Quantcast
  • 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.

News

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

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  $\partial S/\partial V = -p/T$? What has gone wrong here?

+ 0 like - 0 dislike
368 views

Consider two systems, 1 and 2, which can exchange heat and do only the mechanical p-V work. They are isolated from the rest of the world. At equilibrium they have the common pressure, say $\bar p$. Then if we nudge them slightly from equilibrium, then we must have, because of isolation, that (I denote by $E$ the internal energies)
\begin{align*}
\text{total increase in energy } &=(dE_1 - \bar p\;dV_1) + (dE_2 - \bar p\;dV_2)\\
&= d(E_1 + E_2 -\bar p(V_1 + V_2))
\end{align*}
be zero, and hence
$$
E_1 + E_2 = \bar p(V_1 +V_2) + \text{const.}\tag{1}
$$

Now, assuming that the systems are independent,
$$
S_\text{tot}(E_1, E_2, V_1, V_2) = S_1(E_1, V_1)+S(E_2, V_2).\tag{2}
$$

Now, maximizing (2) subject to (1) leads that, at equilibrium,
$$
\frac{\partial S_1}{\partial E_1} = \frac{\partial S_2}{\partial E_2}
$$
which we identify as $1/T$, and
$$
\frac{\partial S_1}{\partial V_1} = -\frac{\bar p}{T},
$$
which contradicts the oft-stated result with the minus sign flipped.

In hindsight, there must be a minus sign on the RHS of the constraint (1), but I don't see why.



Question: What exactly has gone awry in the above reasoning of mine?

asked Mar 8, 2021 in Theoretical Physics by Atom (5 points) [ revision history ]
edited Mar 8, 2021 by Atom

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:
p$\hbar$ysics$\varnothing$verflow
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
...