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 S_m vs T plot (as well as the V_m vs T plot). Since S_m is the first derivative of G_m, we call these first-order transitions. In other phase changes, the discontinuity does not arise until a plot of C_P,m vs T; this is the second derivative of G_m so we call these second-order transitions. In general, an n^th-order phase transition would have a discontinuity in the n^th derivative of G_m, 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 Though now outdated, the Ehrenfest classification is a useful way to think about the differences in the thermodynamics variables seen as systems undergo phase changes.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.