10.4Phase Transitions

The Clausius-Clapeyron equation P = Ae^(−ΔHvap/RT) is rearranged into ‘the linear equation’ ln P = −(ΔHvap/R)(1/T) + ln A by taking a natural log of both sides — a student who does not know what ln undoes cannot follow that step or evaluate ln(P2/P1) in the worked examples.

Deriving the two-point form ln(P2/P1) = (ΔHvap/R)(1/T1 − 1/T2) collapses ln P2 − ln P1 into ln(P2/P1) — the quotient rule for logarithms is the one step that turns two separate ln-equations into the single usable formula every worked example plugs into.

‘Estimating Temperature (or Vapor Pressure)’ solves the Clausius-Clapeyron equation for T2 by isolating 1/T2 on one side and then raising the result to the −1 power to undo the reciprocal — the same isolate-then-invert moves used to solve any exponential/logarithmic equation for a variable trapped inside both a log and a reciprocal.