Gibbs free energy partial derivative

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SUMMARY

The discussion centers on the calculation of the partial derivative of Gibbs free energy (g) with respect to temperature (T) at constant pressure (P). The correct expression derived is that the partial derivative is equal to the negative entropy (s), confirmed through the equation dg = -sdT. The participants clarify that internal energy (u), volume (v), and entropy (s) are not constant during this process, emphasizing the importance of understanding these variables in thermodynamic calculations.

PREREQUISITES
  • Understanding of thermodynamic principles, specifically Gibbs free energy.
  • Familiarity with partial derivatives in calculus.
  • Knowledge of the relationships between internal energy (u), pressure (P), volume (v), and entropy (s).
  • Basic grasp of thermodynamic equations and their applications.
NEXT STEPS
  • Study the derivation of Gibbs free energy and its applications in thermodynamics.
  • Learn about the implications of the Maxwell relations in thermodynamic systems.
  • Explore the concept of entropy and its role in energy transformations.
  • Investigate the conditions under which variables are considered constant in thermodynamic equations.
USEFUL FOR

This discussion is beneficial for students and professionals in thermodynamics, particularly those studying physical chemistry, chemical engineering, and related fields who seek a deeper understanding of Gibbs free energy and its derivatives.

spaghetti3451
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g = u + Pv - Ts

To find the partial derivative of g with respect to T at constant P, we do the following.

dg = du + vdP + Pdv - Tds - sdT and du = Tds - Pdv.

Therefore, dg = vdP - sdT.

At constant pressure, dg = - sdT.
Therefore, the partial derivative is - s.

I think we could have found this result equally well by just taking the derivative of g with respect to T keeping P constant. u, v and s are constant, so the answer is - s.

Which method do you think is the right one?
 
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The method you've detailed above is the right one. It's not true where you say that u, v and s are constant -- they're not.
 
Just to confirm: Are you saying that the first method is correct and the second is not?

Why are u, v and s not constant?
 

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