Question

The value of \(\mathrm{K}_{\mathrm{p}}\) for the reaction, \(2 \mathrm{SO}_{2}+\mathrm{O}_{2} \rightleftharpoons 2 \mathrm{SO}_{3}\) at 700 is \(1.3 \times 10^{-3} \mathrm{~atm}^{-1}\). The value of \(\mathrm{K}_{\mathrm{c}}\) at same temperature will be: (a) \(1.4 \times 10^{-2}\) (b) \(7.4 \times 10^{-2}\) (c) \(5.2 \times 10^{-2}\) (d) \(3.1 \times 10^{-2}\)

Short answer

The value of \( K_c \) is approximately \( 3.1 \times 10^{-2} \).

Step-by-step solution

Understand the Relationship Between Kp and Kc

The relationship between the equilibrium constants \( K_p \) and \( K_c \) is given by the equation \( K_p = K_c (RT)^{\Delta n} \), where \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( \Delta n \) is the change in moles of gas (moles of products - moles of reactants).

Calculate \( \Delta n \)

In the given reaction \( 2 \mathrm{SO}_2 + \mathrm{O}_2 \rightarrow 2 \mathrm{SO}_3 \), there are 2 moles of \( \mathrm{SO}_3 \) as products and a total of 3 moles of reactants (2 moles of \( \mathrm{SO}_2 \) and 1 mole of \( \mathrm{O}_2 \)). Therefore, \( \Delta n = 2 - 3 = -1 \).

Use Kelvin Temperature

Since the problem does not provide the temperature in Kelvin, assume it is at 700 K for calculation. It is essential that temperature \( T \) is always in Kelvin when using the gas constant \( R = 0.0821 \) L atm K^-1 mol^-1.

Rearrange Kp and Kc Relationship

Rearrange the formula for \( K_c \): \[ K_c = \frac{K_p}{(RT)^{\Delta n}} \] Let's calculate this using \( R = 0.0821 \) L atm K^-1 mol^-1, \( T = 700 \) K, and \( \Delta n = -1 \).

Plug Values into the Equation

Substitute the known values into the equation: \[ K_c = \frac{1.3 \times 10^{-3}}{(0.0821 \times 700)^{-1}} \] Simplify this expression to find \( K_c \).

Perform the Calculation

Calculate the denominator: \( (0.0821 \times 700)^{-1} = 1/57.47 \approx 0.0174 \). Therefore, \[ K_c = 1.3 \times 10^{-3} \times 0.0174 \approx 0.0226 \].

Find Closest Match

Compare the calculated \( K_c \) value \( 0.0226 \) to the given options and find the closest match. Option (d) \( 3.1 \times 10^{-2} \) is the closest match.

Key concepts

Equilibrium Constant

In chemical equilibrium, the equilibrium constant (K) is a vital concept. It represents the ratio of the concentration of products to reactants at equilibrium for a reversible chemical reaction. For reactions involving gases, we often use two types of constants: the equilibrium constant in terms of concentration \( K_c \) and the equilibrium constant in terms of pressure \( K_p \). These constants help us understand and predict the position and extent of a reaction without actually running the experiment.
The relationship between \( K_c \) and \( K_p \) is crucial, particularly for gas-phase reactions. The formula connecting them is:

• \( K_p = K_c (RT)^{\Delta n} \)

Where:

• \( R \) is the universal gas constant
• \( T \) is the temperature in Kelvin
• \( \Delta n \) indicates the change in moles of gas

Understanding these terms allows chemists to switch between pressure and concentration views effortlessly, gaining comprehensive insights into how a reaction will behave under different conditions.

Le Chatelier's Principle

Le Chatelier's Principle is a fundamental concept used to predict the behavior of a chemical system at equilibrium in response to external changes. This principle states that if an external stress is applied to a system in equilibrium, the system will adjust itself to counteract the stress and restore a new equilibrium.
Consider a simple chemical equilibrium like the sulfur dioxide reaction: \[ 2 \text{SO}_2 + \text{O}_2 \rightleftharpoons 2 \text{SO}_3 \]Le Chatelier's Principle offers insights into what happens if there's a change in concentration, temperature, or pressure:

• Concentration: Increasing the concentration of a reactant shifts the equilibrium towards the products to reduce the effect of this change.
• Temperature: Changing the temperature affects equilibrium based on whether the reaction is endothermic or exothermic.
• Pressure: For gaseous reactions, an increase in pressure shifts the equilibrium toward the side with fewer gas moles.

Using Le Chatelier's Principle helps in predicting the direction of shift and finding ways to control reaction conditions for desired outcomes.

Gas Constant R

The Gas Constant \( R \) is pivotal in the study of gases and their behaviors in chemical reactions. It appears in numerous fundamental equations in chemistry, including the ideal gas law and the relationship between \( K_p \) and \( K_c \).
The value of \( R \) depends on the units used and is typically measured as:

• 0.0821 L atm K\(^{-1}\) mol\(^{-1}\)
• 8.314 J mol\(^{-1}\) K\(^{-1}\)

In the context of equilibrium constant calculations, \( R \) helps convert between pressure and concentration, vital for accurate predictions. When using the formula \( K_p = K_c (RT)^{\Delta n} \), ensuring \( R \) matches the units of other parameters is critical for successful calculations.
Understanding and utilizing the gas constant effectively enables students and chemists to solve problems involving gaseous reactions and enhances insight into reaction dynamics.