Internal Energy as a function of U(S,V,A,Ni)

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Discussion Overview

The discussion revolves around the function describing internal energy U as a function of entropy S, volume V, area A, and number of particles N. Participants explore the total differential of internal energy and the conditions under which certain variables are held constant.

Discussion Character

  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant presents the internal energy function U(S,V,A,N) and its total differential dU, questioning why certain variables are set to zero.
  • Another participant provides a mathematical breakdown of the total differential, indicating how temperature T, pressure P, and other variables relate to the derivatives of U.
  • A follow-up question asks about the technique used in the derivation and its relation to Legendre transforms.
  • Reference to the Gibbs Formulation and its presence in various thermodynamics and physical chemistry texts is made by another participant.

Areas of Agreement / Disagreement

The discussion includes multiple viewpoints regarding the derivation of the total differential and the conditions applied to the variables. No consensus is reached on the necessity of knowledge of the internal energy equation for deriving the relationships.

Contextual Notes

Participants express uncertainty about the implications of holding certain variables constant and the specific technique used in the derivation, indicating potential limitations in their understanding of the broader context.

Who May Find This Useful

This discussion may be useful for students and professionals interested in thermodynamics, particularly those exploring internal energy and its mathematical formulations.

mcoth420
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A general thermo question...
for the function describing internal energy U(S,V,A,N)

U=TS-PV+γA+μN

please explain how the total differential is

dU=TdS-PdV+γdA+μdN (for a one component system)

Basically why is dT=dP=dγ=dμ=0? Is it because they are intensive or potentials?

Thank you,

M
 

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\begin{array}{l}<br /> U = U(S,V,.N,A...) \\ <br /> dU = {\left( {\frac{{\partial U}}{{\partial S}}} \right)_{V,N,A...}}dS + {\left( {\frac{{\partial U}}{{\partial V}}} \right)_{S,N,A}}dV... \\ <br /> T = {\left( {\frac{{\partial U}}{{\partial S}}} \right)_{V,N,A...}} \\ <br /> P = {\left( {\frac{{\partial U}}{{\partial V}}} \right)_{S,N,A}} \\ <br /> {\rm{etc}} \\ <br /> \end{array}

I will leave you to fill in the bits for moles and area or other quantities.
 
Thank you for your reply...what is this technique called? I am working with Legendre transforms and this is similar...

Another thing...how can you derive T=partialU/partialS or others without knowledge of the internal energy equation?
 
This is the Gibbs Formulation.

Carington : Basic Thermodynamics P 187ff : Oxford University Press

Also in many Physical Chemistry texts.
 

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