Entropy Calculations in Chemical Thermodynamics

Entropy Calculations

Entropy of the Universe

Entropy is a measure of the disorder or randomness in a system. The total entropy change in the universe, ΔSuniv, is a key concept in determining the spontaneity of chemical reactions. It is calculated as the sum of the entropy changes of the system and the surroundings.

• Formula:

$\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr}$

• If $\Delta S_{univ} > 0$, the process is spontaneous.

• If $\Delta S_{univ} < 0$, the process is nonspontaneous.

• If $\Delta S_{univ} = 0$, the process is at equilibrium.

ΔSuniv	Process Type
> 0	Spontaneous
< 0	Nonspontaneous
= 0	Equilibrium

Example: Calculate the total entropy change for a reaction with $\Delta S_{sys} = 2.7\ \mathrm{J/K}$ and $\Delta S_{surr} = -600.0\ \mathrm{J/K}$. Is this reaction spontaneous?

• $\Delta S_{univ} = 2.7\ \mathrm{J/K} + (-600.0\ \mathrm{J/K}) = -597.3\ \mathrm{J/K}$

• Since $\Delta S_{univ} < 0$, the reaction is not spontaneous.

Entropy of the Surroundings

The entropy change of the surroundings, ΔSsurr, is related to the heat exchanged by the system and the temperature at which the process occurs. For processes at constant pressure, the formula is:

$\Delta S_{surr} = -\frac{\Delta H_{sys}}{T}$

• $\Delta H_{sys}$ is the enthalpy change of the system (in J).

• $T$ is the temperature in Kelvin (K).

Example: Determine the change in entropy of the universe for the following reaction at 25°C:

• $2\ \mathrm{Ag}_2\mathrm{O}(s) \rightarrow 4\ \mathrm{Ag}(s) + O_2(g)$

• $\Delta H_{rxn} = +61.4\ \mathrm{kJ}$, $\Delta S_{sys} = +0.18\ \mathrm{kJ/K}$

• Convert $\Delta H_{rxn}$ to J: $61.4\ \mathrm{kJ} = 61400\ \mathrm{J}$

• $T = 25°C = 298\ \mathrm{K}$

• $\Delta S_{surr} = -\frac{61400}{298} = -206\ \mathrm{J/K}$

• $\Delta S_{univ} = 180\ \mathrm{J/K} - 206\ \mathrm{J/K} = -26\ \mathrm{J/K}$

Entropy of the System

Each substance has a standard molar entropy, So, associated with it. The entropy change of a reaction, ΔSrxn, can be calculated using the standard molar entropies of the reactants and products.

• Formula:

$\Delta S_{rxn} = \sum n S^o_{products} - \sum n S^o_{reactants}$

• Standard conditions: 298 K and 1 atm pressure.

• Unlike standard enthalpy of formation, the standard entropy of elements in their natural state is not zero.

Example: Calculate $\Delta S_{rxn}$ for the following reaction at 25°C:

• $2\ \mathrm{NO}(g) + O_2(g) \rightarrow 2\ \mathrm{NO}_2(g)$

• Given: $S^o$ (NO) = 210.8 J/mol·K, $S^o$ (O2) = 205.0 J/mol·K, $S^o$ (NO2) = 240.1 J/mol·K

• $\Delta S_{rxn} = [2 \times 240.1] - [2 \times 210.8 + 1 \times 205.0] = 480.2 - 626.6 = -146.4\ \mathrm{J/K}$

Practice Problem

For the following reaction at 25°C, calculate $\Delta H_{rxn}$, $\Delta S_{rxn}$, and $\Delta S_{univ}$. Determine if the reaction is spontaneous.

• $Fe_2O_3(s) + 3H_2(g) \rightarrow 2Fe(s) + 3H_2O(g)$

• Given: $S^o$ (Fe2O3) = 87.4 J/mol·K, $S^o$ (H2) = 130.7 J/mol·K, $S^o$ (Fe) = 27.3 J/mol·K, $S^o$ (H2O) = 188.7 J/mol·K

Solution Steps:

1. Calculate $\Delta S_{rxn}$ using the formula above.

2. Calculate $\Delta S_{surr}$ using $\Delta H_{rxn}$ (if provided).

3. Sum to find $\Delta S_{univ}$ and determine spontaneity.

Additional info: The notes provide a structured approach to entropy calculations, including formulas, examples, and practice problems relevant to chemical thermodynamics (Ch.19).