Monday, November 12, 2007

K's and Phase Transitions

On Friday's lecture, we discussed how the equilibrium constant K for a given reaction might change as we vary either the temperature or the pressure. From the Gibbs-Helmholtz equation, (∂(∆G/T)/∂T)P = – ∆H/T2 and that ∆G° = – RT lnK, it is straightforward to show that (∂(lnK)/∂T)P = ∆H°/RT2.

To see the effect of pressure, we look at the derivative
(∂(∆G°)/∂P)T =∆V°. But, since the reference point ∆G° (1 bar) is independent of pressure, this derivative is zero and the dependence of K on P to be zero.

From these two results we conclude:

a) K is dependent only on temperature changes
b) this response is very sensitive (it is ln K that changes with T) and
c) the sign of the response is dictated by ∆H° (LeChatelier's principle in disguise)

Lastly, we began a formal discussion on phase diagrams/changes, our last chapter of the quarter, with the Ehrenfest classification of phase transitions. Some phase changes exhibit a discontinuity in the Sm vs T plot (as well as the Vm vs T plot). Since Sm is the first derivative of Gm, we call these first-order transitions. In other phase changes, the discontinuity does not arise until a plot of CP,m vs T; this is the second derivative of Gm so we call these second-order transitions. In general, an nth-order phase transition would have a discontinuity in the nth derivative of Gm, but only first and second-order are seen in practice. Though now outdated, the Ehrenfest classification is a useful way to think about the differences in the thermodynamics variables seen as systems undergo phase changes.

On Wednesday, phase diagrams and coexistence curves before hitting the Clausius-Clapeyron equation.

5 comments:

Ian said...

You integrated incorrectly on #2 Hwk 6

rod said...

Wow, you're quite right. Time to fix it...

Murffalo said...

When reading through the lecture notes while doing some of the homework problems, a few of us noticed there are some equations that don't correlate with what you have on the equation sheet and "Thermodynamics is Change" sheet. For example, in our notes we have dA = SdT - PdV but the equation sheets show it as dA = -SdT - PdV. Is there an error somewhere or something conceptual?

rod said...

I hope it doesn't disappoint anyone that I might occasionally make errors when excitedly thrashing equations all over the board.

And clearly I have completely invented this thermodynamics stuff from scratch. I mean, surely no *textbook* would contain such nonsense as the fundamental differentials...

:)

rod said...

fixed some errors on sample exam 2...