Chemical Kinetics Set-6

Q1. For a first-order reaction A → products, $40\%$ of A reacts in $30\ \text{min}$. The half-life $t_{1/2}$ is approximately:

Q2. Initial rate data for the reaction $A + B \to \text{products}$ are: Exp‑1: $[A]_0=0.10\ \text{M},\ [B]_0=0.20\ \text{M},\ \text{rate}=1.6\times10^{-3}\ \text{M s}^{-1}$; Exp‑2: $[A]_0=0.20\ \text{M},\ [B]_0=0.20\ \text{M},\ \text{rate}=3.2\times10^{-3}\ \text{M s}^{-1}$; Exp‑3: $[A]_0=0.10\ \text{M},\ [B]_0=0.40\ \text{M},\ \text{rate}=3.2\times10^{-3}\ \text{M s}^{-1}$. The rate law and value of the rate constant $k$ are:

Q3. Consider the Lindemann mechanism:
$A + M \xrightleftharpoons[k_{-1}]{k_1} A^* + M$ (activation/deactivation) and $A^* \xrightarrow{k_2} \text{products}$. Given $k_1=2.0\times10^3\ \text{M}^{-1}\text{s}^{-1}$, $k_{-1}=4.0\times10^3\ \text{M}^{-1}\text{s}^{-1}$, $k_2=8.0\ \text{s}^{-1}$ and $[M]=1.0\times10^{-3}\ \text{M}$, the observed rate can be written as $r=k_{\mathrm{obs}}[A]$. The value of $k_{\mathrm{obs}}$ (in $\text{s}^{-1}$) is approximately:

Q4. For the reaction $A+B\to\text{products}$ with $r=k[A][B]$, let $[A]_0=0.10\ \text{M}$, $[B]_0=0.30\ \text{M}$ and $k=0.50\ \text{M}^{-1}\text{s}^{-1}$. The time required for $[A]$ to fall to $0.050\ \text{M}$ is approximately:

Q5. Two elementary reactions follow the Arrhenius equation with parameters:
Reaction I: $A_1=10^{3}\ \text{s}^{-1},\ E_{a1}=50\ \text{kJ mol}^{-1}$; Reaction II: $A_2=10^{20}\ \text{s}^{-1},\ E_{a2}=150\ \text{kJ mol}^{-1}$. Which statement correctly describes which reaction is faster at $T=300\ \text{K}$ and at $T=1000\ \text{K}$?

Q6. A reactant $A$ decomposes by first-order kinetics: $A \rightarrow$ products. If $[A]_0 = 0.80\ \mathrm{M}$ and $[A]=0.60\ \mathrm{M}$ after $20\ \mathrm{min}$, what is $[A]$ after $60\ \mathrm{min}$? (Use $[A]_t=[A]_0e^{-kt}$)

Q7. The initial rate data for the reaction $2A + B \rightarrow$ products are:
Experiment 1: $[A]=0.10\ \mathrm{M},\ [B]=0.10\ \mathrm{M},\ \text{rate}=2.0\times10^{-3}\ \mathrm{M\,s^{-1}}$
Experiment 2: $[A]=0.20\ \mathrm{M},\ [B]=0.10\ \mathrm{M},\ \text{rate}=8.0\times10^{-3}\ \mathrm{M\,s^{-1}}$
Experiment 3: $[A]=0.10\ \mathrm{M},\ [B]=0.20\ \mathrm{M},\ \text{rate}=4.0\times10^{-3}\ \mathrm{M\,s^{-1}}$
Which of the following gives the correct rate law and the value of the rate constant $k$?

Q8. For consecutive first-order reactions $A \xrightarrow{k_1} B \xrightarrow{k_2} C$ with $k_1=0.05\ \mathrm{s^{-1}}$ and $k_2=0.02\ \mathrm{s^{-1}}$, the time $t_{\max}$ at which $[B]$ is maximum is given by $t_{\max}=\dfrac{\ln(k_1/k_2)}{k_1-k_2}$. What is $t_{\max}$ (in s)?

Q9. Two different pathways X and Y for the same elementary step have Arrhenius parameters:
Pathway X: $A_X=1.0\times10^{12}\ \mathrm{s^{-1}},\ E_{a,X}=120\ \mathrm{kJ\,mol^{-1}}$
Pathway Y: $A_Y=1.0\times10^{8}\ \mathrm{s^{-1}},\ E_{a,Y}=60\ \mathrm{kJ\,mol^{-1}}$
Which statement is correct regarding their rate constants $k_X$ and $k_Y$ at $T=300\ \mathrm{K}$ and $T=800\ \mathrm{K}$?

Q10. For the reversible first-order reaction $A \xrightleftharpoons[k_{-1}]{k_1} B$ starting with pure $A$ at $t=0$, the concentration $[A](t)$ approaches its equilibrium value $[A]_{\mathrm{eq}}$ as $[A](t)=[A]_{\mathrm{eq}}+([A]_0-[A]_{\mathrm{eq}})e^{-(k_1+k_{-1})t}$. The time $t_{1/2}$ required for $[A](t)$ to reach the midpoint between $[A]_0$ and $[A]_{\mathrm{eq}}$ is:

Q11. A first-order reaction $A\rightarrow$ products has half-life $t_{1/2}=30\ \mathrm{min}$. If the initial concentration is $[A]_0=0.80\ \mathrm{M}$, how long will it take for $[A]$ to reach $0.20\ \mathrm{M}$? (Use $k=\dfrac{\ln 2}{t_{1/2}}$ and $[A]=[A]_0e^{-kt}$.)

Q12. The rate constant for a reaction is $k_1=1.0\times10^{-3}\ \mathrm{s^{-1}}$ at $T_1=300\ \mathrm{K}$ and $k_2=4.0\times10^{-3}\ \mathrm{s^{-1}}$ at $T_2=350\ \mathrm{K}$. Using the Arrhenius equation $k=Ae^{-E_a/RT}$, the activation energy $E_a$ (in $\mathrm{kJ\ mol^{-1}}$) is approximately:

Q13. Consider the mechanism:
(1) $A+B\rightleftharpoons C$ ; (fast equilibrium, $k_1,k_{-1}$)
(2) $C+A\xrightarrow{k_2}$ products ; (slow)
If step (1) is fast equilibrium and step (2) is rate-determining, which expression represents the overall rate law?

Q14. For the reaction $A+B\rightarrow$ products with rate law $\text{rate}=k[A][B]$, initial concentrations are $[A]_0=0.10\ \mathrm{M}$ and $[B]_0=0.30\ \mathrm{M}$ and $k=1.0\ \mathrm{L\,mol^{-1}s^{-1}}$. Using the integrated relation $\dfrac{1}{[B]_0-[A]_0}\ln\!\left(\dfrac{[B][A]_0}{[A][B]_0}\right)=kt$, the time required for $[A]$ to fall to $0.05\ \mathrm{M}$ is closest to:

Q15. Assertion (A): If the observed activation energy for an overall reaction is negative ($E_a<0$), the measured rate constant decreases with increasing temperature.
Reason (R): An apparent negative $E_a$ can arise when a fast exothermic pre-equilibrium produces an intermediate whose equilibrium concentration falls as temperature rises, so the effective rate constant for the overall process decreases with $T$. (Use $k=Ae^{-E_a/RT}$.)