Thermodynamic Potentials in Proper Variables

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SUMMARY

The discussion focuses on deriving the thermodynamic potentials H (enthalpy), F (Helmholtz free energy), and G (Gibbs free energy) from the internal energy equation U_1 = S^2/a_1 + b_1V(V - 2V_0). Participants utilized Maxwell's relations and the fundamental thermodynamic equations H = U + PV, F = U - TS, and G = U - TS + PV to express these potentials in terms of proper variables. Key steps included finding pressure as p = -∂U/∂V|_S and temperature as T = ∂U/∂S|_V, which facilitated the conversion of variables without needing differentials.

PREREQUISITES
  • Understanding of thermodynamic potentials: Enthalpy, Helmholtz free energy, and Gibbs free energy
  • Familiarity with Maxwell's relations in thermodynamics
  • Knowledge of partial derivatives and their application in thermodynamic equations
  • Basic understanding of the first law of thermodynamics and internal energy
NEXT STEPS
  • Study the derivation of Maxwell's relations in thermodynamics
  • Learn how to apply partial derivatives in thermodynamic contexts
  • Explore the implications of thermodynamic potentials on physical systems
  • Investigate the relationship between temperature, pressure, and volume in thermodynamic processes
USEFUL FOR

This discussion is beneficial for students and professionals in thermodynamics, particularly those studying physical chemistry, mechanical engineering, or any field that requires a deep understanding of thermodynamic potentials and their applications.

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Homework Statement


A solid has energy (for one mole):

##U_1 =\frac{S^2}{a_1}+b_1V(V-2V_0)##

Write H, F, and G in terms of their proper variables.

Homework Equations



Maxwell's relations and
H=U+PV
F=U-TS
G=U-TS+PV

The Attempt at a Solution


[/B]
H, for example:

I have tried writing dV as a sum of partial derivatives times total differentials:

##dV=\frac{\partial V}{\partial S}dS+\frac{\partial V}{\partial P}dP##

I have no idea how to get V in terms of S and P.

I have similar problems for the other cases.

Thanks for any help!
 
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Under relevant equations you mentioned Maxwells equations.
 
Maxwell's relations. I still don't know how that gives anything concrete.
 
Well, probably the first step is to find pressure as ##p=-\frac{\partial U}{\partial V}|_S## and an analogous relation for temperature. For the rest, you don't even need differentials.
 
Last edited:
DrDu said:
Well, probably the first step is to find pressure as ##p=\frac{\partial U}{\partial S}|_V## and an analogous relation for temperature. For the rest, you don't even need differentials.

How did you find that?
 
Sorry, I was already thinking in temperature when writing down this. I corrected the expression in #4.
 
Thanks sir! Your response helped me solve the problem.
 

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