Entropy and Gibbs Free Energy (HSC SSCE Chemistry): Revision Notes

Temperature and Reaction Spontaneity

Understanding how temperature affects whether a chemical reaction will occur spontaneously is fundamental to predicting reaction behaviour. While some reactions proceed immediately when reactants are mixed, others require specific temperature conditions to occur. The relationship between temperature and spontaneity is governed by Gibbs free energy, which combines both energy and entropy considerations.

Types of reactions based on spontaneity

Chemical reactions can be classified into four distinct types based on their spontaneity behaviour and temperature requirements. This classification helps us understand not just whether a reaction will occur, but also what conditions are necessary for it to proceed.

Type 1 reactions occur immediately when reactants are combined at room temperature. These reactions proceed spontaneously without any external energy input beyond initial mixing. Classic examples include the displacement reaction between copper metal and silver nitrate solution, or the vigorous reaction of magnesium with hydrochloric acid:

$\text{Mg(s)} + 2\text{HCl(aq)} \rightarrow \text{MgCl}_2\text{(aq)} + \text{H}_2\text{(g)}$

Type 2 reactions do not proceed at room temperature but occur once given an initial energy "prod" such as ignition with a spark or brief heating. These reactions are characteristically exothermic, meaning they release energy. Once initiated, they generate sufficient heat to sustain the reaction without further external energy input. Common examples include the combustion of fuels like methane (natural gas) or octane (petrol):

$\text{CH}_4\text{(g)} + 2\text{O}_2\text{(g)} \rightarrow \text{CO}_2\text{(g)} + 2\text{H}_2\text{O(g)}$

Although Type 2 reactions are very slow at room temperature, they proceed rapidly at elevated temperatures. Because they can maintain their own temperature once started, they are considered spontaneous reactions.

Type 3 reactions require continuous heating to maintain an elevated temperature throughout the reaction process. These are endothermic reactions that absorb energy from their surroundings. If heating stops, the reaction mixture cools down and the reaction ceases. An example is the thermal decomposition of limestone (calcium carbonate) at approximately $1500~\text{K}$:

$\text{CaCO}_3\text{(s)} \rightarrow \text{CaO(s)} + \text{CO}_2\text{(g)}$

Type 4 reactions do not occur even when maintained at very high temperatures. For instance, the decomposition of water into hydrogen and oxygen does not proceed even at $2000~\text{K}$:

$2\text{H}_2\text{O(g)} \rightarrow 2\text{H}_2\text{(g)} + \text{O}_2\text{(g)}$

These reactions are clearly non-spontaneous under normal conditions.

Refined definition of spontaneous reactions

Based on the classification above, we need a precise definition that distinguishes truly spontaneous reactions from those requiring ongoing energy input. This distinction is critical for understanding thermodynamics and predicting reaction behaviour.

This definition means:

• Type 1 and Type 2 reactions are spontaneous
• Type 3 and Type 4 reactions are non-spontaneous

The key distinction is whether the reaction can sustain itself. Type 2 reactions, though requiring initiation, are exothermic and self-sustaining, making them spontaneous. Type 3 reactions, despite proceeding at high temperatures, require continuous heating and are therefore non-spontaneous.

Standard vs non-standard conditions

The understanding of when and how to apply standard Gibbs free energy values is crucial for accurate predictions. The standard Gibbs free energy change ($\Delta G^\circ$) applies specifically to standard conditions, which means:

• All gases are present at a pressure of $100.0~\text{kPa}$
• All solutes are present at a concentration of $1.00~\text{mol L}^{-1}$
• All species (both reactants and products) are in their standard states

However, in typical laboratory conditions, we usually start with only reactants present, not a mixture of products and reactants. The Gibbs free energy change under these actual conditions is denoted as $\Delta G$ (without the superscript circle).

Interpreting $\Delta G^\circ$ values at $298~\text{K}$

For standard conditions at $298~\text{K}$ (room temperature), the following guidelines help predict reaction behaviour:

• If $\Delta G^\circ < -10~\text{kJ mol}^{-1}$: The reaction is spontaneous and proceeds to virtual completion in the forward direction (until one reactant is consumed)
• If $\Delta G^\circ > +10~\text{kJ mol}^{-1}$: The reaction does not proceed in the forward direction but is spontaneous in the reverse direction
• If $-10~\text{kJ mol}^{-1} < \Delta G^\circ < +10~\text{kJ mol}^{-1}$: The reaction is an equilibrium reaction that proceeds to some extent in both forward and reverse directions but not to completion

Calculating $\Delta G^\circ$ at different temperatures

While standard thermodynamic data tables typically provide values at $298~\text{K}$, we often need to determine spontaneity at other temperatures. Understanding this calculation is essential for predicting industrial reaction conditions and optimizing chemical processes.

The key insight is that both $\Delta H^\circ$ and $\Delta S^\circ$ change only slightly with temperature, so we can treat them as temperature-independent for practical calculations.

Procedure for temperature-dependent calculations

To calculate $\Delta G^\circ$ at a temperature other than $298~\text{K}$:

Step 1: Calculate $\Delta H^\circ$ at $298~\text{K}$ using standard enthalpy of formation data:

$\Delta H^\circ = \sum \Delta H_f^\circ(\text{products}) - \sum \Delta H_f^\circ(\text{reactants})$

Step 2: Calculate $\Delta S^\circ$ at $298~\text{K}$ using standard entropy data:

$\Delta S^\circ = \sum S^\circ(\text{products}) - \sum S^\circ(\text{reactants})$

Step 3: Assume that $\Delta H^\circ$ and $\Delta S^\circ$ are independent of temperature (i.e., their values at $298~\text{K}$ apply at other temperatures)

Step 4: Calculate $\Delta G^\circ$ at the desired temperature using:

$\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ$

where $T$ is the absolute temperature in kelvin.

Worked example: methane-steam reaction

This example demonstrates why the industrial production of hydrogen from methane requires high temperatures—the reaction is thermodynamically unfavourable at moderate temperatures but becomes spontaneous at sufficiently high temperatures.

Two important generalisations

Examining the Gibbs free energy equation $\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ$ reveals two valuable generalisations about how temperature affects spontaneity. These generalisations allow us to make quick predictions without detailed calculations.

Generalisation 1: Effect of entropy sign on temperature dependence

If $\Delta S^\circ$ is positive: As temperature increases, the term $T\Delta S^\circ$ becomes more positive, making $\Delta G^\circ$ more negative (since we subtract $T\Delta S^\circ$). This means reactions with positive entropy changes become more likely to be spontaneous at higher temperatures.

If $\Delta S^\circ$ is negative: As temperature increases, $T\Delta S^\circ$ becomes more negative, but we subtract it (making $\Delta G^\circ$ less negative or more positive). This means reactions with negative entropy changes become less likely to be spontaneous at higher temperatures.

Generalisation 2: Magnitude of entropy change

If $\Delta S^\circ$ is small compared to $\Delta H^\circ$: The temperature term $T\Delta S^\circ$ has little effect, so $\Delta G^\circ$ changes only slightly with temperature. Spontaneity is largely determined by $\Delta H^\circ$ and remains relatively constant across a temperature range.

If $\Delta S^\circ$ is large: The temperature term $T\Delta S^\circ$ dominates, causing $\Delta G^\circ$ to change significantly with temperature. Spontaneity is strongly temperature-dependent, and reactions may switch from non-spontaneous to spontaneous (or vice versa) as temperature changes.

These generalisations explain why:

• Decomposition reactions (which increase the number of gas molecules) often require high temperatures
• Combustion reactions (already exothermic) remain spontaneous across wide temperature ranges
• Some reactions that won't proceed at room temperature become spontaneous when heated

Remember!