Helmholtz and Gibbs Energies

In thermodynamics, two important state functions help us understand the criteria for spontaneity in chemical processes: Helmholtz energy (A) and Gibbs energy (G). These functions allow us to determine whether a process is spontaneous under certain conditions.

Helmholtz Energy (A)

The Helmholtz energy is defined at constant volume ($V = const.$) and is given by:

\[A = U - TS\]

where U is the internal energy of the system, T is the temperature, and S is the entropy of the system.

 

Criterion for equilibrium

To derive a criterion for equilibrium in terms of the Helmholtz free energy, let's consider that both V = const. and T = const.:

\[dA = dU - TdS - SdT\]

\[dU = dQ - pdV = dQ_{rev} = TdS\]

\[ \implies dA = TdS - TdS = 0\]

This is a condition for equilibrium, meaning that if $dA = 0 $ the system is in equilibrium.

More generally, in thermodynamic systems, $ dA_{T,V} \leq 0 $ indicates that at $V = const.$ and $T = const.$, the Helmholtz free energy decreases for spontaneous processes.

 

Relationship between Helmholtz energy and work

From the First ($dU = dQ + dW$) and the Second ($dS \geq \frac{dQ}{T}$) laws of thermodynamics, we have the inequality:

\[ dU \leq TdS + dW \]

This inequality implies that the total change in internal energy, $ dU $, is bounded by the amount of heat added to the system ($ TdS $) and the work done ($dW$). Rearranging this expression allows us to analyze the work done by the system.

The maximum work ($dW_{\text{max}}$) obtainable when the temperature is held constant (indicating a maximum possible work under reversible conditions) is:

\[ dW \leq dU - TdS \]

This represents 

\[ dA = dU - TdS \]

Thus, we define:

\[ dW_{\text{max}} = dA \]

The expression indicates that $ dA $ represents the energy available for work, or the maximum work extractable from the system under constant temperature and reversible conditions.

 

Gibbs Energy (G)

The Gibbs energy is defined at constant pressure ($p = const.$) and is given by:

\[G = H - TS\]

where $H$ is the enthalpy of the system, $T$ is the temperature, and $S$ is the entropy of the system.

 

Criterion for equilibrium

Similarly to the Helmholtz energy, we can derive an equilibrium criterion based on the Gibbs energy as well. Consider that both $p = const. (dp = 0)$ and $T = const. (dT = 0)$:

\[dU = dQ - pdV = TdS - pdV\]
 

\[dH = dU + pdV + Vdp = TdS + Vdp \implies dH = TdS\]

Therefore, for the Gibbs energy we derive:

\[dG = dH - TdS - SdT \implies dG = dH - TdS \implies dG = TdS - TdS = 0\]
 

This is a condition for equilibrium, meaning that if $dG = 0 $ the system is in equilibrium. Again, in genera, $ dG_{T,p} \leq 0 $ indicates that at $p = const.$ and $T = const.$, the Gibbs free energy decreases for spontaneous processes.

At constant pressure and temperature, the direction of a spontaneous reaction is indicated by a decrease in Gibbs energy.

 

Relationship between Gibbs energy and work

Combining the First law of thermodynamics ($dU = dQ + dW$) with the expression for enthalpy ($dH = dU + d(pV)$) we get:

\[ dH = dQ + dW + d(pV) \]
 

Substituting further in the Gibbs energy equation and considering $T = const.$ and $p = const.$:

\[ dG = dH - TdS - SdT = dQ + dW + d(pV) - TdS - SdT \implies \]
\[ dG = TdS + pdV + dW_{add} + d(pV) - TdS - SdT\]

Canceling the positive and negative terms and setting $dT = 0$, and $dp = 0$ (the process occurs at $T = const.$ and $p = const.$) gives:

Therefore,  

\[ dG = dW_{\text{add}} \]

The equation indicates that Gibbs energy is the free energy in the system available for carrying non-expansion work.

 

Spontaneous Change and Reaction Types

There are two types of reactions that influence spontaneity:

Endothermic reactions: These require an input of heat, which is often overcompensated by an increase in the system's entropy to make the reaction spontaneous.

$ dH \geq 0 $ and $dG = dH - TdS$

This represents spontaneity when: $ dH < TdS $

Exothermic reactions: These generally release energy and are more likely to be spontaneous, provided the entropic term is not too negative.

$ dH < 0$ and $dG < 0 $

 

Applying Gibbs Energy to Chemical Reactions

In chemical reactions, especially those occurring at constant pressure (such as many biological and atmospheric processes), Gibbs free energy is crucial in determining spontaneity. If the change in Gibbs energy for the reaction is negative (i.e. the Gibbs energy of the products must be less than that of the reactants), the reaction proceeds spontaneously:

\[\Delta G = G_{products} - G_{reactants} < 0 \]

 

Standard molar Gibbs energies and Gibbs energies of formation

\[ \Delta_{\text{rxn}} G^{\circ} = \Delta_{\text{rxn}} H^{\circ} - T \Delta_{\text{rxn}} S^{\circ} \]
\[ \Rightarrow \Delta_{\text{rxn}} G^{\circ} = \sum \nu_i \Delta G^{\circ}_{\text{products}} - \sum \nu_i \Delta G^{\circ}_{\text{reactants}} \]
\[ \Rightarrow \Delta G^{\circ} = \sum_j \nu_j \Delta G^{\circ} (j) \]

 

Summary

The Helmholtz and Gibbs energies are essential thermodynamic functions in predicting the direction and extent of spontaneous processes. While Helmholtz energy is suited for constant volume processes, Gibbs energy is more commonly applied in constant pressure conditions, making it particularly valuable in chemistry and biology.