`Delta G` is energy availabe to do useful work and is thus a meaure of "Free energy". Show mathemactically that `Delta G` is a measure of free energy. Find the unit of `Delta G`. IF a reaction has positive enthalpy change and positive entropy change, under what condition will the reaction be spontaneous ?

Explanation

Gibbs free energy is that thermodynamic quantity of a system, the decreases in whose value during a process is equal to the maximum possible useful work that can be obtained from the system.
Mathematically, this results may be derived as follows
The relationship between heat absorbed by a system q, the change in its internal energy, `Delta U` and the work done by the system is given by the equation of the first law of thermodynamics, therefore
`q = Delta U + Q_("expansion") + W_("non-expansion")` ...(i)
Under constant pressure condition, the expansions work is given by `p Delta V`.
`:. q = Delta U+pDelta V + W_("non-expansion") " " (because Delta U+ p Delta V =Delta H)`
`= Delta H + W_("non-expansion")` ...(ii)
For a reversible change taking place at constant temperature,
`Delta S = (q_("rev"))/(T)` or `q_("rev") = T Delta S` ...(iii)
Substituting the value of q from Eq. (iii) in Eq. (ii) we get
`T Delta S = Delta H + W_("non-expansion")`
or `Delta H - T Delta S = -W_("non-expansion")` ...(iv)
For a change taking place under conditions of constant temperature and pressure,
`Delta G = Delta H - T Delta S`
Substituting this value in equation (iv), we get
`Delta G = -W_("non-expansion")` ...(v)
Thus, free energy change can be taken as a measure of work other than the work of expansion. For most changes , the work of expansion can not be converted to other useful work, whereas the non-expansion work is convertible to useful work.
Rearranging equation (v), it may write as
`-Delta G = W_("non-expansion") = W_("useful")`
As `-Delta G = W_("useful")` therefore, `Delta G` has the same units as those of work i.e., joule
`Delta G = Delta H - T Delta S`.
If `Delta H` = positive and `Delta S` = positive, then
`Delta G` will be negative i.e., process will be spontaneous only when `T Delta S gt Delta H` in magnitude, which will be so when temperature is high.