Unit 9: Equilibrium and Advanced Thermodynamics—Balance in Chemical Reactions

Section 5: Spontaneity and Gibbs Free Energy

As the previous sections have shown, chemical reactions that increase entropy tend to be spontaneous. Section 3 demonstrated that exothermic reactions increase entropy by allowing a wider distribution of energy quanta. Section 4 illustrated how reactions that create liquids and gases also tend to increase entropy.

In summary:

• Negative ΔH favors a spontaneous reaction.
• Positive ΔS favors a spontaneous reaction.
• Positive ΔH opposes a spontaneous reaction.
• Negative ΔS opposes a spontaneous reaction.

What now remains is to combine these two factors to create a complete picture—a way to definitively determine if a reaction will be spontaneous. When ΔH and ΔS both favor spontaneity, the reaction is always spontaneous. When they both oppose spontaneity, the reaction is never spontaneous. In other reactions, ΔH and ΔS "disagree," and one favors spontaneity while the other opposes it. In this situation, we must consider the actual values of ΔH and ΔS and perform a simple calculation.

In the 1870s, American mathematician Josiah Willard Gibbs (1839–1903) developed the concept of Gibbs free energy (given the variable G) and an equation that determines whether or not a reaction will happen spontaneously. The Gibbs free energy of a system depends on the enthalpy, entropy, and absolute temperature of the system (the derivation of this equation is beyond the scope of this text):

G = H - TS

The change in Gibbs free energy is calculated as follows:

ΔG = ΔH - TΔS

For a spontaneous reaction, ΔG will be negative; for a nonspontaneous reaction, ΔG will be positive. Notice that when both ΔH and ΔS favor a spontaneous reaction (ΔH is negative, ΔS is positive), the sign of ΔG must necessarily be negative, indicating a spontaneous reaction. And when both ΔH and ΔS oppose a spontaneous reaction (ΔH is positive, ΔS is negative), the sign of ΔG must necessarily be positive, indicating a nonspontaneous reaction.

When ΔH and ΔS conflict, plugging in actual values for ΔH, ΔS, and T will determine ΔG and its sign. Let's consider the explosive reaction of sodium metal and chlorine gas to produce table salt:

2Na(s) + Cl2(g) → 2NaCl(s)

ΔH = -822.24 kJ/mol     ΔS = -181.3 J/K•mol     T = 298 K

The ΔG for this reaction is -768.2 kJ/mol, so this reaction is spontaneous at this temperature. Although the entropy of the system decreases in this reaction, it is more than offset by the heat released to the surroundings. Here's another example:

H2O(s) → H2O(l)

ΔH = +6.01 kJ/mol     ΔS = +22 J/K•mol     T = 298 K

For this simple reaction—the melting of ice—the value of ΔG is -0.546 kJ/mol at 25°C (298 K), which is room temperature. This result comes as no surprise, as the melting point of ice is 0°C (273 K). At room temperature, the increase in the system's entropy outweighs the endothermic nature of the reaction, and the reaction happens spontaneously. Note that dropping the temperature below the melting point changes the sign of ΔG to positive; at -25°C (248 K), ΔG = +0.554 kJ/mol. Again, this conforms to experience; below its melting point, a solid will not melt. Indeed, the reverse reaction is spontaneous at these lower temperatures; liquid water turns to ice. The value of ΔG for the freezing reaction (at 248 K) is the same but with the sign reversed: -0.554 kJ/mol.

To summarize:

ΔSΔH ΔGSpontaneous?
+--Always
-++Never
--+ at high T, - at low TAt low T
++- at high T, + at low TAt high T

What is ΔG exactly at the melting point? The ΔG when the temperature is 273 K is 0 kJ/mol. Neither the forward nor the reverse reaction is spontaneous. At this particular temperature, the reaction is held in a kind of limbo between the forward and reverse. This state is called "equilibrium," which we will discuss in greater detail in later sections.

Glossary

Gibbs free energy

The value that indicates the thermodynamic stability of a system. All systems spontaneously move toward a minimum in Gibbs free energy.