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Gibbs free energy and enthalpy are closely related.
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If one has experimental data such as gibbs free energy change versus temperature, the enthalpy change
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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.
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Divide both side of equation gives (G over T) equals (H over T) - S. Differentiate both side with respect to T.
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In this equation, the first part d (H over T) over dT is like this.
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H over T can be regard as the product of two function, H and 1 over T.
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Therefore, differentiation results in (1 over T) times (dH over dT) + H times (d (1 over T) over dT).
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The differentiation of the last part is -1 over T squared.
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Thus d (H over T) over dT becomes (1 over T) times (dH over dT) - H over (T squared).
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Now, the original equation can be written with it.
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It is (1 over T) times (dH over dT) - H over (T squared) - (dS over dT).
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dH over dT at constant pressure is Cp.
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Now the equation becomes Cp over T - H over (T squared) - (dS over dT).
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Now let's look at the second part (dS over dT). Here, let's start from the definition of entropy.
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dS is reversible heat divided by temperature T. At constant pressure,
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the heat is the same with enthalpy and it is Cp dT. Multiplying both sides by T results in this equation.
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TdS is Cp dT. Rearrange the equation. Then, dS over dT at constant pressure is Cp over T.
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Insert this result into the above equation. Then d (G over T) over dT becomes - H over (T squared).
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Let's rearrange this result. Divide both sides by -1 over (T squared).
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Then, the left side is d (G over T) over dT over (-1 over (T squared), and the right side is just H.
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Look at the bottom of the left side. It is -1 over (T squared) times dT.
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Then it is d (1 over T). So, the equation becomes like this, d (G over T) over d (1 over T) equals H.
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These two equations are called Gibbs-Helmholtz equation.
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They are just different forms but equivalent equations. These equations are applicable to
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a closed system of fixed composition at constant pressure.
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Graphical interpretation of Gibbs-Helmholtz equation is shown here.
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If we have G over T versus 1 over T graph, the slope is then the enthalpy.
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For chemical reactions also, if we have Gibbs free energy versus temperature data,
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then the enthalpy change can be calculated like this.