State Functions, Euler's Criterion

After welcoming in Week 4 today, we finished our cyclic process question: isothermal expansion followed by adiabatic compression, ending in isochoric cooling. Once again we saw, through calculation rather than assertion, that q and w are path functions -- and nonzero -- while U and H are state functions, making ∆U=∆H=0 for the cycle. In the process we found that this area was negative, giving positive work (done on system by surroundings).

Our second problem addressed how we might calculate ∆H for any case in which the heat capacity is temperature-dependent. As in the ideal gas case, we can simply integrate the expression dH=C_PdT; but unlike that example, the heat capacity is not constant. Still, the procedure is nearly identical. This is an important step towards generalizing our thermodynamic approach to systems beyond ideal gases.

Chapter 3 lays much of the mathematical foundation for the rest of the quarter. Hopefully you are beginning to see the importance of state functions and we will build heavily on this idea. But before today, I had simply asserted which properties were state functions and which were path functions. Now, with the introduction of Euler's Criterion for exactness, we have a mathematical litmus test: Iff the differential dZ is exact, then Z is a state function. Later, we will turn this principle on its head (in the form of Maxwell's relations) and generate a handful of remarkable thermodynamic relations that are far from obvious.

On Wednesday, backwards sixes will fly as we begin to build our Big Box of Mathemagics.