Show that (du/dv)T = T(dp/dt)v - p
Using Tds = du + pdv and a Maxwell relation
The Attempt at a Solution
I've solved the problem, but I'm not entirely sure my method is correct.
Tds = du + pdv ---> du = Tds - Pdv
- Using dF=(dF/dx)ydx +(dF/dy)xdy
- Therefore Tds - Pdv = (du/dT)v+(du/dv)Tdv
- Divide by dv:
(du/dT)vdT/dv + (du/dv)T = T(ds/dv)T - p
Now, to get the right answer, this term:
must equal zero, but I'm not sure why - please can somebody explain?
Then you simply insert Maxwell relation -(ds/dv)T = -(dp/dT)v
and rearrange to get the correct answer.
Many thanks for any help!