Gibbs Free Energy and Spontaneity
What Makes a Process Spontaneous?
Definition
Spontaneity
A chemical reaction or physical process is spontaneous if it occurs without external intervention under a given set of conditions.
To determine spontaneity, we use the Gibbs free energy equation:
$$\Delta G = \Delta H - T\Delta S$$ where:
- $ \Delta G $: Gibbs free energy change ($kJ \, mol^{-1}$)
- $ \Delta H $: Enthalpy change ($kJ \, mol^{-1}$)
- $ \Delta S $: Entropy change ($kJ \, K^{-1} \, mol^{-1}$)
- $ T $: Temperature (K)
Spontaneity and $ \Delta G $:
- If $ \Delta G< 0 $: The process is spontaneous.
- If $ \Delta G = 0 $: The system is at equilibrium.
- If $ \Delta G > 0 $: The process is non-spontaneous under the given conditions.
Example
- Combustion of methane ($ \text{CH}_4 + 2\text{O}_2 \to \text{CO}_2 + 2\text{H}_2\text{O} $) is highly exothermic ($ \Delta H< 0 $) and increases disorder ($ \Delta S >0 $).
- Hence, $ \Delta G $ is negative, making the reaction spontaneous at all temperatures.
Common Mistake
- Many students mistakenly assume that all exothermic reactions ($ \Delta H< 0 $) are spontaneous.
- While this is often true, you must also consider $ \Delta S $ and $ T $.
- For example, freezing water is exothermic but non-spontaneous above 0°C because the entropy decreases ($ \Delta S < 0 $).
Temperature Dependence of Spontaneity
The interplay between $ \Delta H $, $ \Delta S $, and $ T $ determines whether a reaction is spontaneous. Let’s break this down into four scenarios:
| $\Delta H$ | $\Delta S$ | $\Delta G$ | Spontaneity |
|---|---|---|---|
| < 0 | > 0 | Always negative | Always spontanous |
| > 0 | < 0 | Always positive | Always non-spontaneous |
| < 0 | < 0 | Negative at low $T$, positive at high $T$ | Spontaneous at low $T$ |
| > 0 | > 0 | Negative at high $T$, positive at low $T$ | Spontaneous at high $T$ |
Determining the Temperature at Which $ \Delta G = 0 $
- At the threshold temperature, $ \Delta G = 0 $, and the system is at equilibrium.
- Rearranging the Gibbs free energy equation gives:
$$T = \frac{\Delta H}{\Delta S}$$
This temperature marks the transition between spontaneity and non-spontaneity.
Example
- Consider the decomposition of calcium carbonate ($ \text{CaCO}_3 \to \text{CaO} + \text{CO}_2 $) with $ \Delta H = 178 \, \text{kJ mol}^{-1} $ and $ \Delta S = 161 \, \text{J K}^{-1} \text{mol}^{-1} $.
- Convert $ \Delta S $ to kJ: $ 161 \, \text{J K}^{-1} = 0.161 \, \text{kJ K}^{-1} $.
- Threshold temperature:$T = \frac{\Delta H}{\Delta S} = \frac{178}{0.161} = 1106 \, \text{K}$.
- The reaction becomes spontaneous above 1106 K.
Tip
Always convert entropy values from J to kJ when using the Gibbs free energy equation to maintain consistent units.
