For first order decomposition of $N_2O_5$ at constant volume, $p_i$ is initial pressure and $p_t$ is pressure at any time $t$, during the reaction. Which of the following statement/s is incorrect?

(A) Half life does not depend on initial pressure
(B) Total pressure decreases with time
(C) rate constant will be the same if initial pressure is doubled.
(D) $k=\frac{2.303}{t}\log(\frac{p_t}{p_i})$

Choose the correct answer from the options given below:

(B) and (D) only

The correct answer is Option (4) → (B) and (D) only

Let’s analyze the first-order decomposition of $N_2O_5$. The balanced chemical equation for this reaction is:

$2N_2O_5(g) \rightarrow 2N_2O_4(g) + O_2(g) \quad \text{or} \quad N_2O_5(g) \rightarrow 2NO_2(g) + \frac{1}{2}O_2(g)$

Evaluation of Statements

(A) Half life does not depend on initial pressure: CORRECT

For a first-order reaction, the half-life ($t_{1/2}$) is given by $t_{1/2} = \frac{0.693}{k}$. Since it only depends on the rate constant ($k$) and not on the initial concentration (or initial pressure $P_i$), this statement is true.

(B) Total pressure decreases with time: INCORRECT

Look at the stoichiometry: 2 moles of gas decompose to produce 3 moles of gas (2 moles of $N_2O_4$ + 1 mole of $O_2$). Since the number of moles of gaseous products is greater than the moles of gaseous reactants, the total pressure ($P_t$) increases as the reaction progresses.

(C) Rate constant will be the same if initial pressure is doubled: CORRECT

The rate constant ($k$) is a characteristic of the reaction at a specific temperature. It depends on temperature (Arrhenius equation) but is independent of the initial concentration or pressure of the reactants.

(D) $k = \frac{2.303}{t} \log\left(\frac{P_i}{P_t}\right)$: INCORRECT

For a first-order reaction in terms of pressure, the formula is $k = \frac{2.303}{t} \log\left(\frac{P_i}{P_{N_2O_5}}\right)$, where $P_{N_2O_5}$ is the partial pressure of the reactant at time $t$.

However, the $P_t$ mentioned in the prompt usually refers to Total Pressure. In this decomposition, the total pressure increases, making $P_i < P_t$. The specific integrated rate law for this stoichiometry is actually:

$k = \frac{2.303}{t} \log\left(\frac{P_i}{2P_i - P_t}\right)$

The expression provided in option (D) is mathematically incorrect for this specific reaction.

Conclusion

The incorrect statements are (B) and (D).