Partial derivative in thermodynamics

In summary: Lso known as the theorem of changing order of partial derivatives, it states that if the second partial derivatives of a function exist and are continuous in a region, then the order of differentiation can be changed without changing the result. This is known as the Chain Rule in Calculus and can be found in most textbooks or online resources.
  • #1
orgohell
3
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
 
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  • #2
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)
 

FAQ: Partial derivative in thermodynamics

What is a partial derivative in thermodynamics?

A partial derivative in thermodynamics is a mathematical concept that describes the rate of change of a thermodynamic property with respect to one variable, while keeping all other variables constant. It allows us to analyze how a system changes when one of its variables changes.

How is a partial derivative calculated?

A partial derivative is calculated by taking the derivative of a function with respect to one variable, while treating all other variables as constants. This can be done using standard calculus techniques such as the chain rule and product rule.

What is the significance of partial derivatives in thermodynamics?

Partial derivatives play a crucial role in thermodynamics as they allow us to understand the behavior of a system under changing conditions. They help us to analyze how different variables affect a system and make predictions about its behavior.

Can partial derivatives be negative in thermodynamics?

Yes, partial derivatives can be negative in thermodynamics. This indicates that the variable in question is decreasing as the other variables are held constant. For example, a negative partial derivative of pressure with respect to volume would indicate that as the volume decreases, the pressure increases.

What is the difference between a partial derivative and a total derivative in thermodynamics?

A partial derivative describes the change in a function with respect to one variable, while keeping all other variables constant. A total derivative, on the other hand, describes the change in a function with respect to all variables. In thermodynamics, a total derivative is used to analyze changes in a system as a whole, while partial derivatives are used to analyze changes in specific variables.

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