## Van't Hoff Equation

Alyssa Wilson 2A
Posts: 65
Joined: Fri Sep 28, 2018 12:18 am

### Van't Hoff Equation

I'm a little confused on why the Van't Hoff equation shows the temperature dependence of K. I understand that it relates the difference in standard Gibbs free energy to K, because it's easy to measure, but why does it show that the temperature depends on K?

David S
Posts: 54
Joined: Fri Sep 28, 2018 12:15 am

### Re: Van't Hoff Equation

The Van't Hoff equation is meant to show the Temperature dependance of K, and while it is derived from an expression that relates G to K, in its ultimate for the equation excludes G.

Derivation:

1) Given $\Delta G = -RTln(K)$ and $\Delta G = \Delta H - T\Delta S$:

$\Delta H - T\Delta S = -RTln(K)$

2) Divide both sides by -RT to isolate ln(K):

$ln(K) = \frac{-\Delta H}{RT} +\frac{\Delta S}{R}$

3) Use the above equation, plug in K1 at T1 to make the equation for initial K and plug in K2 at T2 to make the equation for Final K. Subtract the final K equation by initial K equation:

$ln(K_{2}) - ln(K_{1}) = \frac{-\Delta H}{RT_{2}} + \frac{\Delta H}{RT_{1}} +\frac{\Delta S}{R} -\frac{\Delta S}{R}$

4) Simplify the equation by using ln(a)-ln(b) = ln(a/b) and cancelling out the entropy terms:

$ln(\frac{K_{2}}{K_{1}}) = \frac{-\Delta H}{RT_{2}} +\frac{\Delta H}{RT_{1}}$

5) Now you factor out -delta H/R to get the final form that we use:

$ln(\frac{K_{2}}{K_{1}}) = \frac{-\Delta H}{R}(\frac{1}{T_{2}}-\frac{1}{T_{1}})$

Note how the process of deriving this equation eventually allows us to compute a final K @ Tfinal from an initial K @ Tini knowing only the enthalpy of reaction.