Correction to today's lecture

I inadvertently omitted a part of an equation derived today in class.

Recall that we had for the pure-component chemical potential:

d(mu/T)/dT = -h / T^2

If we assume h is constant with temperature, we can integrate from Tm to T, where Tm is the pure melting temperature, to get:

mu(T)/T - mu(Tm)/Tm = h * (1/T - 1/Tm)

Or, rearranging

mu(T) = h * (1 - T/Tm) + mu(Tm) * (T/Tm)

In class, I left off the rightmost term. When we apply this equation to both the pure liquid and crystal and take the difference, we get:

mu_L(Tm) - mu_X(Tm) = (h_L - h_X) * (1 - T/Tm) + [mu_L(Tm) - mu_X(Tm)] * (T/Tm)

However, mu_L(Tm) = mu_X(Tm) by the conditions for phase equilibrium at Tm, so the rightmost term vanishes.

MSS