Deriving Internal Energy from Volume with Constant N: Thermodynamics Proof

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SUMMARY

The discussion focuses on deriving an expression for internal energy (U) as a function of volume (V) under constant particle number (N) in thermodynamics. The key equation derived is (dU/dV) = (TdS/dV) - P, which transitions to (dU/dV) = (dP/dT) - P through the application of Maxwell's relations. The relationship dS/dV = dP/dT is established from the Helmholtz free energy differential, dA = -SdT - PdV, confirming the interdependence of thermodynamic variables.

PREREQUISITES
  • Understanding of thermodynamic principles, specifically internal energy and Helmholtz free energy.
  • Familiarity with Maxwell's relations in thermodynamics.
  • Knowledge of partial derivatives and their application in thermodynamic equations.
  • Basic concepts of control variables in thermodynamic systems (T, V, N).
NEXT STEPS
  • Study Maxwell's relations in detail to understand their implications in thermodynamics.
  • Explore the derivation and applications of Helmholtz free energy in thermodynamic systems.
  • Learn about the implications of constant particle number (N) in thermodynamic equations.
  • Investigate the relationship between entropy (S) and volume (V) in various thermodynamic processes.
USEFUL FOR

This discussion is beneficial for students and professionals in thermodynamics, particularly those studying physical chemistry, chemical engineering, and related fields focusing on energy relationships in thermodynamic systems.

LeT374
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Hi all, I have a small question about a proof.

Question:
Under control variables T, V, and N, derive an expression to relate internal energy as a function of volume. Assume that N is constant throughout.

Thoughts:
Starting with dU = TdS - PdV + udN.
Cancel out dN --> dU = TdS - PdV
Divide by dV --> (dU/dV) = (TdS/dV) - P
In my answer key,
It jumps from the above equation to (dU/dV) = (TdP/dT) - P
I don't understand why dS/dV was replaced by dP/dT, how was that relationship derived?

Thanks.
 
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Maxwell's relations.

You know that the Helmholtz free energy dA is
dA = -SdT - PdV
Since dA is an exact differential, dS/dV=dP/dT. In fact, you can get a similar relationship between the properties for each of the four fundamental equations.

Here's more on Maxwell's relations
http://chsfpc5.chem.ncsu.edu/~franzen/CH431/lecture/lec_13_maxwell.htm"
 
Last edited by a moderator:
Ah, next chapter in class. Thanks!
 

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