Chemistry | Entropy and spontaneity (Additional higher level)

Reactivity 1.4: Entropy and Spontaneity (Additional Higher Level)

Welcome to one of the most exciting (and challenging!) parts of thermodynamics. In previous sections (Reactivity 1.1–1.3), we learned about enthalpy (\(\Delta H\)), which tells us whether a reaction releases heat (exothermic) or absorbs heat (endothermic).

But here’s the big question: Does being exothermic guarantee a reaction will happen? Nope! Ice melting is spontaneous, but it’s endothermic (\(\Delta H > 0\)).

This HL chapter introduces the true driving force behind all chemical and physical change: Spontaneity. To determine if a process is spontaneous, we must consider two factors: enthalpy (\(\Delta H\)) and a new concept called Entropy (\(\Delta S\)).

1. The Concept of Entropy (\(S\))

Entropy (\(S\)) is often described as disorder, but a more accurate and chemical definition is the dispersal of energy or the number of ways energy can be distributed within a system. High entropy means energy is spread out and particles have high freedom of movement.

The Second Law of Thermodynamics

This is the fundamental rule governing the universe:

In any spontaneous process, the total entropy of the universe must increase.

$$ \Delta S_{\text{universe}} = \Delta S_{\text{system}} + \Delta S_{\text{surroundings}} > 0 $$

For a reaction to happen on its own (spontaneously), the overall "messiness" of the universe must increase.

Factors Affecting Entropy

Processes that lead to greater dispersal of energy (higher entropy) are generally favoured:

• Phase Changes: Moving from a restricted state to a more free state increases entropy.

Solid (low S) < Liquid (medium S) < Gas (high S)

• Increase in Number of Particles: If one reactant molecule breaks down into two or more product molecules, the total number of ways energy can be distributed increases.

Example: \(2 \text{O}_3 (g) \rightarrow 3 \text{O}_2 (g)\). \(\Delta S\) is positive.

• Temperature Increase: Higher temperature means particles move faster and have more kinetic energy, increasing the number of possible energy distributions.
• Mixing/Dissolving: Mixing two substances (like dissolving salt in water) increases the volume accessible to the particles, increasing the overall disorder.

Analogy: Think of your study desk. It takes effort (energy input) to keep it neat (low entropy). If you leave it alone, it naturally moves towards a state of disarray (high entropy). This spontaneous movement toward disorder reflects the natural tendencies of matter and energy.

2. Calculating Standard Entropy Change (\(\Delta S^{\circ}\))

Just like standard enthalpy change (\(\Delta H^{\circ}\)), we can calculate the standard entropy change of a reaction (\(\Delta S^{\circ}_{\text{reaction}}\)) using standard molar entropies (\(S^{\circ}\)) provided in data booklets.

Note: Unlike \(\Delta H^{\circ}_f\), the standard molar entropy \(S^{\circ}\) of an element in its standard state is not zero. All substances (except a perfect crystal at 0 K) have some inherent entropy because they possess energy.

Step-by-Step Calculation:

The calculation follows the same logic as Hess's Law for enthalpy:

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

Where \(n\) and \(m\) are the stoichiometric coefficients from the balanced equation.

Units Check! Standard molar entropy is usually given in \( \text{J} \text{K}^{-1} \text{mol}^{-1} \). This is crucial later when mixing units with enthalpy!

3. Gibbs Free Energy (\(\Delta G\)) – The Decider

Since we can’t easily measure the entropy change of the entire universe (\(\Delta S_{\text{universe}}\)), we use a powerful state function called Gibbs Free Energy (\(\Delta G\)), which focuses only on the system.

Gibbs Free Energy is the energy available in a system to do useful work. More importantly, it is the master indicator of spontaneity within the system itself.

Definition of Spontaneity via \(\Delta G\):

• If \(\Delta G < 0\) (negative): The process is spontaneous (or feasible). It will occur on its own under the specified conditions.
• If \(\Delta G > 0\) (positive): The process is non-spontaneous. It requires a continuous external input of energy to happen.
• If \(\Delta G = 0\): The system is at equilibrium.

Mnemonic: Grandma is Grumbling (Negative) = Going (Spontaneous)!

4. The Gibbs Equation

Gibbs linked the three key thermodynamic quantities—enthalpy, entropy, and temperature—into one essential equation for HL Chemistry:

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

Where:

• \(\Delta G\) = Change in Gibbs Free Energy (usually in \(\text{kJ} \text{mol}^{-1}\))
• \(\Delta H\) = Change in Enthalpy (usually in \(\text{kJ} \text{mol}^{-1}\))
• \(T\) = Absolute Temperature (in Kelvin, K)
• \(\Delta S\) = Change in Entropy (usually in \(\text{J} \text{K}^{-1} \text{mol}^{-1}\))

⚠️ Critical Calculation Warning: Units! ⚠️

Enthalpy (\(\Delta H\)) is almost always given in kilojoules (\(\text{kJ}\)), while Entropy (\(\Delta S\)) is almost always given in joules (\(\text{J}\)).

You must convert one term before calculating \(\Delta G\). It is safest to convert \(\Delta S\) from \(\text{J} \text{K}^{-1} \text{mol}^{-1}\) to \(\text{kJ} \text{K}^{-1} \text{mol}^{-1}\) by dividing by 1000.

5. Temperature Dependence of Spontaneity

The term \(-T \Delta S\) is often called the entropy contribution to free energy. Because of the temperature \(T\), the spontaneity of a reaction can often change depending on whether it is hot or cold.

The four possible sign combinations for \(\Delta H\) and \(\Delta S\) dictate whether the reaction is spontaneous at all temperatures, never spontaneous, or dependent on temperature:

Case 1: The Favourable Combination

• \(\Delta H\) is Negative (Exothermic)
• \(\Delta S\) is Positive (Increased disorder)
• Conclusion: \(\Delta G\) will always be negative. The reaction is Spontaneous at All Temperatures.

Example: Burning fuels. They release energy (\(\Delta H < 0\)) and create lots of gas (\(\Delta S > 0\)).

Case 2: The Unfavourable Combination

• \(\Delta H\) is Positive (Endothermic)
• \(\Delta S\) is Negative (Decreased disorder)
• Conclusion: \(\Delta G\) will always be positive. The reaction is Never Spontaneous (Requires continuous energy input).

Example: Separating water into hydrogen and oxygen gas. This requires energy and increases order.

Case 3: Spontaneity at Low Temperatures (Enthalpy Dominates)

• \(\Delta H\) is Negative (Exothermic)
• \(\Delta S\) is Negative (Decreased disorder)
• Spontaneity: Spontaneous only when \(T\) is low.

Explanation: At low T, the small positive penalty from \(-T \Delta S\) is outweighed by the large negative benefit from \(\Delta H\).

Case 4: Spontaneity at High Temperatures (Entropy Dominates)

• \(\Delta H\) is Positive (Endothermic)
• \(\Delta S\) is Positive (Increased disorder)
• Spontaneity: Spontaneous only when \(T\) is high.

Explanation: At high T, the large positive penalty from \(\Delta H\) is outweighed by the even larger negative term from \(-T \Delta S\) (since \(\Delta S\) is positive, \(-T \Delta S\) is a large negative number).

Example: Melting ice at room temperature. Melting is endothermic (\(\Delta H > 0\)) but creates disorder (\(\Delta S > 0\)). It only happens when T is high enough (above 273 K).

Determining the Crossover Temperature (\(T_{\text{equilibrium}}\))

For Cases 3 and 4, there is a specific temperature where the reaction transitions from spontaneous to non-spontaneous (or vice versa). At this crossover point, the system is at equilibrium, meaning \(\Delta G = 0\).

To find the temperature \(T\) at which spontaneity changes, set \(\Delta G = 0\) in the Gibbs equation:

$$ 0 = \Delta H - T \Delta S $$

Rearranging to solve for \(T\):

$$ T = \frac{\Delta H}{\Delta S} $$

Remember to use consistent units (e.g., both \(\Delta H\) and \(\Delta S\) in \(\text{kJ} \text{mol}^{-1}\))!