How to obtain maxwell relations

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SUMMARY

The discussion focuses on deriving Maxwell relations, specifically how to compute the derivative ds/dL at constant temperature using the equation T ds = dE - F dL. The key conclusion is that ds/dL equals dF/dT, which is established through the function H = E - Ts and the equality of mixed second partial derivatives of H. The relationship is confirmed by the equation ∂2H/∂T∂L = ∂2H/∂L∂T, leading to the Maxwell relation - (∂s/∂L) = ∂F/∂T.

PREREQUISITES
  • Understanding of thermodynamic variables such as temperature (T), entropy (S), and energy (E).
  • Familiarity with partial derivatives and their application in thermodynamics.
  • Knowledge of Maxwell relations and their significance in physical chemistry.
  • Basic grasp of the concept of functions in thermodynamics, particularly the function H = E - Ts.
NEXT STEPS
  • Study the derivation of Maxwell relations in thermodynamics.
  • Learn about the implications of mixed partial derivatives in thermodynamic functions.
  • Explore the application of the function H = E - Ts in various thermodynamic problems.
  • Investigate the physical significance of the relationship ds/dL = dF/dT in real-world systems.
USEFUL FOR

Students and professionals in physics and physical chemistry, particularly those studying thermodynamics and seeking to deepen their understanding of Maxwell relations and their applications.

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Hey, I have had a lot of trouble understanding how one obtains a Maxwell relation.

So let's say in general I know(from a specific problem)

T ds = dE - F dL

where F is a tension and L is a length, E is the energy T is the temperature and S is the entropy of a system.

In a specific problem I am asked to compute

ds/dL at constant Temp.

I have an expression for F and it turns out that ds/dL = dF/dT but I don't know how one arrives at this point.

Any help is appreciated, thanks.
 
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Write it as dE = T ds + F dL. Define a function H = E - Ts, for which dH = - s dT + F dL. The Maxwell relation follows from the fact that the mixed second partial derivatives of H are equal:

∂2H/∂T∂L = ∂2H/∂L∂T which is - (∂s/∂L) = ∂F/∂T

(Did you have the signs right?)
 

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