A. $\Delta G$ is net energy available to do useful work and is thus a measure of "free energy". Show mathematically that $\Delta G$ is a measure of free energy.

B. 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?

• We know that,

$\Delta S_{Total}=\Delta S_{sys}+\Delta S_{surr}$

• If the system is in thermal equilibrium with the surrounding, then the temperature of the surrounding is same as that of the system.

Also, increase in enthalpy of the surrounding is equal to decrease in the enthalpy of the system.

Therefore, entropy change of surroundings,

$\Delta S_{surr}=\frac{\Delta H_{surr}}{T}=\frac{-\Delta H_{sys}}{T}$

$\Delta S_{Total}=\Delta S_{sys}+\frac{-\Delta H_{sys}}{T}$

$T\Delta S_{Total}=T\Delta S_{sys}-\Delta H_{sys}$

For spontaneous change, $\Delta S_{Total}>0$

$\therefore T\Delta S_{Total}-\Delta H_{sys}>0$

$=>-(\Delta H_{sys}-T\Delta S_{sys})>0$

But, $\Delta H_{sys}-T\Delta S_{sys}=\Delta G_{sys}$

$\therefore -\Delta G_{sys}>0$

$=>\Delta G_{sys}=\Delta H_{sys}-T\Delta S_{sys}<0$

• $\Delta H_{sys}$ is the enthalpy change of a reaction, $T\Delta S_{sys}$ is the energy which is not available to do useful work.
• So $\Delta G_{sys}$ is the net energy available to do useful work and is thus a measure of the ‘free energy’. For this reason, it is also known as the free energy of the reaction.
• $\Delta G$ has the unit of energy because both $\Delta H$ and $T\Delta S_{sys}$ are terms of energy.

The unit of free energy change is same as the unit of energy i.e., joule $\left(J\right)$.

If a reaction has positive enthalpy change and positive entropy change then the reaction will be spontaneous at high temperatures.

$\Delta G_{sys}=\Delta H_{sys}-T\Delta S_{sys}$

For positive values of enthalpy change and entropy change, large values of temperature will make $\Delta G_{sys}<0$.