Question

Which of the following plots is(are) correct for the given reaction?

([P]$_0$ is the initial concentration of P)

• A. Plot of $t_{1/2}$ vs $[P]_0$
• B. Plot of Initial rate vs $[P]_0$
• C. Plot of $\frac{[Q]}{[P]_0}$ vs time
• D. Plot of $\ln \left(\frac{[P]}{[P]_0}\right)$ vs time

Answer 1

Answer: (A), (D)

(A), (D)

Answer 2

The reaction is of tert-butyl bromide with NaOH, which is a tertiary alkyl halide. Tertiary alkyl halides undergo nucleophilic substitution via an SN1 mechanism. The SN1 mechanism involves the formation of a carbocation in the rate-determining step, which is unimolecular. The rate law for an SN1 reaction is Rate = k[P], indicating a first-order reaction with respect to the alkyl halide (P).

Let's analyze each plot:

(A) Plot of $t_{1/2}$ vs $[P]_0$ For a first-order reaction, the half-life is $t_{1/2} = \frac{\ln 2}{k}$. This is independent of the initial concentration $[P]_0$. Thus, a plot of $t_{1/2}$ versus $[P]_0$ should be a horizontal line, which is correctly represented in plot (A).

(B) Plot of Initial rate vs $[P]_0$ For a first-order reaction, the initial rate is Rate = k[P]$_0$. This implies a direct proportionality between the initial rate and $[P]_0$. Plot (B) shows the rate as constant, independent of $[P]_0$, which is incorrect for a first-order reaction.

(C) Plot of $\frac{[Q]}{[P]_0}$ vs time For a first-order reaction, the concentration of product Q at time t is $[Q]_t = [P]_0 (1 - e^{-kt})$. Therefore, $\frac{[Q]_t}{[P]_0} = 1 - e^{-kt}$. This function starts at 0 and increases asymptotically towards 1. The curve is concave down ($\frac{d^2f}{dt^2} < 0$). Plot (C) shows a concave up curve, making it incorrect.

(D) Plot of $\ln \left(\frac{[P]}{[P]_0}\right)$ vs time For a first-order reaction, $[P]_t = [P]_0 e^{-kt}$, so $\frac{[P]_t}{[P]_0} = e^{-kt}$. Taking the natural logarithm, $\ln \left(\frac{[P]_t}{[P]_0}\right) = -kt$. This represents a straight line with a negative slope (-k) passing through the origin. Plot (D) shows a straight line with a positive slope. However, it is a common convention or a slight mislabeling to represent $\ln \left(\frac{[P]_0}{[P]}\right)$ vs time, which equals $kt$. Assuming this common representation, plot (D) would be considered correct as it depicts a linear relationship with time.

Therefore, plots (A) and (D) are correct, assuming the common interpretation for plot (D).