Entropy is a measure of disorder

Entropy (S) is a thermodynamic quantity that measures the degree of disorder or randomness in a system. It quantifies the number of ways particles can be arranged and how energy is distributed among those particles.

Key points about entropy:

• A high entropy value indicates a high level of disorder.
• Entropy is always positive and increases as disorder increases.
• Systems naturally tend towards states of higher entropy.

Factors affecting entropy

Three factors influence the entropy of a substance:

1. Physical state

The physical state of a substance significantly impacts its entropy. In general, entropy increases as a substance transitions from solid to liquid to gas.

• Solids have the lowest entropy due to their highly ordered structure, with particles confined to fixed positions.
• Liquids have higher entropy than solids, as their particles have more freedom of movement.
• Gases have the highest entropy, with particles moving randomly and occupying a larger volume.

Examples:

• The combustion of magnesium ribbon in air: 2Mg_(s)_ + O_2(g)_ ➔ 2MgO_(s)_ This reaction involves a gas reactant (O_2_) forming a solid product (MgO), resulting in a decrease in entropy.

• The reaction between ethanoic acid and ammonium carbonate: 2CH_3_COOH_(aq)_ + (NH_4_)2_CO_3(s) ➔ 2CH_3_COONH_4(aq)_ + H_2_O_(l)_ + CO_2(g)_

This reaction produces a gas (CO_2_), leading to an increase in entropy.

2. Number of particles

Entropy increases with the number of particles in a system, as a larger number of particles leads to more possible arrangements and energy distributions.

Example:

The decomposition of nitrogen tetroxide: N₂O_4(g) ➔ 2NO_2(g)

This reaction doubles the number of gas molecules, resulting in an entropy increase.

3. Dissolution

Dissolving a substance increases its entropy because the dissolved particles gain more freedom of movement compared to their solid state.

Example:

Dissolving ammonium nitrate in water: NH_4_NO_3(s)_ ➔ NH_4_^+^(aq)** + NO_3_^-^(aq)**

Entropy and reaction feasibility

Particles naturally tend towards more disordered states, as increased entropy provides greater energetic stability. This drive for disorder can make certain reactions feasible (able to proceed spontaneously) even when the enthalpy change is endothermic.

For example, the reaction between sodium hydrogencarbonate and hydrochloric acid is endothermic but still feasible:

NaHCO_3(s) + HCl_(aq) ➔ NaCl_(aq) + CO_2(g) + H₂O_(l)

The reaction produces gaseous carbon dioxide and liquid water, which have higher entropy than the solid reactant. This entropy increase overcomes the endothermic enthalpy change, allowing the reaction to occur spontaneously at room temperature.

This example demonstrates that enthalpy changes alone do not determine whether a reaction will occur spontaneously; reaction feasibility depends on the balance between enthalpy and entropy changes.

Calculating entropy changes for reactions

During a reaction, the entropy change of the system (ΔS_system) can be calculated using the formula:

ΔS_system = S_products $-$ S_reactants

Where:

• S_products is the sum of the standard entropies of all products.
• S_reactants is the sum of the standard entropies of all reactants.

Standard entropy (S^⦵) is the entropy of 1 mole of a substance under standard conditions (100 kPa pressure and 298 K temperature). Its units are J K^-1 mol^-1.

The steps for calculating ΔS_system are:

1. Find the sum of standard entropies for the products (ΣS^⦵(products)).
2. Find the sum of standard entropies for the reactants (ΣS^⦵(reactants)).
3. Subtract the sum of reactant entropies from the sum of product entropies.

If the balanced equation has coefficients other than 1, multiply the standard entropy of each substance by its coefficient.

Worked example 1 - Calculating entropy change for a reaction

Calculate the entropy change for the reaction where hydrogen chloride reacts with ammonia to produce ammonium chloride at standard conditions:

HCl_(g) + NH_3(g) ➔ NH₄Cl_(s)

Given:

• S^⦵[HCl_(g)] = 186.9 J K^-1 mol^-1
• S^⦵[NH_3(g)] = 192.8 J K^-1 mol^-1
• S^⦵[NH₄Cl_(s)] = 94.6 J K^-1 mol^-1

Step 1: Calculate the entropy of the products

S_products = S^⦵[NH₄Cl] = 94.6 J K^-1 mol^-1

Step 2: Calculate the entropy of the reactants

S_reactants = S^⦵[HCl] + S^⦵[NH₃]

= 186.9 + 192.8 = 379.7 J K^-1 mol^-1

Step 3: Equation

ΔS_system = S_products $-$ S_reactants

Step 4: Substitution and correct evaluation

ΔS_system = 94.6 $-$ 379.7 = $-$285.1 J K^-1 mol^-1

The negative ΔS_system indicates a decrease in entropy, reflecting the increased order as two gases form a solid.

Total entropy change

To determine if a reaction is feasible, we need to consider the total entropy change of both the system and its surroundings:

ΔS^⦵_total = ΔS^⦵_system + ΔS^⦵_surroundings

The entropy change of the surroundings can be calculated using the following equation:

ΔS^⦵_surroundings = $-\frac{\mathrm{\Delta }{H}^{\ominus }}{T}$

Where:

• ΔH^⦵ = standard enthalpy change of the reaction (J mol^-1).
• T is the temperature (K).

Worked example 2 - Calculating total entropy change

Calculate the total entropy change, in J K^-1 mol^-1, for the reaction of ammonia with hydrogen chloride:

NH_3(g) + HCl_(g) ➔ NH₄Cl_(s)

Given:

• ΔH^⦵ = $-$315 kJ mol^-1 (at 298 K)
• ΔS^⦵_system = $-$285.1 J K^-1 mol^-1 (calculated in worked example 1)

Step 1: Conversion of kJ mol^-1 into J mol^-1

To convert from kJ mol^-1 into J mol^-1, multiply by 1,000

$-$315 kJ mol^-1 = $-$315,000 J mol^-1

Step 2: Calculate ΔS^⦵_surroundings

ΔS^⦵_surroundings = $-\frac{\mathrm{\Delta }{H}^{\ominus }}{T}=-\frac{\mathrm{315,000}}{298}$ = +1,057 J K^-1 mol^-1

Step 3: Calculate ΔS^⦵_total

ΔS^⦵_total = ΔS^⦵_system + ΔS^⦵_surroundings = $-$285.1 + 1,057 = +772 J K^-1 mol^-1

The positive total entropy change suggests that this reaction is likely to be feasible under standard conditions.

Worked example 3 - Calculating total entropy change

Calculate the total entropy change, in J K^-1 mol^-1, for the dissolution of ammonium nitrate crystals in water under standard conditions:

NH_4_NO_3(s)_ ➔ NH_4_^+^(aq)** + NO_3_^-^(aq)**

Given:

• ΔH = +25.70 kJ mol^-1^ (at 298 K)
• S^⦵[NH_4_NO_3_] = 151.1 J K^-1^ mol^-1^
• S^⦵[NH_4_^+^(aq)] = 113.4 J K^-1^ mol^-1^
• S^⦵[NO_3_^-^(aq)] = 146.4 J K^-1^ mol^-1^

Step 1: Conversion of kJ mol^-1 into J mol^-1

To convert from kJ mol^-1^ into J mol^-1^, multiply by 1,000

25.70 kJ mol^-1 = 25,700 J mol^-1

Step 2: Calculate ΔS^⦵_surroundings

ΔS^⦵_surroundings = $-\frac{\mathrm{\Delta }{H}^{\ominus }}{T}=-\frac{\mathrm{25,700}}{298}=-$86.24 J K^-1 mol^-1

Step 3: Calculate ΔS^⦵_system

ΔS^⦵_system = ΔS^⦵_products $-$ ΔS^⦵_reactants = 113.4 + 146.4 $-$ 151.1 = +108.7 J K^-1 mol^-1

Step 4: Calculate ΔS^⦵_total

ΔS^⦵_total = ΔS^⦵_system + ΔS^⦵_surroundings = 108.7 $-$ 86.24 = +22.5 J K^-1 mol^-1

Worked example 4 - Calculating total entropy change

Calculate the total entropy change, in J K^-1 mol^-1, for the reaction between barium hydroxide and ammonium chloride:

Ba(OH)2**•8H_2_O_(s)_ + 2NH_4_Cl_(s)_ ➔ BaCl_2(s)_ + 10H_2_O_(l)_ + 2NH_3(g)_

Given:

• ΔH = +164.0 kJ mol^-1^ (at 298 K)
• S^⦵[Ba(OH)2**•8H_2_O] = 427.0 J K^-1^ mol^-1^
• S^⦵[NH_4_Cl] = 94.6 J K^-1^ mol^-1^
• S^⦵[BaCl_2_] = 123.7 J K^-1^ mol^-1^
• S^⦵[H_2_O] = 69.9 J K^-1^ mol^-1^
• S^⦵[NH_3_] = 192.3 J K^-1^ mol^-1^

Step 1: Conversion of kJ mol^-1 into J mol^-1

To convert from kJ mol^-1^ into J mol^-1^, multiply by 1,000

164.0 kJ mol^-1 = 164,000 J mol^-1

Step 2: Calculate ΔS^⦵_surroundings

ΔS^⦵_surroundings = $-\frac{\mathrm{\Delta }{H}^{\ominus }}{T}=-\frac{\mathrm{164,000}}{298}=-$550.3 J K^-1 mol^-1

Step 3: Calculate ΔS^⦵_system

ΔS^⦵_system = ΔS^⦵_products $-$ ΔS^⦵_reactants **

ΔS^⦵_system *= (*123.7 + (10 x 69.9) + (2 x 192.3)) $-$(427.0 + (2 x 94.6) = +591.1 J K^-1 mol^-1

Step 4: Calculate ΔS^⦵_total

ΔS^⦵_total = ΔS^⦵_system + ΔS^⦵_surroundings = 591.1 $-$ 550.3 = +40.8 J K^-1 mol^-1

The positive ΔS_total_ indicates that the reaction is spontaneous, despite being endothermic. This spontaneity is driven by the significant increase in entropy of the system, which arises from the formation of liquid water and gaseous ammonia as products.