Question

Deduce the Vant Hoff equation.

Answer 1

This equation gives that quantitative temperature dependence of equilibrium constant (K). The relation between standard free energy change (∆G°) and equilibrium constant is

∆G° = – RT In K ………………..(1)

We know that, ∆G° = ∆H° – T∆S° …………(2)

Substituting (1) in equation (2)

– RT In K = ∆H° – T∆S°

Rearranging, In K = `(Delta "H"^0)/"RT" + (Delta "S"^0)/"RT"`    ...(3)

Differentiating equation (3) with respect to temperature,

`(("d"("ln"  "K"))/"dT") = (Delta "H"^0)/"RT"^2`    ....(4)

Equation (4) is known as differential form of van,t Hoff equation.

On integrating the equation (4), between T₁ and T₂ with their respective equilbrium consatnts K₁ and K₂.

`int_("K"_1)^("K"_2) "d" ("ln K") = (Delta "H"^0)/"R" int_("T"_1)^("T"_2) "dT"/"T"^2`

`["ln K"]_("K"_1)^("K"_2) = (Delta "H"^0)/"R" [- 1/"T"]_("T"_1)^("T"_2)`

`"ln K"_2 - "ln K"_1 = (Delta "H"^0)/"R" - [1/"T"_2 + 1/"T"_1]`

`"ln" "K"_2/"K"_1 = (Delta "H"^0)/"R" [("T"_2 - "T"_1)/("T"_2"T"_1)]`

`log  "K"_2/"K"_1 = (Delta "H"^0)/(2.303 "R")[("T"_2 - "T"_1)/("T"_2"T"_1)]`   ....(5)

Equation 5 is known as integrated form of Van’t Hoff equation.