# Chemical potential as a function of temperature

I have considered an ideal Fermi gas. Then, we can obtain an expression for chemical potential as a function of temperature. I want to understand the physical significance to it or what it really means. Isn't chemical potential generally a function of temperature for all kinds of gases?

• Possible duplicate at physics.stackexchange.com/q/69664 though I think this is more about how you would explain what is happening with words right? Mar 6 '15 at 9:47

I think the best way to think about it is in terms of entropy ($S$). The chemical potential $\mu$ is related to the entropy $S$ by $\mu = -T \frac{\partial S}{\partial N} .$

The entropy $S$=$S(N,V,T)$ (or (N,V,E), or etc...) is a function of N. Chemical potential is a useful concept because it tells you how the entropy changes due to changes in N, the number of particles or whatever in your system.

Then statements like "particles will go from high chemical potential to low chemical potential" are just code for "the entropy isn't maximized right now, so the particles will move around so that the entropy is maximized".

Yes, the chemical potential is generally a function of temperature. As such, the chemical potential in the specific case of a Fermi gas is a function of temperature.

In the Fermi gas, the chemical potential is "repelled" by the region of higher density of states as the temperature increases. This is because the former step-function (the T=0 fermi function) broadens out on both sides equally. And so, to maintain a constant number of particles, the location of the center has to move away from the high density of states region.

For a Fermi gas, the higher density of states is at higher energy and so the chemical potential decreases with increasing temperature (at least for small T) like $$\mu(T)\sim E_F-\alpha T^2\;,$$ where $$E_F$$ is the $$T=0$$ chemical potential and $$\alpha$$ is a constant.

• Can you please help me understand this statement diagrammatically if possible."In the Fermi gas, the chemical potential is "repelled" by the region of higher density of states as the temperature increases. This is because the former step-function (the T=0 fermi function) broadens out on both sides equally. And so, to maintain a constant number of particles, the location of the center has to move away from the high density of states region." Mar 6 '15 at 8:14

Consider the total differential of the Helmholtz Free energy $F$ with

$dF = -SdT +\mu dN$

with chemical potential $\mu$, entropy $S$ and particle number $N$. A necessary condition to have such a total differential is the relation:

$\frac{\partial \mu}{\partial T} = \frac{\partial (-S)}{\partial N} = - s_{mol}$.

This relation proves clearly that chemical potential is function of temperature $T$ if the molar entropy $s_{mol}$ is nonzero. And even a gas at equilibrium has a nonzero molar entropy. This equation has significance in several processes of physical chemistry.

At equilibrium, substance A has two phases which are as follows:

$$\alpha \rightleftharpoons \beta$$

Choose the correct relations between the temperature and chemical potential of these phases.

Ans: If two phases of a given substance are equilibrium with each other, then the temperature and chemical potential of these phases are the same. Therefore, the correct relationship between temperature and chemical potential of $$\alpha$$ and $$\beta$$ phases is

$$T_{\alpha}=T_{\beta} \quad ; \quad \mu_{\alpha}=\mu_{\beta}$$