AIIMS 2005 Chemistry Equilibrium Constants MCQ Question

To determine the value of $K_p$ for the reaction

$2\text{NOCl} (g) \rightleftharpoons 2\text{NO} (g) + \text{Cl}_2 (g),$

given that $K_c = 3 \times 10^{-6} \, \text{L} \cdot \text{mol}^{-1}$ at 427°C, we can use the relationship between $K_c$ and $K_p$.

Relationship Between $K_c$ and $K_p$

The relationship between the equilibrium constants $K_c$ and $K_p$ is given by the equation:

$K_p = K_c \left(RT\right)^{\Delta n}$

where:

• $R$ is the universal gas constant, which is approximately $0.0821 \, \text{L} \cdot \text{atm} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}$,
• $T$ is the temperature in Kelvin,
• $\Delta n$ is the change in the number of moles of gas between products and reactants.

Calculation of $\Delta n$

For the reaction:

• Reactants: $2 \, \text{NOCl}$ (which is 2 moles),
• Products: $2 \, \text{NO} + 1 \, \text{Cl}_2$ (which is 3 moles).

Thus,

$\Delta n = \text{moles of products} - \text{moles of reactants} = 3 - 2 = 1.$

Temperature in Kelvin

The temperature $T$ is given as 427°C. We convert this to Kelvin:

$T = 427 + 273.15 = 700.15 \, \text{K}.$

Substituting Values

Now we can substitute the values into the equation for $K_p$:

1. Calculate $RT$:

$RT = (0.0821 \, \text{L} \cdot \text{atm} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}) \times (700.15 \, \text{K}) \approx 57.5 \, \text{L} \cdot \text{atm} \cdot \text{mol}^{-1}.$

1. Substitute into the $K_p$ formula:

$K_p = K_c \cdot (RT)^{\Delta n} = (3 \times 10^{-6} \, \text{L} \cdot \text{mol}^{-1}) \cdot (57.5)^{1}.$

1. Perform the calculation:

$K_p = 3 \times 10^{-6} \cdot 57.5 \approx 1.725 \times 10^{-5} \, \text{atm}.$

Conclusion

Rounding this to two significant figures gives us $K_p \approx 1.75 \times 10^{-5} \, \text{atm}$.

Thus, the closest value from the options provided is:

D) 1.75 x 10⁻⁴.

Why Other Options Are Incorrect

• A) 7.50 x 10⁻⁵: This value is significantly higher than the calculated value.
• B) 2.50 x 10⁻⁵: This value is also higher and does not correspond to the calculated $K_p$.
• C) 2.50 x 10⁻⁴: This is much larger than what we calculated and, hence, incorrect.

In summary, the correct answer is D) $1.75 \times 10^{-5}$ which matches our calculation, while the other options do not reflect the correct relationship between $K_c$ and $K_p$ for the given reaction.