Question

For the reaction \(\mathrm{H}_{2} \mathrm{O}(g)+\mathrm{Cl}_{2} \mathrm{O}(g) \longrightarrow 2 \mathrm{HClO}(g),\) you know \(\Delta S_{\mathrm{rxn}}^{\circ}\) and \(S^{\circ}\) of \(\mathrm{HClO}(g)\) and of \(\mathrm{H}_{2} \mathrm{O}(g) .\) Write an expression that can be used to determine \(S^{\circ}\) of \(\mathrm{Cl}_{2} \mathrm{O}(g)\)

Short answer

The expression is \[S^{\circ}(\text{Cl}_2 \text{O}(g)) = 2S^{\circ}(\text{HClO}(g)) - S^{\circ}(\text{H}_2 \text{O}(g)) - \Delta S_{\text{rxn}}^{\circ}\].

Step-by-step solution

Write the reaction

The given reaction is: \[ \text{H}_2 \text{O}(g) + \text{Cl}_2 \text{O}(g) \rightarrow 2 \text{HClO}(g) \]

Recall the formula for entropy change of the reaction

The standard entropy change of the reaction, \(\Delta S_{\text{rxn}}^{\circ}\), is given by: \[ \Delta S_{\text{rxn}}^{\circ} = \sum S^{\circ}_{\text{products}} - \sum S^{\circ}_{\text{reactants}} \]

List the standard entropy values for reactants and products

Given: \(S^{\circ}\) of \(\text{H}_2 \text{O}(g)\) and \(\text{HClO}(g)\) To find: \(S^{\circ}\) of \(\text{Cl}_2 \text{O}(g)\)

Substitute the entropy terms into the formula

Substitute the values into the entropy change formula: \[ \Delta S_{\text{rxn}}^{\circ} = 2S^{\circ}(\text{HClO}(g)) - (S^{\circ}(\text{H}_2 \text{O}(g)) + S^{\circ}(\text{Cl}_2 \text{O}(g))) \]

Isolate standard entropy of Cl2O(g)

Rearrange the formula to solve for \(S^{\circ}(\text{Cl}_2 \text{O}(g))\): \[ S^{\circ}(\text{Cl}_2 \text{O}(g)) = 2S^{\circ}(\text{HClO}(g)) - S^{\circ}(\text{H}_2 \text{O}(g)) - \Delta S_{\text{rxn}}^{\circ} \]

Key concepts

Thermodynamics

Thermodynamics is the study of energy, heat, and work within physical systems. It's a branch of physics that helps us understand how energy is transformed and transferred. One of the key principles in thermodynamics is the First Law of Thermodynamics, which states that energy cannot be created or destroyed; it can only change form. The Second Law of Thermodynamics introduces the concept of entropy, stating that systems will tend towards increased disorder or randomness over time. In our context, when we look at chemical reactions, thermodynamics plays a critical role in predicting whether a reaction will occur spontaneously and how much energy will be required or released.

Entropy Change

Entropy change, denoted as \( \Delta S \), measures the change in disorder or randomness in a system during a process or reaction. For a given chemical reaction, the standard entropy change (\( \Delta S^{\circ}_{\text{rxn}} \)) is calculated using the standard entropies (\( S^{\circ} \)) of products and reactants. The formula to calculate the standard entropy change of reaction is:
\[ \Delta S^{\circ}_{\text{rxn}} = \sum S^{\circ}_{\text{products}} - \sum S^{\circ}_{\text{reactants}} \]
Here, the sum of the standard entropies of the products is subtracted by the sum of the standard entropies of the reactants. This tells us how the disorder in the system changes as the reaction proceeds. In the exercise, knowing the standard entropies of water vapor and hypochlorous acid gas, along with the entropy change of the reaction, allows us to find the standard entropy of dichlorine monoxide gas.

Reaction Entropy

Reaction entropy is the change in entropy that occurs when a specific chemical reaction takes place. It's an important aspect of thermodynamics because it helps predict whether a reaction will be spontaneous. A reaction with a positive \( \Delta S \) suggests an increase in disorder, which is generally favorable. In our exercise, we are given the reaction for the formation of hypochlorous acid gas from water vapor and dichlorine monoxide gas.
The reaction equation is:
\[ \text{H}_2 \text{O}(g) + \text{Cl}_2 \text{O}(g) \rightarrow 2 \text{HClO}(g) \]
Using the standard entropy values of \( \text{H}_2 \text{O}(g) \) and \( \text{HClO}(g) \), and the known overall standard entropy change for the reaction, we can determine the unknown standard entropy of \( \text{Cl}_2 \text{O}(g) \). Substituting these into our formula for standard entropy change, and then rearranging to solve for the unknown, we derive:
\[ S^{\circ}( \text{Cl}_2 \text{O}(g) ) = 2S^{\circ}( \text{HClO}(g) ) - S^{\circ}( \text{H}_2 \text{O}(g) ) - \Delta S_{\text{rxn}}^{\circ} \].
This formula allows us to calculate the missing standard entropy, completing our understanding of the reaction's thermodynamics.