Spontaneity

The second law tells us that a spontaneous reaction increases the entropy of the universe; that is ΔS_univ > 0.  To determine this we need both ΔS_sys and ΔS_surr.  However, we are usually only concerned with what happens in a particular system, and the calculation of ΔS_surr can be difficult.  For that reason we have invented yet another thermodynamic function to allow us to tell whether a reaction is spontaneous if we consider only the system itself.

Lets start with the mathematical expression for the second law for a spontaneous process and rearrange it a bit:

ΔS_univ = ΔS_sys + ΔS_surr > 0

Substituting -ΔH_sys / T for ΔS_surr we have

ΔS_univ = ΔS_sys - ΔH_sys / T

Now we multiply each side of the equation by T, which gives

TΔS_univ = - ΔH_sys + TΔS_sys  

Now we have a criterion for a spontaneous process that is expressed only in terms of the properties of the system and we can ignore the surroundings.  We make one more change to the above equation by multiplying it by -1.

-TΔS_univ =  ΔH_sys - T ΔS_sys

This equation says that for a process carried out at a temperature T, if (ΔH_sys - T ΔS_sys) is less than zero, the process must be spontaneous.  The quantity ΔH_sys - T ΔS_sys is called Gibbs free energy(G), or simply free energy:

G = H - TS

All quantities in this equation pertain to the system, and T is the temperature of the system.  You can see that G has units of energy.  Like H and S, G is a state function.  The change in free energy of a system for a constant-temperature process is

ΔG = ΔH - TΔS

 This equation is known as the Gibbs-Helmholtz equation.

The following rules are useful in deciding if a reaction will be spontaneous.

• When ΔG^o is more negative than about -10 kJ, the reaction is spontaneous.
• When ΔG^o is more positive than about 10 kJ, the reaction is nonspontaneous.
• When ΔG^o has a value between about -10 kJ and +10 kJ the reaction is at equilibrium.