Free energy is a concept that we have invented to incorporate the entropy change for the surroundings into a change in a state function of the system. Free energy is a way for us to think about the 2nd Law of thermodynamics only from the perspective of the system.
The second law states that for a process to be spontaneous it must increase the entropy of the universe.
\[\Delta S_{\rm univ} = \Delta S_{\rm sys} + \Delta S_{\rm surr} > 0 \]
If we invoke certain conditions, we can rewrite this equation. First we will look at constant pressure, which is a natural condition for chemistry. At constant pressure the heat is simply the change in entropy of the system. Since the change in the entropy of the surrounding is related to the heat, we can now write this in terms of the enthalpy change of the system. We can also look at conditions of constant temperature so the entropy change of the surroundings can simply be related to the enthalpy change of the system.
\[\Delta S_{\rm surr} = {-q \over T} = {- \Delta H_{\rm sys} \over T}\]
We can now rewrite our 2nd Law equation
\[\Delta S_{\rm total} = \Delta S_{\rm sys} + \Delta S_{\rm surr} = \Delta S_{\rm sys} - {\Delta H_{\rm sys} \over T} > 0\]
Now everything is in terms of the system (so we can drop the system subscript). This is, in a way, a small victory. We could be finished here, but our 2nd Law equation is still in entropy terms. We don't like to think in terms of entropy we would rather think in terms of energy. We can get to energy units if we simply multiply through by T.
\[T \Delta S - \Delta H > 0\]
We could again be done here. However, we now have an energy that increases for spontaneous processes. That seems counter intuitive. So we multiply the whole equation by a minus. This yields
\[\Delta H - T \Delta S < 0\]
Introducing a new State Function (variable) - Gibb's Free Energy We now have an energy term for the system which will decrease for a spontaneous process (at constant temperature and pressure). We can define this new state variable to reflect this energy. The new variable is the Gibb's Free Energy, \(G\):
\[ G = H - TS \]
This is the definition of the Gibb's Free Energy which will allow us to predict the spontaneity of a reaction using all system-based variables. This allows us to rewrite the second law based on only a system state function, \(\Delta G\). For a spontaneous process for an isolated system (at constant temperature and pressure) the change in free energy must be negative.
\[{\rm for\;spontaneous\;change: }\hskip20pt\Delta G < 0 \]
We can use these ideas for find the standard Gibb's Energy change for a chemical change. Free energy is a state function and therefore can be calculated via the free energies of formation of the reactants and products just like the enthalpy of reaction was:
\[\Delta G_{\rm rxn}^\circ =\sum{n\Delta G^\circ_{\rm f} (\rm products)} - \sum{n\Delta G^\circ_{\rm f} (\rm reactants)}\]
Most thermodynamic tables include \(\Delta H^\circ_{\rm f}\), \(\Delta G^\circ_{\rm f}\), and \(\Delta S^\circ\), but sometimes you might NOT have \(\Delta G^\circ_{\rm f}\) (like on an exam) and you should know how to calculate \(\Delta G\) from \(\Delta H\) and \(\Delta S\) using the familiar equation:
\[\Delta G = \Delta H - T\Delta S\]
The standard version looks like this:
\[\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ \]