in thermodynamics there are many extremely useful relations obtained from derivatives of thermodynamic quantities. here are some exercises on the mathematical manipulations involved. a summary is available in the reference manual: https://courses.physics.illinois.edu/phys213/sp2018/reference/mathematics.html . for simplicity, here we only describe relations with fixed numbers of particles.
The entropy S(U,V) is a function of U and V in equilibrium. Using the chain rule, (dS(U,V) dS(U,V) = dU + dᏙ dU V dV Standard Gimu, V (ascu,V) ;) 0,7), Standard Vapor pre This is always true; it's a mathematical identity. The "Fundamental Relation of Thermodynamics" says that in equilibrium: Tds = dU + pdV where T is temperature, S is entropy, U is internal energy, p is pressure, and V is volume. This describes how, in equilibrium, S depends on changes in U and V. Here are other derivatives to evaluate, always assuming equilibrium so you can use the Fundamental Relation. The answers should be simple formulas expressed in terms of the variables S, T, U, V, and p. 1) What's (ds(U,V)) ? Submit Help 2) What's DU (S,V) ds $:V )y? Submit Help 3) what's (dues,V), Submit + 4) Let's look at Helmhotz free energy, F(T,V) = U – ST. What's (dF(T,V) )T Submit Help 5) what's (FTV)v? Submit 6) Let's look at Gibbs free energy: G(T,p) =U - TS+pV. What's ( dG(Tp) dT ? Submit 7) what's (dG (TP))T? Submit