Problem 73 - A Complete Resource Book in Chemistry for JEE Main

Question

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

Step-by-Step Solution

Answer

Kp increases with increase of x: (c) is correct.

Step 1: Clarify the Equilibrium Constant Expression

The given expression is \(K_p = \frac{4x^2 p}{1 - x^2}\). This shows the relationship between the equilibrium constant \(K_p\), the extent of decomposition \(x\), and the pressure \(p\). Notice that \(K_p\) depends on both \(x\) and \(p\).

Step 2: Analyze Changes in Kp with x

Evaluate how \(K_p\) changes with \(x\). In the numerator, \(4x^2\) indicates that \(K_p\) is directly proportional to \(x^2\). In the denominator, \(K_p\) is inversely proportional to \(1 - x^2\). Therefore, as \(x\) increases, \(K_p\) increases since both increments in the numerator and decreases in the denominator cause an overall increase in \(K_p\).

Step 3: Analyze Changes in Kp with P

Evaluate how \(K_p\) changes with \(p\). The equilibrium constant is directly proportional to \(p\) in the expression. This means if pressure \(p\) increases, \(K_p\) also increases. But since \(K_p\) is a constant based on temperature, it actually should not change when only \(p\) changes, suggesting a potential conflict with the expression. Generally, the direct relationship shouldn't influence a correctly defined constant at constant temperature.

Step 4: Identify the Correct Statement

Review the options considering the calculations from Steps 2 and 3. Option (c), which states that \(K_p\) increases with an increase of \(x\), aligns with the finding that \(K_p\) indeed increases as extent of decomposition \(x\) increases.

Key Concepts

Understanding Chemical Equilibrium

Many students find chemical equilibrium to be a bit tricky. Let's dive into this concept together. Chemical equilibrium is a state in which the rates of the forward and backward reactions are equal.
This means that the concentrations of reactants and products remain constant over time, but not necessarily equal in amount. It's essential to recognize that while the reaction appears static, molecules are continuously reacting in both directions.
When a reaction reaches equilibrium, we can use either the concentration constant ( K_c ) or the pressure constant ( K_p ) to describe the system. These constants rely heavily on factors like temperature and can tell us the ratio of product to reactant at equilibrium.

• For reactions involving gases, K_p might be more pertinent, because it considers the partial pressures of the chemicals.
• K_p remains constant at a given temperature and isn't typically influenced by changes in pressure.

You might wonder how this aligns with the formula from our problem. Remember, a true equilibrium constant ( K_p ) doesn't vary with pressure or the extent of reaction ( x ) at constant temperature. Thus, when a change is spotted, it might signify that the reaction hasn't fully adjusted yet, or there is an error in interpretation.

Exploring Le Chatelier's Principle

Le Chatelier's Principle offers insight into how a system at equilibrium responds to changes or disturbances. Let's break it down! This principle states that when a change is applied to a system at equilibrium, the system adjusts to counteract the effect of that change.
In simpler terms, a disturbance pushes the equilibrium to shift in the direction that will restore balance.
For instance, if you increase the concentration of a reactant, the equilibrium shifts towards the side of the product to use up the excess reactant. Similarly, if pressure is increased, a gaseous reaction might shift towards the side with fewer gas molecules to decrease pressure.

• The principle helps chemists predict how altering conditions, like temperature, concentration, or pressure, can affect the progress of a reaction.
• However, it's important to recognize that Le Chatelier's Principle doesn't apply to equilibrium constants themselves.

In regards to our decomposition reaction, as pressure increases, one might think K_p changes based on Le Chatelier's Principle. However, K_p is fixed for a given temperature, so while the reaction balance tips, the equilibrium constant doesn't change, again highlighting why K_p must be evaluated carefully.

Diving into Reaction Kinetics

Reaction kinetics is the study of the rate at which a chemical reaction proceeds. In essence, it's about figuring out how fast reactants turn into products.
This is crucial in understanding the dynamics of a reaction and how equilibrium is eventually established.
The rate of a chemical reaction can be impacted by factors like concentration, temperature, pressure, and the presence of catalysts. Kinetics and equilibrium work hand-in-hand, as the speed at which equilibrium is reached depends on reaction kinetics.

• Reaction rates can be determined by measuring how rapidly a reactant is used up or how fast a product is formed.
• Kinetics doesn't determine K_p directly, but it can influence how quickly we reach equilibrium.

Through reaction kinetics, chemists aim to control or predict the behavior of a reaction under different conditions. However, in our given reaction, understanding that K_p is temperature bound rather than kinetics-bound is crucial. When analyzing how equilibrium constants change, remember the broader influence of reaction kinetics, despite K_p remaining a permanent fixture for a specific temperature.