AIIMS 2004 Chemistry Rate Laws MCQ Question

To solve this problem, we need to analyze the kinetics of the given reaction:

$$\text{N}_2\text{O}_5(g) \rightarrow 2\text{NO}_2(g) + \frac{1}{2}\text{O}_2(g$$

Understanding the Reaction Order

This reaction is a first-order reaction with respect to N₂O₅. In first-order reactions, the rate of reaction is proportional to the concentration of one reactant. The general form of the rate law for a first-order reaction can be expressed as:

$$\frac{d[N_2O_5]}{dt} = -k[N_2O_5]$$

Integrating the Rate Law

To find the relationship between the concentration of N₂O₅ at any time $t$ and its initial concentration $[N_2O_5]_0$, we integrate the rate law. The integration of the equation yields:

1. Rearranging the rate law gives us:

$$\frac{d[N_2O_5]}{[N_2O_5]} = -k \, dt$$

1. Integrating from $[N_2O_5]_0$ at $t = 0$ to $[N_2O_5]$ at time $t$:

$$\int_{[N_2O_5]_0}^{[N_2O_5]} \frac{d[N_2O_5]}{[N_2O_5]} = -k \int_0^t dt$$

1. This results in:

$$\ln[N_2O_5] - \ln[N_2O_5]_0 = -kt$$

1. Rearranging this gives us:

$$\ln[N_2O_5] = \ln[N_2O_5]_0 - kt$$

This equation is in the form of option D:

Correct Answer:

$$\ln[N_2O_5] = \ln[N_2O_5]_0 - kt$$

Explanation of the Other Options

Now, let’s discuss why the other options are incorrect:

Option A:

$$[N_2O_5] = [N_2O_5]_0 + kt$$

• This expression suggests a linear increase in concentration over time, which is incorrect for a first-order reaction. In first-order kinetics, the concentration decreases exponentially, not linearly.

Option B:

$$[N_2O_5]_0 = [N_2O_5]_0 e^{-kt}$$

• This option is misleading; it implies that the initial concentration is equal to itself multiplied by an exponential decay term. While it might appear valid, it is not the correct form of the equation. The correct relationship should involve the concentration at time $t$ rather than the initial concentration equating to a function of itself.

Option C:

$$\log[N_2O_5] = \log[N_2O_5]_0 + kt$$

• This is incorrect because it indicates that the logarithm of the concentration increases linearly with time, which is characteristic of zero-order reactions. For first-order kinetics, the natural logarithm of the concentration decreases linearly with time.

Conclusion

In summary, the correct answer is D:

$$\ln[N_2O_5] = \ln[N_2O_5]_0 - kt$$

This equation accurately describes the relationship between the concentration of N₂O₅ over time for a first-order reaction, while the other options do not reflect the behavior of first-order kinetics.