AOT - Gibbs Free Energy (Lesson)

Gibbs Free Energy

Up to this point, two different thermodynamic factors have been discussed which have direct bearing on the spontaneity of chemical reactions. From previous lessons, it has been shown that enthalpy affects spontaneity because exothermic reactions are typically favorable reactions; however, some endothermic reactions are known to be quite spontaneous as well. Likewise, it has been shown that chemical reactions that result in an increase in entropy are also favorable processes, but, as in the case of endothermic processes, reactions that result in a decrease in entropy are quite prevalent as well. So how can the spontaneity of a chemical or physical process be truly determined? The answer lies in yet another state function referred to as Gibbs free energy. This concept allows for a definite evaluation of spontaneity by calculating and examining a single value.

Gibbs Free Energy

The change in Gibbs free energy for a system is defined as and calculated using the following equation:

$LaTeX: \Delta G=\Delta H-T\Delta S$ $\Delta G=\Delta H-T\Delta S$

As the equation above implies, both the change in enthalpy as well as the change in entropy of the system is taken into consideration when determining the third parameter of Gibbs free energy. Whether a process is spontaneous or not can be evaluated using the following scenarios

Table
$LaTeX: \Delta G$ $\Delta G$	Spontaneous Reaction
$LaTeX: -\Delta G$ $-\Delta G$	Reaction is spontaneous in the forward direction
$LaTeX: +\Delta G$ $+\Delta G$	Reaction is non-spontaneous in the forward direction
$LaTeX: \Delta G=0$ $\Delta G=0$	Reaction is at equilibrium

Because Gibbs free energy is a state function in the same manner as enthalpy and entropy, it can be calculated using tabulated values as has been previously shown. The equation to calculate the standard Gibbs free energy is as follows:

$LaTeX: \Delta G^{^{\circ}}_{rxn}$ $\Delta G^{^{\circ}}_{rxn}$ = $LaTeX: \Sigma\Delta G_f^{^{\circ}}\left(products\right)-\Sigma\Delta G_f^{^{\circ}}\left(reactants\right)$ $\Sigma\Delta G_f^{^{\circ}}\left(products\right)-\Sigma\Delta G_f^{^{\circ}}\left(reactants\right)$

For example, the standard Gibbs free energy change for the reaction shown below can be calculated using the Gibbs free energy of formation values that are provided in the table.

2 Al (s) + Fe₂O₃ (s) $LaTeX: \longrightarrow$ $\longrightarrow$ Al₂O₃ (s) + 2 Fe (s)

Table
Substance	$LaTeX: \Delta G^{^{\circ}}_f$ $\Delta G^{^{\circ}}_f$ (kJ/mol)
Al (s)	0
Fe₂O₃ (s)	-742.2
Al₂O₃ (s)	-1582.3
Fe (s)	0

$LaTeX: \Delta G^{^{\circ}}_{rxn}$ $\Delta G^{^{\circ}}_{rxn}$ = $LaTeX: \lbrace\Delta G_f^{^{\circ}}\left\lbrack Al_2O_3(s\right)]+2\Delta G_f^{^{\circ}}\left\lbrack Fe\left(s\right)\right\rbrack\}-\left\lbrace2\Delta G_f^{^{\circ}}\left\lbrack Al\left(s\right)\right\rbrack+\Delta G_f^{^{\circ}}\left\lbrack Fe_2O_3\left(s\right)\right\rbrack\right\rbrace$ $\lbrace\Delta G_f^{^{\circ}}\left\lbrack Al_2O_3(s\right)]+2\Delta G_f^{^{\circ}}\left\lbrack Fe\left(s\right)\right\rbrack\}-\left\lbrace2\Delta G_f^{^{\circ}}\left\lbrack Al\left(s\right)\right\rbrack+\Delta G_f^{^{\circ}}\left\lbrack Fe_2O_3\left(s\right)\right\rbrack\right\rbrace$

$LaTeX: \Delta G^{^{\circ}}_{rxn}$ $\Delta G^{^{\circ}}_{rxn}$ = [1(-1582.3) + 2(0)] LaTeX: - $-$ [2(0) + 1(-742.2)]

$LaTeX: \Delta G^{^{\circ}}_{rxn}$ $\Delta G^{^{\circ}}_{rxn}$ = -840.1 kJ/mol

You Try It!

In the following self-assessment activity, calculate $LaTeX: \Delta G_{rxn}^{^{\circ}}$ $\Delta G_{rxn}^{^{\circ}}$ . Click on the plus sign to check your answer!

Influence of Temperature on Spontaneity

When investigating the Gibbs free energy relationship, it becomes readily apparent that it has a temperature dependence. This relationship can be thought of as a combination of two different terms: an enthalpy term ( $LaTeX: \Delta$ $\Delta$ H) and an entropy term (T $LaTeX: \Delta$ $\Delta$ S). Temperature affects the Gibbs free energy change primarily through the entropy term, and as temperature increases for a chemical reaction, the entropy term becomes more and more influential. The implications of this are that, for some reactions, spontaneity may change depending on the temperature at which the reaction proceeds. The table below reveals all possibilities in terms of combinations of the signs for $LaTeX: \Delta$ $\Delta$ H and $LaTeX: \Delta$ $\Delta$ S as well as how temperature factors into the spontaneity.

$LaTeX: \Delta G=\Delta H-T\Delta S$ $\Delta G=\Delta H-T\Delta S$

Table
$LaTeX: \Delta$ $\Delta$ H	$LaTeX: \Delta$ $\Delta$ S	$-$ T $LaTeX: \Delta$ $\Delta$ S	$LaTeX: \Delta$ $\Delta$ G	Reaction Characteristics
$-$	$+$	$-$	$-$	Spontaneous at all temperatures
$+$	$-$	$+$	$+$	Non-spontaneous at all temperatures
$-$	$-$	$+$	$+$ or $-$	Spontaneous at low T; non-spontaneous at high T
$+$	$+$	$-$	$+$ or $-$	Spontaneous at high T; non-spontaneous at low T

Using the Gibbs free energy equation, the above scenarios can be understood. When $LaTeX: \Delta$ $\Delta$ H is a negative value and $LaTeX: \Delta$ $\Delta$ S is a positive value there are no temperatures where $LaTeX: \Delta$ $\Delta$ G can be positive. Since temperature is expressed in the absolute Kelvin scale with no negative temperatures possible, a positive subtracted from a negative will always yield a negative value indicating a spontaneous reaction. The second scenario is essentially the opposite case. When $LaTeX: \Delta$ $\Delta$ H is positive and $LaTeX: \Delta$ $\Delta$ S is negative, only positive values of $LaTeX: \Delta$ $\Delta$ G are possible, and the reaction will always be non-spontaneous regardless of temperature.

The last two scenarios show the dependence of temperature. If both $LaTeX: \Delta$ $\Delta$ H and $LaTeX: \Delta$ $\Delta$ S are negative, then the entropy term needs to be minimized in order for a reaction to be spontaneous. Only at low temperatures (T) will this occur in order for $LaTeX: \Delta$ $\Delta$ G to remain negative. In the case where both $LaTeX: \Delta$ $\Delta$ H and $LaTeX: \Delta$ $\Delta$ S are positive, the entropy term is needed to overcome the unfavorable, positive $LaTeX: \Delta$ $\Delta$ H, and the reaction will only be spontaneous at high temperatures (T).

It is important to note, however, that while $LaTeX: \Delta$ $\Delta$ G is highly dependent upon temperature, $LaTeX: \Delta$ $\Delta$ H and $LaTeX: \Delta$ $\Delta$ S have only a negligible temperature dependence.

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