or $\mu = \mu(p)$? What system are you looking at---a gas, a liquid, a solid? – a-cyclohexane-molecule Nov 25 '17 at 5:40
• @a-cyclohexane-molecule Thanks for your reminding. I'm looking for an explicit expression relating chemical potential and partial pressure for a gas-liquid system. – jsxs Nov 25 '17 at 6:43