Find Chemical Potential of Ideal Gas: Gibbs-Duhen Relation

[Dr. 104]

Homework Statement

Find the chemical potential for an ideal gas as a function of temperature and pressure. Use the "Gibbs-Duhen relation".

Homework Equations

$$\mu=\frac{\partial U}{\partial N}$$
$$dU=TdS-pdV+\sum\limits_{i}\mu_{i}dN_{i}$$
$$U=Q+W$$
Gibbs-Duhen relation: $$0=SdT-Vdp+\sum\limits_{i}N_{i}d\mu_{i}$$
Ideal gas law: $$pV=Nk_{b}T$$

The Attempt at a Solution

Well first I just tried putting the second equation (above) into the first one, but that just resulted in $\mu$ again, so that was a dead end.
I also tried plugging in everything I could from the ideal gas equation into the second equation (above) and the Gibbs-Duhen equation, because the problem asks for an ideal gas, hoping something would pop out after that, but I had no luck.

Thermodynamics has always been my weakest subject in physics.
I have several problems like this to do, so I'm not actually looking for a solution for this particular problem, I'm looking for general information on how to solve this type of problem that will help me on all of them.
I have no trouble like this with other areas of physics, but there's something about thermodynamics that my mind just doesn't get.. Maybe I just haven't learned it properly but I don't feel like there's any consistent set of fundamental equations, or underlying theory that I can cling to when I'm lost.

 

[member 553083]

Any advice or resources would be greatly appreciated. Thank you.A:The chemical potential of an ideal gas is given by the equation$$ \mu = \frac{\partial U}{\partial N} = \frac{RT}{N} $$where $U$ is the internal energy, $R$ is the gas constant, and $T$ is the temperature.This equation can be derived by combining the ideal gas law (which states that $pV = Nk_BT$, where $p$ is the pressure, $V$ is the volume, $k_B$ is Boltzmann's constant) and the Gibbs-Duhem relation (which states that $dU = TdS - pdV + \sum_i \mu_idN_i$).Using the ideal gas law, we can rearrange the equation to get $VdP = Nk_BdT$. Substituting this expression into the Gibbs-Duhem relation and solving for $\mu_i$ yields$$ \mu = \frac{RT}{N}. $$Hope this helps!