Problem 7 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

\(\mathrm{XeF}_{6}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{XeOF}_{4}+2 \mathrm{HF}\) equilibrium constant \(=K_{1}\). \(\mathrm{XeO}_{4}+\mathrm{XeF}_{6} \rightleftharpoons \mathrm{XeOF}_{4}+\mathrm{XeO}_{3} \mathrm{~F}_{2}\) equilibrium constant \(=K_{2} .\) Then equilibrium constant for the following reaction will be: \(\mathrm{XeO}_{4}+2 \mathrm{HF} \rightleftharpoons \mathrm{XeO}_{3} \mathrm{~F}_{2}+\mathrm{H}_{2} \mathrm{O}\) (a) \(\frac{K_{1}}{K_{2}}\) (b) \(K_{1}+K_{2}\) (c) \(\frac{K_{2}}{K_{1}}\) (d) \(K_{2}-K_{1}\)

Step-by-Step Solution

Answer

\(\frac{K_{2}}{K_{1}}\)

Step 1: Write down the given reactions and equilibrium constants

The given chemical reactions with their respective equilibrium constants are:1. \(\mathrm{XeF}_{6}+\mathrm{H}_{2}O \rightleftharpoons \mathrm{XeOF}_{4}+2\mathrm{HF}\) with equilibrium constant \(K_{1}\).2. \(\mathrm{XeO}_{4}+\mathrm{XeF}_{6} \rightleftharpoons \mathrm{XeOF}_{4}+\mathrm{XeO}_{3}\mathrm{~F}_{2}\) with equilibrium constant \(K_{2}\).

Step 2: Write down the target reaction

The target reaction for which we need to find the equilibrium constant is:\(\mathrm{XeO}_{4}+2\mathrm{HF} \rightleftharpoons \mathrm{XeO}_{3}\mathrm{~F}_{2}+\mathrm{H}_{2}O\)

Step 3: Analyze the reactions to find a relationship

Looking at the given reactions, we can notice that the target reaction is obtained by reversing Reaction 1 and adding it to Reaction 2. Reversing a reaction will inverse its equilibrium constant, while adding two reactions together will result in the multiplication of their equilibrium constants.

Step 4: Write the equilibrium constant for the target reaction

If we reverse Reaction 1, the equilibrium constant becomes \(\frac{1}{K_{1}}\). We then multiply this reversed Reaction 1 with Reaction 2 to get the target reaction:\(\frac{1}{K_{1}} \times K_{2} = \frac{K_{2}}{K_{1}}\).This is the equilibrium constant for the target reaction.

Key Concepts

Physical Chemistry

Physical chemistry is a branch of chemistry that deals with the investigation of how matter behaves on a molecular and atomic level, and how chemical reactions occur. Understanding the principles of physical chemistry can help students grasp how and why substances combine or separate to form other substances, and how they interact with energy. An equilibrium state, an essential concept in this field, is when the rate of the forward reaction equals the rate of the reverse reaction, and the concentrations of reactants and products remain constant over time.

Within this realm, the study of gas laws, thermodynamics, kinetics, and quantum mechanics are crucial for comprehending how chemical reactions occur and how they can be quantified. These studies not only provide insight into the natural world but also are foundational in developing new materials, pharmaceuticals, and energy sources.

Equilibrium Constant Calculation

The calculation of equilibrium constants is a fundamental aspect of chemical equilibrium in physical chemistry. An equilibrium constant, denoted as 'K', is a numerical value that expresses the ratio of the concentrations of products to reactants at equilibrium. Each chemical reaction has its unique equilibrium constant, which can be used to predict the direction and extent of the reaction under a given set of conditions.

Calculating an equilibrium constant involves writing the expression based on the balanced chemical equation and using the concentrations or partial pressures of the reactants and products. It's important to remember that only species in the aqueous or gaseous states are included in the equilibrium expression, and solids and pure liquids are omitted as their concentrations are constants.

When reactions are combined or reversed, the equilibrium constants for these new reactions are calculated by manipulating the original constants—multiplication for sequential reactions and reciprocal for a reversed reaction.

Chemical Reaction Analysis

Analyzing chemical reactions is essential to determine the behavior of reactants and products during a chemical change. It involves understanding stoichiometry, which is the quantitative relationship between reactants and products in a balanced chemical equation, and kinetics, which is the study of the rate at which a chemical process occurs. For reactions at equilibrium, recognizing how different reactions contribute to a net reaction is essential for determining the overall equilibrium constant.

When analyzing complex reactions, one must deduce how to obtain a desired reaction through a combination of known reactions. This process usually includes reversing and adding reactions while considering the corresponding changes in their equilibrium constants. This technique allows the determination of the equilibrium constant of a target reaction, which can predict the concentrations of species at equilibrium and thus the feasibility of the reaction.

The ability to manipulate reactions and their constants is also crucial in industries where controlled reactions are necessary for product formation, such as pharmaceuticals and materials engineering.