Problem 5 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

At constant temperature, the equilibrium constant \(\left(K_{\mathrm{p}}\right)\) for the decomposition reaction: \(\mathrm{N}_{2} \mathrm{O}_{4} \rightleftharpoons 2 \mathrm{NO}_{2}\) is expressed by \(K_{\mathrm{p}}=\frac{4 x^{2} P}{1-x^{2}}\), where \(P=\) total pressure at equilibrium, \(x=\) extent of decomposition. Which one of the following statements is true? (a) \(K_{\mathrm{p}}\) increases with increase of \(P\). (b) \(K_{\mathrm{v}}\) increases with increase of \(x\). (c) \(K\), increases with decrease of \(x\). (d) \(K\), remains constant with change in \(P\) and \(x\).

Step-by-Step Solution

Answer

The true statement is (d) \(K\) remains constant with change in \(P\) and \(x\), because the equilibrium constant, \(K_{\mathrm{p}}\), only depends on temperature and not on the initial concentrations or the total pressure of the reactants and products at equilibrium.

Step 1: Understanding the equilibrium constant expression

The equilibrium constant for the reaction is given by the expression \(K_{\mathrm{p}}=\frac{4 x^{2} P}{1-x^{2}}\), where \(P\) is the total pressure at equilibrium and \(x\) is the extent of decomposition of \(N_2O_4\) into \(NO_2\). To analyze how \(K_{\mathrm{p}}\) behaves in relation to \(P\) and \(x\), one needs to understand how each term in the expression \(4 x^{2} P\) and \(1 - x^{2}\) affects \(K_{\mathrm{p}}\).

Step 2: Exploring the effect of total pressure (P) on Kp

To examine the effect of total pressure \(P\) on the value of the equilibrium constant \(K_{\mathrm{p}}\), we keep \(x\) constant and observe the relationship of \(P\) to \(K_{\mathrm{p}}\). Since the numerator of the expression for \(K_{\mathrm{p}}\) is directly proportional to \(P\) (as \(4x^2P\) increases with increasing \(P\)), and the denominator remains constant when \(x\) is constant, it shows that \(K_{\mathrm{p}}\) increases with an increase in \(P\). This makes option (a) a possibility.

Step 3: Analyzing the effect of extent of decomposition (x) on Kp

Now, let's look at the effect of the extent of decomposition \(x\) on \(K_{\mathrm{p}}\). If \(x\) increases then both the numerator \(4 x^{2} P\) and the denominator \(1-x^{2}\) will change. However, the numerator increases more rapidly than the denominator decreases because the numerator is proportional to the square of \(x\), while the denominator only decreases to a minimal extent until \(x\) gets very close to 1. Therefore, \(K_{\mathrm{p}}\) will generally increase with an increase in \(x\), ruling out option (c) and making option (b) a possibility.

Step 4: Determining the answer based on the equilibrium constant property

The equilibrium constant for a given reaction at a specific temperature is always constant. It does not change with varying pressure or extent of decomposition, so long as the temperature remains the same. Neither an increase in \(P\) nor a change in \(x\) at constant temperature would affect the value of \(K_{\mathrm{p}}\). Therefore, the correct statement is that \(K_{\mathrm{p}}\), or simply \(K\), remains constant with a change in \(P\) and \(x\), if temperature remains constant. This means option (d) is the true statement, while (a) and (b) are incorrect.

Key Concepts

Chemical Equilibrium

In the realm of physical chemistry, chemical equilibrium refers to a state where the concentrations of reactants and products in a chemical reaction remain unchanged over time. At this juncture, the rate at which the reactants convert to products is equal to the rate at which the products revert back to reactants.

It's important to note that equilibrium does not mean the reactants and products are in equal concentration, but rather that their concentrations have stabilized in a fixed ratio. The exercise at hand demonstrates chemical equilibrium through the decomposition reaction of dinitrogen tetroxide into nitrogen dioxide, symbolized as \( N_2O_4 \rightleftharpoons 2 NO_2 \). Here, the system reaches equilibrium when the rate of decomposition of \( N_2O_4 \) equals the rate of formation of \( NO_2 \).

Le Chatelier's Principle

Le Chatelier's Principle is a pivotal concept in chemical equilibrium that predicts how a change in conditions, such as pressure, temperature, or concentration, will affect the position of equilibrium. According to this principle, if an external change is imposed on a system at equilibrium, the system will adjust itself to partially counteract the change and re-establish equilibrium.

For example, increasing pressure on a system will shift the equilibrium towards the side with fewer moles of gas. In the context of our textbook exercise, Le Chatelier's Principle would imply that altering the pressure or the extent of decomposition (\(x\)) would shift the equilibrium position, but not the equilibrium constant (\(K\)), as it is solely dependent on temperature.

Physical Chemistry Problems

Solving physical chemistry problems typically involves understanding complex concepts and applying mathematical equations to real-world chemical scenarios. Problems often challenge students to interpret formulas and predict the behavior of chemical systems under various conditions.

Key to addressing these problems is a solid comprehension of concepts like equilibrium, Le Chatelier's Principle, and thermodynamics. The problem provided from the textbook is a classic example, requiring the application of the equilibrium constant expression to determine how changes in pressure and decomposition extent affect the system.

Equilibrium Constant Expression

The equilibrium constant expression quantitatively describes the position of equilibrium for a chemical reaction. For the reaction at hand, \( K_{\mathrm{p}} \) is used to represent the equilibrium constant in terms of partial pressures. Defined by the expression \(K_{\mathrm{p}}=\frac{4 x^{2} P}{1-x^{2}}\), it establishes a relationship between the extent of decomposition \(x\), the total pressure \(P\), and the position of equilibrium.

It's essential to realize that \(K_{\mathrm{p}}\) is invariant with changes in pressure or the extent of decomposition as long as the temperature is constant. Therefore, it's a fundamental tool in predicting the outcome of equilibrium reactions and understanding how various factors can influence, but not change, the equilibrium constant.