G vs. H

Gibbs free energy and enthalpy are closely related. If one has experimental data such as gibbs free energy change versus temperature, the enthalpy change can be obtained from delta G over T versus 1 over T curve. Let’s start from the definition of G. G is H - TS. Divide both side of equation gives (G over T) equals (H over T) - S. Differentiate both side with respect to T. In this equation, the first part d (H over T) over dT is like this. H over T can be regard as the product of two function, H and 1 over T. Therefore, differentiation results in (1 over T) times (dH over dT) + H times (d (1 over T) over dT).

The differentiation of the last part is -1 over T squared. Thus d (H over T) over dT becomes (1 over T) times (dH over dT) - H over (T squared). Now, the original equation can be written with it. It is (1 over T) times (dH over dT) - H over (T squared) - (dS over dT). dH over dT at constant pressure is Cp. Now the equation becomes Cp over T - H over (T squared) - (dS over dT). Now let’s look at the second part (dS over dT). Here, let’s start from the definition of entropy. dS is reversible heat divided by temperature T. At constant pressure, the heat is the same with enthalpy and it is Cp dT.

Multiplying both sides by T results in this equation. TdS is Cp dT. Rearrange the equation. Then, dS over dT at constant pressure is Cp over T. Insert this result into the above equation. Then d (G over T) over dT becomes - H over (T squared). Let’s rearrange this result. Divide both sides by -1 over (T squared). Then, the left side is d (G over T) over dT over (-1 over (T squared), and the right side is just H. Look at the bottom of the left side. It is -1 over (T squared) times dT. Then it is d (1 over T). So, the equation becomes like this, d (G over T) over d (1 over T) equals H.

These two equations are called Gibbs-Helmholtz equation. They are just different forms but equivalent equations. These equations are applicable to a closed system of fixed composition at constant pressure. Graphical interpretation of Gibbs-Helmholtz equation is shown here. If we have G over T versus 1 over T graph, the slope is then the enthalpy. For chemical reactions also, if we have Gibbs free energy versus temperature data, then the enthalpy change can be calculated like this.