1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
|
---
title: "Fundamental thermodynamic relation"
date: 2021-07-07
categories:
- Physics
- Thermodynamics
layout: "concept"
---
The **fundamental thermodynamic relation** combines the first two
[laws of thermodynamics](/know/concept/laws-of-thermodynamics/),
and gives the change of the internal energy $U$,
which is a [thermodynamic potential](/know/concept/thermodynamic-potential/),
in terms of the change in
entropy $S$, volume $V$, and the number of particles $N$.
Starting from the first law of thermodynamics,
we write an infinitesimal change in energy $\dd{U}$ as follows,
where $T$ is the temperature and $P$ is the pressure:
$$\begin{aligned}
\dd{U} &= \dd{Q} + \dd{W} = T \dd{S} - P \dd{V}
\end{aligned}$$
The term $T \dd{S}$ comes from the second law of thermodynamics,
and represents the transfer of thermal energy,
while $P \dd{V}$ represents physical work.
However, we are missing a term, namely matter transfer.
If particles can enter/leave the system (i.e. the population $N$ is variable),
then each such particle costs an amount $\mu$ of energy,
where $\mu$ is known as the **chemical potential**:
$$\begin{aligned}
\dd{U} = T \dd{S} - P \dd{V} + \mu \dd{N}
\end{aligned}$$
To generalize even further, there may be multiple species of particle,
which each have a chemical potential $\mu_i$.
In that case, we sum over all species $i$:
$$\begin{aligned}
\boxed{
\dd{U} = T \dd{S} - P \dd{V} + \sum_{i}^{} \mu_i \dd{N_i}
}
\end{aligned}$$
## References
1. H. Gould, J. Tobochnik,
*Statistical and thermal physics*, 2nd edition,
Princeton.
|
