# Why does Gibbs free energy have to be negative?

Sep 26, 2014

For a reaction to happen spontaneously, the total entropy of the system and surroundings must increase:

$\Delta {S}_{o v e r a l l} = \Delta {S}_{s u r} + \Delta {S}_{s y s} > 0$

The entropy of the system changes by $\frac{\Delta {H}_{s y s}}{T}$, and because $\Delta {H}_{s y s} = - \Delta {H}_{s u r}$, entropy change of the surroundings can be calculated from the equation

$\Delta {S}_{s u r} = - \frac{\Delta H}{T}$

Substituting this for $\Delta {S}_{s u r}$ gives

$\Delta {S}_{o v e r a l l} = \frac{- \Delta H}{T} + \Delta {S}_{s y s} > 0$

Multiplying through by $- T$ gives

$\Delta G = - T \Delta {S}_{o v e r a l l} = \Delta H - T \Delta {S}_{s y s} < 0$