Problem 98 - Chemistry An Atoms First Approach

Question

Consider the reaction: $$\mathrm{H}_{2} \mathrm{S}(g)+\mathrm{SO}_{2}(g) \longrightarrow 3 \mathrm{S}(g)+2 \mathrm{H}_{2} \mathrm{O}(l)$$ for which \(\Delta H\) is \(-233 \mathrm{kJ}\) and \(\Delta S\) is \(-424 \mathrm{J} / \mathrm{K}.\) a. Calculate the free energy change for the reaction \((\Delta G)\) at \(393 \mathrm{K}.\) b. Assuming \(\Delta H\) and \(\Delta S\) do not depend on temperature, at what temperatures is this reaction spontaneous?

Step-by-Step Solution

Answer

(a) The Gibbs free energy change (ΔG) at 393 K is -66,368 J. (b) The reaction is spontaneous at temperatures greater than 549.06 K.

Step 1: Identify the Given Variables

Enthalpy Change (ΔH) = -233 kJ Entropy Change (ΔS) = -424 J/K Temperature (T) = 393 K Note: Enthalpy and entropy changes should be in the same unit, so we will need to convert ΔH to J (joules).

Step 2: Convert ΔH to J

1 kJ = 1000 J ΔH = -233 kJ × 1000 J/kJ = -233,000 J

Step 3: Calculate ΔG using the Gibbs Free Energy Formula

The formula for Gibbs free energy change (ΔG) is: ΔG = ΔH - TΔS Now, plug in the values for ΔH, T, and ΔS: ΔG = -233,000 J - (393 K × -424 J/K)

Step 4: Calculate ΔG

ΔG = -233,000 J + (393 K × 424 J/K) ΔG ≈ -233,000 J + 166,632 J ΔG ≈ -66,368 J (a). Therefore, the Gibbs free energy change (ΔG) at 393 K is -66,368 J.

Step 5: Determine the Temperature at which the Reaction is Spontaneous (ΔG < 0)

For the reaction to be spontaneous, ΔG should be negative (ΔG < 0). Using the formula for ΔG, we can set up an inequality: ΔG = ΔH - TΔS < 0 T > ΔH/ΔS We will keep the ΔH and ΔS in their original units (kJ and J/K): T > -233 kJ / (-0.424 kJ/K)

Step 6: Calculate the Temperature

T > 549.06 K (b). So, the reaction is spontaneous at temperatures greater than 549.06 K.

Key Concepts

Enthalpy Change

In chemistry, enthalpy change (\(\Delta H\)) refers to the heat exchanged in a chemical reaction when pressure is constant. It is an essential component to determine the energy changes that occur during a reaction.
For the reaction of hydrogen sulfide (\(\mathrm{H}_2\mathrm{S}(g)\)) and sulfur dioxide (\(\mathrm{SO}_2(g)\)) with the formation of sulfur and water, the enthalpy change is given as \(-233 \text{kJ}\). This negative \(\Delta H\) value indicates an exothermic process, meaning that heat is released into the surroundings.

• A negative \(\Delta H\): The reaction is exothermic and releases energy.
• A positive \(\Delta H\): The reaction is endothermic and absorbs energy.

Understanding enthalpy helps in predicting whether the energy released (or absorbed) is significant enough to drive chemical processes in a given direction.

Entropy Change

Entropy change (\(\Delta S\)) quantifies the disorder or randomness in the system during a reaction. It measures how much the entropy, or disorder, of a system increases or decreases.
For our specific reaction, \(\Delta S\) is given as \(-424 \text{J/K}\), which signifies a decrease in entropy. This means the products are more ordered than the reactants—there is a decrease in randomness.

• A positive \(\Delta S\): Greater disorder or randomness in the products compared to the reactants.
• A negative \(\Delta S\): Greater order or less randomness in the products compared to the reactants.

Entropy changes help assess whether reactions are likely to be spontaneous because they provide insight into the distribution of energy at a microscopic level.

Spontaneous Reactions

A reaction is considered spontaneous if it occurs naturally without needing continuous external energy. The spontaneity of chemical reactions is determined by the Gibbs free energy change (\(\Delta G\)). The formula to calculate this is:

1. \(\Delta G = \Delta H - T\Delta S\)
2. \(+ \Delta G\): Non-spontaneous (reaction absorbs energy, requires input).
3. \(- \Delta G\): Spontaneous (reaction releases energy, occurs on its own).

In the specific reaction provided, the Gibbs free energy change was found to be around \(-66,368 \text{J}\) at a temperature of \(393 \text{K}\). This negative value suggests the reaction is spontaneous at this temperature.
Spontaneity also depends on the temperature:

• At temperatures above \(549.06 \text{K}\), the reaction is spontaneous.
• Such conditions ensure that the energy of the surroundings and increased entropy are enough for the reaction to proceed naturally.

Understanding spontaneity is crucial for knowing how and when a chemical reaction will proceed without needing extra energy.