Help with proof of thermodynamics equation

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SUMMARY

The discussion centers on deriving the thermodynamic equations related to a wire under tension, specifically demonstrating that the partial derivative of Helmholtz free energy (F) with respect to length (L) equals tension (T), and the partial derivative of F with respect to temperature (T) equals negative entropy (S). The equations provided include fundamental thermodynamic relationships such as dF=-SdT-pdV and dU=TdS-pdV. The confusion arises from the dual use of the letter T for both temperature and tension, which necessitates careful differentiation in calculations.

PREREQUISITES
  • Understanding of thermodynamic concepts, specifically Helmholtz free energy.
  • Familiarity with partial derivatives in the context of thermodynamics.
  • Knowledge of fundamental thermodynamic equations, including dU=TdS-pdV.
  • Ability to differentiate between variables with similar notations, such as temperature and tension.
NEXT STEPS
  • Study the derivation of Helmholtz free energy and its applications in thermodynamics.
  • Learn how to manipulate and interpret partial derivatives in thermodynamic equations.
  • Research the implications of work done on a system, specifically in the context of stretching materials.
  • Explore the differences between various thermodynamic potentials, including Gibbs free energy and internal energy.
USEFUL FOR

Students studying thermodynamics, particularly those tackling complex problems involving thermodynamic systems and free energy calculations. This discussion is also beneficial for educators seeking to clarify common misconceptions in thermodynamic notation.

ksingh1990
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1. Homework Statement
Suppose a thermodynamic system consists of a wire of length L under a tension T show that:
(dF/dL)=T
and
(dF/dT)=-S
(both are partial derivatives)


2. Homework Equations
F=U-TS
H=U+PV
G=U+PV-TS
dU=TdS-pdV
dF=-SdT-pdV
dH=TdS+Vdp
dG=-SdT+Vdp
dQ=TdS
dW=pdV


I'm getting confused with the T for temperature and T for tension. Please explain to me how to do this as I have no idea about where I should begin.
Thank You
 
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How about modifying the differential equation dF=-S\,dT-P\,dV to incorporate work done by stretching the wire? Also, you can probably ignore P\,dV work in this problem.
 

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