Chemical potential of an ideal gas problem

Click For Summary
SUMMARY

The chemical potential of an ideal gas can be expressed as μ = c_{P}T - c_{V}T ln T - RT ln V - s_{0}T + const. To derive this, one must first find the entropy S = S(T,V) for an ideal gas, which is given by S = nc_{V}ln T + nRln V. The Gibbs free energy G is calculated using the equation G = U - TS + PV, where U for an ideal gas is a constant (Nfk/2). The relationship between the heat capacities is established as c_{P} = c_{V} + nR, leading to the conclusion that nk_{B}T = c_{P}T - c_{V}T.

PREREQUISITES
  • Understanding of thermodynamic concepts such as chemical potential and Gibbs free energy.
  • Familiarity with the equations of state for ideal gases, specifically PV = nRT.
  • Knowledge of heat capacities, particularly the relationship between c_{P} and c_{V}.
  • Basic understanding of entropy and its dependence on temperature and volume.
NEXT STEPS
  • Study the derivation of the Gibbs free energy for ideal gases in detail.
  • Learn about the implications of the Maxwell relations in thermodynamics.
  • Explore the concept of entropy in greater depth, particularly for different thermodynamic processes.
  • Investigate the applications of chemical potential in real-world systems, such as phase transitions.
USEFUL FOR

Students and professionals in chemistry and physics, particularly those studying thermodynamics and the behavior of ideal gases. This discussion is beneficial for anyone looking to deepen their understanding of chemical potential and related thermodynamic concepts.

arenaninja
Messages
25
Reaction score
0
Hey everyone. I hope someone can help. I'm off on this by several factors so I'm wondering what I may be inferring incorrectly.

Homework Statement


Express the chemical potential of an ideal gas in termps of T and V:
\mu = c_{P}T - c_{V}T\ln T - RT\ln V - s_{0}T + const

Homework Equations


(Hint: Find the entropy S = S(T,V); use G = U - TS + PV and write \mu = G/n)

The Attempt at a Solution


For S = S(T,V) of an ideal gas we have:
S = nc_{V}\ln T + nR\ln V
Now we attempt to find G:
G = U - TS + PV
Recognize that U for an ideal gas is a constant (Nfk/2), and we have:
G = -nc_{V}T\ln T - nRT\ln V + \frac{nfk_{B}}{2} + PV

As you can see, I'm missing two terms. I'm not sure how PV would translate into those two terms. So overall I'm not faring very well in this problem.

Any hints? Insights? Corrections?
 
Last edited:
Physics news on Phys.org
What is PV for an ideal gas?
 
PV = nRT
Also, for ideal gases:
c_{P} = c_{V} + nR
Ohhh I see (I think). So the last term:
nk_{B}T = c_{P}T - c_{V}T

I'm guessing c_{V}T = s_{0}T. Great!

Thank you very much!
 

Similar threads

Thermodynamics: Possible process between a van der Waals gas and an ideal gas
  • · Replies 1 ·
Replies
1
Views
2K
Thermodynamics: deriving expression for S = S(T, V, N) - constant problems
  • · Replies 19 ·
Replies
19
Views
2K
Relationship between pressure & chem potential in ideal gas
  • · Replies 5 ·
Replies
5
Views
2K
[Doubt] Chemical Potential of an Ideal Gas
  • · Replies 3 ·
Replies
3
Views
2K
Equation of state for a real gas
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad How to see this form for the chemical potential of an ideal gas?
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Helmholtz entropy of ideal gas mixture is additive?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Does Callen's Entropy Expression for a General Ideal Gas Contain an Error?
  • · Replies 19 ·
Replies
19
Views
3K
How is Chemical Potential Affected by Altitude in an Ideal Gas?
  • · Replies 1 ·
Replies
1
Views
4K
Finding chemical potential with given thermodynamic relation
  • · Replies 2 ·
Replies
2
Views
2K