# I Fundamental thermodynamics relation

1. Jul 13, 2017

### davidge

I was looking at the fundamental equation $dU = Tds - Pdv + \sum_i \ \mu_i \ dN_i$ and I was thinking of how many different ways one has for deriving it.

I know I have to look through a book on Thermodynamics. I actually have done that some time ago and I will do that again. But the following seems to me not to violate any principle of physics. So I just like to know whether it is valid.

So one way of thinking about the fundamental equation above seems like

Suppose we increase the energy of a system by a tiny amount. This corresponds (in average) to an increase of the kinetic energy of a typical molecule of the system. As the system expands after we added the tiny amount of energy to it, that increment on the kinetic energy of the particle is lost by the work the particle has done. The last term in the equation could the regarded as due chemical interactions of molecule-to-molecule, and the plus sign indicates that the potential energy associated with the interaction becomes less negative as molecules moves a little bit apart from each other.

2. Jul 13, 2017

### Useful nucleus

I think your intuitive idea is OK. The variation in the internal energy could arise due to different types of works and/or heat addition. You indicated 3 of them (heat, mechanical work, and chemical work). But there many others and each work will extend dU by another term. For example elastic work (σdε), polarization work (EdD), ....

For chemical work, you can simply think of it as the energy needed to add one more particle to a system that has already N particle. Chemical reactions, as you mentioned, are complex and involve variations in many types of work and/or heat.

3. Jul 13, 2017

### davidge

Thanks, @Useful nucleus. So is it correct to add up all the other contributions into the fundamental equation? If so, why is it usually presented in the form I wrote above (i.e. the terms not included)?

4. Jul 14, 2017

### Useful nucleus

Yes, all kinds of possible work processes can be summed up into the equation you wrote above. Texts usually reserve the term fundamental equation to U=U(S,V,N,...) or S=S(U,V,N,...) instead of their first differentials like the one you wrote above. Authors claim that for the purposing of studying thermodynamics, let's just focus on the three historically most important work and heat terms. In other fields where thermodynamics is applied the other terms can be introduced as needed. For example when you study elasticity, one certainly meets σdε. A favorite of mine is Herbert Callen's introduction to thermodynamics where he warns the reader about these other terms and from time to time gives problems on magnetic work.

I also found the following IUPAC technical report to be very useful in discussing many of these work terms. See table 1:

https://pdfs.semanticscholar.org/4377/766430bebdf5b1b18b88f61493d9ce47d466.pdf