Partial derivative in thermodynamics

Click For Summary
SUMMARY

The discussion centers on the application of partial derivatives in thermodynamics, specifically the expression (∂Cv/∂V)T=0. Participants clarify that the ability to switch the order of differentiation in the proof is justified by Schwarz's theorem, which is applicable when the function U meets certain continuity conditions. The proof transitions from [(∂/∂V)(∂U/∂T)V]T to [(∂/∂T)(∂U/∂V)T]V, demonstrating the interchangeability of the derivatives under these conditions.

PREREQUISITES
  • Understanding of partial derivatives in calculus
  • Familiarity with thermodynamic concepts such as heat capacity
  • Knowledge of Schwarz's theorem and its implications
  • Basic principles of multivariable calculus
NEXT STEPS
  • Study the application of Schwarz's theorem in calculus
  • Learn about the implications of Stokes' theorem in multivariable calculus
  • Explore the properties of heat capacity in thermodynamics
  • Review techniques for calculating partial derivatives
USEFUL FOR

Students and professionals in physics, particularly those studying thermodynamics, as well as mathematicians focusing on calculus and its applications in physical sciences.

orgohell
Messages
2
Reaction score
0
So I have a proof and I can't follow the process, I think its because I haven't learned how to do partial derivatives or I've forgotten, anyways can someone tell me if this is a rule in calculus

(∂Cv/∂V)T=0

I've gotten to
[(∂/∂V)(∂U/∂T)V]T

and the proof I have goes to
[(∂/∂T)(∂U/∂V)T]V
-is this a rule? they were able to switch the denominators of the derivatives which is can see but then they switched the constants as well If so what is the name so I can read up on it, I can't seem to find it with google
 
Chemistry news on Phys.org
orgohell said:
So I have a proof and I can't follow the process, I think its because I haven't learned how to do partial derivatives or I've forgotten, anyways can someone tell me if this is a rule in calculus

(∂Cv/∂V)T=0

I've gotten to
[(∂/∂V)(∂U/∂T)V]T

and the proof I have goes to
[(∂/∂T)(∂U/∂V)T]V
-is this a rule? they were able to switch the denominators of the derivatives which is can see but then they switched the constants as well If so what is the name so I can read up on it, I can't seem to find it with google

That's allowed because U is a function that is in the conditions of Schwarz's theorem (which can be derived from stokes' theorem)
 

Similar threads

High School Question about the definition of a partial derivative
  • · Replies 13 ·
Replies
13
Views
2K
Chemistry Why can we differentiate this entropy total derivative with repect to Temperature?
  • · Replies 5 ·
Replies
5
Views
4K
Undergrad Velocity of a free particle using Landau's approach
  • · Replies 20 ·
Replies
20
Views
1K
Graduate Question on Dirac's derivatives of the 4-velocity w.r.t. coordinates
  • · Replies 53 ·
2
Replies
53
Views
4K
Temperature of two subsystems at equilibrium
  • · Replies 7 ·
Replies
7
Views
1K
Undergrad Change of Variables in Thermodynamics
  • · Replies 4 ·
Replies
4
Views
3K
Multivariable calculus proof involving the partial derivatives of an expression
  • · Replies 4 ·
Replies
4
Views
1K
Undergrad How Does Temperature Affect Wire Tension at Constant Length?
  • · Replies 4 ·
Replies
4
Views
2K
Issue deriving the third law of thermodynamics
  • · Replies 11 ·
Replies
11
Views
1K
Help with derivation of electric field of a moving charge
  • · Replies 5 ·
Replies
5
Views
4K