physical chemistry - Chemical Equilibrium - Why does changing the temperature shift the equilibrium?

I know that raising the temperature in a reversible chemical reaction causes the equilibrium to shift to the endothermic side.

I know that $\Delta G = \Delta H − T\Delta S $ but I don't know how to put them together to show why the equilibrium would shift.

To summarise, I already know what happens when, and I'm interested in why.

Answer

I think the best way to think of equilibria intuitively is in terms of rates of reaction. At equilibrium, the forward and the reverse reactions are happening at the same rate.

If you increase the temperature, what happens to the rates of the forward and reverse reactions?

Using the Arrhenius equation

$$k = Ae^{-E_\mathrm{a}/(RT)}$$

you can see that as temperature increases, the rate will increase. The amount it increases depends on the activation energy ($E_\mathrm{a}$).

The activation energy on the endothermic side of the reaction will always be larger than the exothermic side. This is because the transition state is always higher in energy than either of the reactants/products, and the reactants for the endothermic reaction are by definition lower in energy than the products! This means that the endothermic reaction will be sped up more than the exothermic reaction. This means the reaction is now out of equilibrium and the endothermic reaction will happen until the reaction is back in equilibrium again.