Find Chemical Potential of Ideal Gas: Gibbs-Duhen Relation

In summary, the chemical potential of an ideal gas can be determined using the Gibbs-Duhem relation and the ideal gas law, and it is given by $\mu = \frac{RT}{N}$.
  • #1
Dr. 104
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Homework Statement


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

Homework Equations


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

The Attempt at a Solution


Well first I just tried putting the second equation (above) into the first one, but that just resulted in [itex]\mu[/itex] 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.
 
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  • #2
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!
 

1. What is the Gibbs-Duhen relation?

The Gibbs-Duhen relation is a fundamental thermodynamic equation that relates the chemical potential of a component in a mixture to the partial molar volume and partial molar entropy of that component.

2. How is the Gibbs-Duhen relation used to find the chemical potential of an ideal gas?

The Gibbs-Duhen relation states that the change in chemical potential for an ideal gas is equal to the product of the ideal gas constant and the change in temperature. This allows us to calculate the chemical potential of an ideal gas at a specific temperature using known values for the ideal gas constant and the temperature.

3. What is an ideal gas?

An ideal gas is a theoretical concept that describes a gas that follows the ideal gas law, which states that the pressure, volume, and temperature of a gas are all directly proportional to each other. In reality, no gas is truly ideal, but many gases behave similarly to an ideal gas under certain conditions.

4. What is the significance of finding the chemical potential of an ideal gas?

The chemical potential of an ideal gas can provide important information about the behavior of the gas and its interactions with other components in a mixture. It is also a key parameter in thermodynamic calculations and can help determine the equilibrium state of a system.

5. Can the Gibbs-Duhen relation be applied to non-ideal gases?

Yes, the Gibbs-Duhen relation can be applied to both ideal and non-ideal gases. However, for non-ideal gases, additional equations or models may need to be used in conjunction with the Gibbs-Duhen relation to accurately calculate the chemical potential.

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