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With the 2nd Law of Thermodynamics we can say $$[1]\space\space \Delta{S}_{universe} = \Delta{S}_{surroundings} + \Delta{S}_{system} > 0$$

Then using $$[2]\space\space \Delta{S}_{system} = \frac{q_{rev}}{T}$$

with (at constant pressure) $$[3]\space\space q = \Delta{H}$$

and (from the 1st Law of Thermodynamics) $$[4]\space\space \Delta{H}_{system} + \Delta{H}_{surroundings} = 0$$

we can get $$[4]\space\space \Delta{S}_{system} = \frac{\Delta{H}_{system}}{T}$$ $$[5]\space\space \Delta{S}_{surroundings} = \frac{\Delta{H}_{surroundings}}{T} = \frac{-\Delta{H}_{system}}{T}$$.

Substituting these back into [1] would give $$[6] \space\space \Delta{S}_{universe} = \frac{-\Delta{H}_{system}}{T} + \frac{\Delta{H}_{system}}{T} > 0$$

which simplifies to $$0 > 0 $$

which is not true.

I'm assuming I did something wrong with this, but cannot figure out what. Where is the issue?