Step 3 An exact differential can only be written in one unique way, so if we have two different expressions for the same function U = U(S, V) then we can identify the corre- sponding terms with each other. So, for what we have above, it must be that au T = and - p = (15) as av V s Step 4 Write down the generic expression for the mixed partial derivatives being equal to each other: a au a au (16) av as V/ s as av / s/ v Step 5 Use the results from Step 3 aT ap 17) = av as S VQuestion 4 (8 pts) Maxwell relations (MR) arise in thermodynamics and relate partial derivatives to each other. So-called state functions which are given uppercase letters in thermodynamics, like U, H, A, G and so forth, are functions and so have exact differentials du, dH, etc. Suppose f(x, y) is a function of two variables; as it is a function then its differential must be exact. Then df is an exact differential and (12) x x We just follow these easy steps. Step 1 Write the total differential of the thermodynamic state function. For internal energy, U, it would be du = Tds - pdv (13) L Step 2 Write the total differential, generically, as a function of the same two variables (look at the "d" terms to decide which two variables) U = U(S, V): au au du = ds + dv as av (14) V S (This is just a generic form for a function of two variables- look at the units to assure yourself it's correct.)