Van't Hoff Equation

[Caroline C 1G]

Could someone explain the steps involved in deriving lnK=(-deltaH/RT)+(deltaS/R) from deltaH-TdeltaS=-RTlnK?

[Lauren Seidl 1D]

You start with deltaH - TdeltaS = -RTlnK. Then divide both sides by -RT to get (-deltaH/RT) + (deltaS/R) = lnK , or flip it around to get lnK = (-deltaH/RT) + (deltaS/R).

[Mika Sonnleitner 1A]

From ΔH - TΔS = -RTln(K), you simply divide by -RT to get ln K = -(ΔH/RT) + (ΔS/R).
However, this form of the Van't Hoff equation is not that commonly used. To derive the more widespread version of the Van't Hoff equation, you add two Van't Hoff equations in order to calculate the equilibrium constant K at different temperatures (as long as ΔH is known).

1. Van't Hoff equations for two different temperatures
ln K1 = -(ΔH/RT1) + (ΔS/R)
ln K2 = -(ΔH/RT2) + (ΔS/R)

2. subtract ln K1 from ln K2
ln K2 - ln K1 = ln (K2/K1) = -(ΔH/RT2) + (ΔS/R) -[-(ΔH/RT1) + (ΔS/R)]
The two (ΔS/R) factors cancel out, and the resulting equation is:
ln (K2/K1) = -(ΔH/RT2) + (ΔH/RT1), which can also be written as:
ln (K2/K1) = (-ΔH/R)[(1/T2)-(1/T1)]

These derivations are under the assumptions that ΔS and ΔH are constant.