• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,348 answers , 22,725 comments
1,470 users with positive rep
818 active unimported users
More ...

  derivative w.r.t. conjugate variables (Thermodynamics)

+ 2 like - 0 dislike

Is there a general relation to derivatives of natural variables in thermodynamics with respect to their conjugate variable?

e.g. \(\left(\frac{\partial S}{\partial T}\right)_p =\) ?

something like the Maxwell relations?

asked Feb 3, 2017 in General Physics by kellekai (10 points) [ no revision ]

1 Answer

+ 2 like - 0 dislike

Well, yes. But the list is very big. For your example we have that

$$ \left( \frac{\partial P }{ \partial T } \right)\Big|_{S} = -  \left( \frac{\partial P }{ \partial S } \right)\Big|_{T}  \left( \frac{\partial S }{ \partial T } \right)\Big|_{P}$$

but the LHS due to Maxwell's relations is the same as $ \left( \frac{\partial S}{ \partial V } \right)\Big|_{P}$. But for this guy we have that

$$ \left( \frac{\partial P }{ \partial V } \right)\Big|_{S} = -  \left( \frac{\partial P }{ \partial S } \right)\Big|_{V}  \left( \frac{\partial S }{ \partial V } \right)\Big|_{P}$$

and from Maxwell we have that $-  \left( \frac{\partial P }{ \partial S } \right)\Big|_{V} =  \left( \frac{\partial T }{ \partial V } \right)\Big|_{S} $. Now for the RHS of this we have

 $$ \left( \frac{\partial T }{ \partial V } \right)\Big|_{S} = -  \left( \frac{\partial T }{ \partial S } \right)\Big|_{V}  \left( \frac{\partial S }{ \partial V } \right)\Big|_{T}$$

Finally, again from Maxwell we know that $\left( \frac{\partial S }{ \partial V } \right)\Big|_{T} = \left( \frac{\partial P }{ \partial T } \right)\Big|_{V} = -\frac{\alpha}{\kappa}$ where

$$ \alpha = \frac{1}{V}\left( \frac{\partial V}{\partial T} \right)\Big|_{P}  $$ 


$$ \kappa = -\frac{1}{V}\left( \frac{\partial V}{\partial P} \right)\Big|_{T}  $$ 

I hope this helps.

answered Feb 3, 2017 by conformal_gk (3,625 points) [ no revision ]

Thanks, yes it's helpful :) the trick u use to split the partial derivatives didn't found it's way to my mind...

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights