### Gibbs Free Energy

As we discussed in the previous section, reaction favorability can be ambiguous when entropy and enthalpy are in conflict. In particular, the increase in disorder (ΔS°) for a system and the surroundings can be of different signs. For instance, boiling water converts a liquid into a gas, generating a very positive ΔS°system value for the reaction. However, at room temperature, energy must be extracted from the surroundings, giving a negative ΔS°surroundings value. The total entropy change for the universe can be found by adding the two values: if the result is negative, the reaction is not favorable, and vice versa. The entropy change for the surroundings can be found by dividing the opposite of ΔH° by the temperature in Kelvins: ΔS°surroundings = -ΔH° / T ; the lower the temperature, the greater the increase in disorder. The negative value represents that energy is being transferred out of the system. We could use the equation

ΔS°universe = ΔS°system + -ΔH° / T

to find the overall favorability for the reaction, but there is an easier way. In the 1800's, J. Willard Gibbs manipulated the above equation to form a new equation, using a new value, G, to represent the free energy of the system. The proof of this manipulation is too long to include here, but may be found in other texts and on the Internet. His revised equation became:

ΔG°system = ΔH°system - TΔS°system

where T is the temperature in Kelvin, and ΔG is the change in the Gibbs free energy. If ΔG is positive, then the reaction is product-favored (not favorable) and will not occur at the given temperature. If ΔG is negative, then the reaction will occur at the specified temperature. This is one of the most important equations in chemistry, since it lets us predict the favorability of any reaction at any temperature, knowing only the enthalpy and entropy changes.

Since ΔG must be negative, we can make some generalizations based on the signs of ΔH and ΔS:

• ΔH negative, ΔS positive: Since ΔH is already negative and the "- TΔS" group will be negative as well (T is never negative), ΔG will be negative and the reaction will be product-favored at any temperature.
• ΔH negative, ΔS negative: ΔH is negative but the "- TΔS" group is positive, so the reaction favorability depends on the temperature. If T is small, then ΔG is negative (the negative ΔH overpowers the positive - TΔS) and the reaction is favorable at low temperatures. At high temperatures, the positive "- TΔS" group is largely positive, and ΔG is positive as well, leading an unfavorable reaction.
• ΔH positive, ΔS positive: This is the opposite scenario as above: at low temperatures, the negative "- TΔS" group is too small to overcome the positive ΔH value, so ΔG is positive and the reaction is reactant-favored. At high temperatures, however, the "- TΔS" group is very negative and overpowers the ΔH value, leading to a negative ΔG and a product-favored reaction.
• ΔH positive, ΔS negative: In this configuration, ΔH is positive, and the "- TΔS" group is always positive as well, so the value for ΔG is always positive. Therefore, the reaction is unfavorable whatever the temperature.

This table finally answers the question we asked back in the "Chemical Reactions" chapter: why do some reactions always or never occur, while others are dependent on temperature? The entropy and enthalpy changes are responsible for the varying behaviors of chemical reactions. Using the Gibbs equation above, the outcome of a reaction can be determined.

However, at standard conditions, the process of predicting reaction favorability is even easier. Just as with the H and S values for enthalpy and entropy, the ΔG° value can be computed with the following equation, given ΔG°formation values for all species:

ΔG°system = Σ ΔG°formation(products) - Σ ΔG°formation(reactants)

Using the values for ΔG°formation in a reference text, you can predict reaction favorability by simply doing some quick addition and subtraction! However, this method only works at the standard temperature of 25 °C. You can find values for ΔH°formation, S°, and ΔG°formation for many species in our extensive "Reference" section in the Thermodynamics page.

As a final note on the Gibbs free energy equation, for reactions in which ΔH°reaction and ΔS°system have opposite signs, there is a temperature for which ΔH°system and the "- TΔS°reaction" group have the same value. In this case, the value for ΔG° is zero: the reaction is neither product- nor reactant-favored, so it is in equilibrium and does not go at all. You can find the equivalence temperature by setting ΔG° to zero and solving for T. This will be only an approximate value, because the thermodynamic values all assume 25 °C; if you change the temperature, these values are no longer completely accurate.

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