Problem 50 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

Calculate \(\Delta_{\mathrm{f}} H^{\circ}\) for aqueous chloride ion from the following data: \(\frac{1}{2} \mathrm{H}_{2}(\mathrm{~g})+\frac{1}{2} \mathrm{Cl}_{2}(\mathrm{~g}) \rightarrow \mathrm{HCl}(\mathrm{g}), \quad \Delta_{\mathrm{f}} H^{\mathrm{o}}\) \(=-92.4 \mathrm{~kJ}\) \(\mathrm{HCl}(\mathrm{g})+n \mathrm{H}_{2} \mathrm{O}(\mathrm{l}) \rightarrow \mathrm{H}^{+}(\mathrm{aq})+\mathrm{Cl}^{-}(\mathrm{aq})\) \(\Delta H^{\circ}=-74.8 \mathrm{~kJ}\) \(\Delta_{\mathrm{f}} H^{\circ}\left(\mathrm{H}^{+}\right.\), aq. \()=0.0 \mathrm{~kJ}\) (a) \(0.0\) (b) \(+83.6 \mathrm{~kJ}\) (c) \(+167.2 \mathrm{~kJ}\) (d) \(-167.2 \mathrm{~kJ}\)

Step-by-Step Solution

Answer

\(\Delta_{\mathrm{f}} H^{\circ} \text{ for } \mathrm{Cl}^{-}(\mathrm{aq}) = -92.4 \mathrm{kJ} + (-74.8 \mathrm{kJ}) = -167.2 \mathrm{kJ}\)

Step 1: Write the Hess's Law Statement

According to Hess's Law, if a reaction is the sum of two or more other reactions, the heat change for the total reaction is the sum of the heat changes for the individual reactions.

Step 2: Write the Formation Reaction for Chloride Ion

Write the thermochemical equation for the formation of a chloride ion in water. It must be the combination of the given reactions which includes the formation of gaseous HCl and its dissolution in water to create hydrochloric acid.

Step 3: Express the Formation Reaction

Combine the two provided reactions to form the reaction: \(\frac{1}{2} H_2(g) + \frac{1}{2} Cl_2(g) + n H_2O(l) \rightarrow H^+(aq) + Cl^-(aq)\).

Step 4: Calculate the Standard Enthalpy of Formation for Chloride Ion

Sum the enthalpies of the given reactions, keeping in mind that the standard enthalpy of formation for the hydrogen ion in aqueous solution is zero, to find the standard enthalpy of formation for the chloride ion.

Key Concepts

Understanding Hess's Law

In chemistry, Hess's Law is a principle that relates to the total enthalpy change in a chemical reaction. It states that the total enthalpy change, regardless of the number of steps or intermediate stages of the reaction, is the same as if the reaction were to happen in a single step. This is because enthalpy is a state function, which means its value is determined only by the current state of the system and not by the path the system took to reach that state.

Now, let's take a real-world analogy to simplify this concept: imagine building a house. The total cost to build it will be the same whether you pay for it in one lump sum or in smaller payments over time. Similarly, when it comes to chemical reactions, the total heat change (like the total cost of the house) doesn't care about the