Partial integration misunderstanding

[dingo_d]

Homework Statement

I'm doing some thermodynamics and I didn't quite get one thing:

I have a function: $$U=U(T,\ V(T,p))$$ and I need to find the expression:

$$\left(\frac{\partial U}{\partial T}\right)_p$$

The Attempt at a Solution

Now what I found in the book and what Mathematica gave me is:

$$\left(\frac{\partial U}{\partial T}\right)_p=\left(\frac{\partial U}{\partial T}\right)_V+\left(\frac{\partial U}{\partial V}\right)_T\left(\frac{\partial V}{\partial T}\right)_p$$


But I can't see why that is :\ I tried looking with the formulas for mixed partial derivatives but I must be missing sth out :\

Can someone point me in the right direction?

 

[Hootenanny]

Homework Statement

I'm doing some thermodynamics and I didn't quite get one thing:

I have a function: $$U=U(T,\ V(T,p))$$ and I need to find the expression:

$$\left(\frac{\partial U}{\partial T}\right)_p$$

The Attempt at a Solution

Now what I found in the book and what Mathematica gave me is:

$$\left(\frac{\partial U}{\partial T}\right)_p=\left(\frac{\partial U}{\partial T}\right)_V+\left(\frac{\partial U}{\partial V}\right)_T\left(\frac{\partial V}{\partial T}\right)_p$$


But I can't see why that is :\ I tried looking with the formulas for mixed partial derivatives but I must be missing sth out :\

Can someone point me in the right direction?

Whilst it isn't important to the question, this isn't partial integration, it is partial differentiation.

It is useful in this case to simplify the problem a little first. Start by treating U as a function of two variables. If

$$U = U\left(T,V\right)$$ ,

(i.e. ignore the dependence of V on T) can you work out

$$\left(\frac{\partial U}{\partial T}\right)_p$$

?

 

[dingo_d]

Oh doh! diferentiation! Woops, my bad.

Is it:

$$\left(\frac{\partial U}{\partial T}\right)_p=\left(\frac{\partial U}{\partial V}\right)_p\left(\frac{\partial V}{\partial T}\right)_p$$?

Ok so I have:$$dU=\left(\frac{\partial U}{\partial V}\right)_TdV+\left(\frac{\partial U}{\partial T}\right)_VdT$$

Now I can diferentiate that with respect to T at constant p:$$\left(\frac{\partial U}{\partial T}\right)_p=\left(\frac{\partial U}{\partial V}\right)_T\left(\frac{\partial V}{\partial T}\right)_p+\left(\frac{\partial U}{\partial T}\right)_V\underbrace{\left(\frac{\partial T}{\partial T}\right)_p}_1$$

Oh! That's it! XD Thnx :D