Chemical Kinetics Set-5

Q1. For the first-order decomposition $A \rightarrow$ products with rate constant $k = 0.693\ \mathrm{s^{-1}}$ and initial concentration $[A]_0 = 0.40\ \mathrm{M}$, the time required for $[A]$ to fall to $0.10\ \mathrm{M}$ is:

Q2. A reaction follows rate law $-\dfrac{d[A]}{dt}=k[A]^2$. In one experiment with $[A]_0=0.10\ \mathrm{M}$ the concentration falls to $0.05\ \mathrm{M}$ in $100\ \mathrm{s}$. If the reaction is repeated with $[A]_0=0.20\ \mathrm{M}$, the time required for $[A]$ to fall to $0.10\ \mathrm{M}$ is:

Q3. Consider the mechanism
(1) $A + B \xrightleftharpoons[k_{-1}]{k_1} C\quad$(fast equilibrium)
(2) $C + B \xrightarrow{k_2} P\quad$(slow).
Using the pre-equilibrium approximation, which of the following rate expressions for formation of $P$ is predicted?

Q4. For the bimolecular reaction with rate law $r=k[A][B]$, $B$ is kept in large excess so that pseudo-first-order kinetics applies. When $[B]=0.10\ \mathrm{M}$ the half-life of $A$ is $200\ \mathrm{s}$; when $[B]=0.20\ \mathrm{M}$ the half-life of $A$ is $100\ \mathrm{s}$. The true bimolecular rate constant $k$ is:

Q5. A first-order reaction has half-life $t_{1/2}=100\ \mathrm{s}$ at $300\ \mathrm{K}$ and $t_{1/2}=25\ \mathrm{s}$ at $310\ \mathrm{K}$. Assuming Arrhenius behaviour $k=A\exp\!\left(-\dfrac{E_a}{RT}\right)$, estimate the activation energy $E_a$ (in kJ mol$^{-1}$). Use $R=8.314\ \mathrm{J\,mol^{-1}\,K^{-1}}$.

Q6. A first-order reaction A → products has a half-life $t_{1/2}=20\ \text{min}$. If the initial concentration is $[A]_0=0.80\ \text{M}$, what is $[A]$ after $30\ \text{min}$? (Use $[A]_t=[A]_0e^{-kt}$ and $t_{1/2}=\dfrac{\ln2}{k}$.)

Q7. Initial rate data for a reaction are: Exp(1) $[A]=0.10\ \mathrm{M},\ [B]=0.10\ \mathrm{M},\ r=2.0\times10^{-4}\ \mathrm{M s}^{-1}$; Exp(2) $[A]=0.20\ \mathrm{M},\ [B]=0.10\ \mathrm{M},\ r=8.0\times10^{-4}\ \mathrm{M s}^{-1}$; Exp(3) $[A]=0.10\ \mathrm{M},\ [B]=0.20\ \mathrm{M},\ r=8.0\times10^{-4}\ \mathrm{M s}^{-1}$. If the rate law is $r=k[A]^m[B]^n$, determine $m,n$ and the initial rate when $[A]=0.020\ \mathrm{M},\ [B]=0.10\ \mathrm{M}$.

Q8. For the bimolecular reaction $A+B\rightarrow$ products the rate law is $r=k[A][B]$ with $k=20\ \mathrm{M}^{-1}\text{s}^{-1}$. If $[A]_0=0.010\ \mathrm{M}$ and $[B]_0=0.50\ \mathrm{M}$ and $[B]$ is in large excess (so $[B]\approx\text{constant}$), how long will it take for $[A]$ to decrease to $25\%$ of its initial value? (Treat this as pseudo-first-order with $k_{\text{obs}}=k[B]$.)

Q9. Consider the mechanism: (1) $A\xrightarrow{k_1}I$, (2) $I+B\xrightarrow{k_2}P$, (3) $I\xrightarrow{k_{-1}}A$. Given $k_1=2.0\ \text{s}^{-1}$, $k_2=1.0\times10^{6}\ \text{M}^{-1}\text{s}^{-1}$, $k_{-1}=100\ \text{s}^{-1}$ and initial concentrations $[A]_0=1.0\times10^{-4}\ \text{M},\ [B]_0=1.0\times10^{-3}\ \text{M}$, use the steady-state approximation for $I$ (with $[I]\approx\dfrac{k_1[A]}{k_2[B]+k_{-1}}$) to calculate the initial rate $r_0=\dfrac{d[P]}{dt}\big|_{t=0}=k_2[I][B]$.

Q10. A radical chain mechanism proceeds by: Initiation $I\rightarrow 2R\cdot$ at constant rate $r_i$, Propagation $R\cdot + B\xrightarrow{k_p}P + R\cdot$, Termination $2R\cdot\xrightarrow{k_t}D$. Assuming steady state for $[R\cdot]$ and that initiation yields two radicals per initiation, one obtains $[R\cdot]=\sqrt{r_i/k_t}$ and $r_p=k_p[B][R\cdot]$. Given $r_i=1.0\times10^{-8}\ \mathrm{M s}^{-1}$, $k_p=2.0\times10^{7}\ \mathrm{M}^{-1}\text{s}^{-1}$, $k_t=4.0\times10^{9}\ \mathrm{M}^{-1}\text{s}^{-1}$ and $[B]=0.10\ \mathrm{M}$, what is the rate of product formation $r_p$?

Q11. For a first-order reaction the half-life is $t_{1/2}$. The time required for $75\%$ of the reactant to be consumed is

Q12. A reaction $2\mathrm{A}\rightarrow$ products follows second-order kinetics: $\dfrac{1}{[\mathrm{A}]} = kt + \dfrac{1}{[\mathrm{A}]_0}$. If $[\mathrm{A}]_0=0.20\ \mathrm{M}$ and after $50\ \mathrm{s}$ the concentration $[\mathrm{A}]$ is $0.05\ \mathrm{M}$, the half-life $t_{1/2}$ of the reaction is approximately

Q13. The initial rates for the reaction between A and B were measured as follows: Exp I: $[A]=0.10\ \mathrm{M},\ [B]=0.10\ \mathrm{M},\ \text{rate}=2.0\times10^{-3}\ \mathrm{M\,s^{-1}}$; Exp II: $[A]=0.20\ \mathrm{M},\ [B]=0.10\ \mathrm{M},\ \text{rate}=4.0\times10^{-3}\ \mathrm{M\,s^{-1}}$; Exp III: $[A]=0.20\ \mathrm{M},\ [B]=0.20\ \mathrm{M},\ \text{rate}=1.6\times10^{-2}\ \mathrm{M\,s^{-1}}$. The rate law and rate constant $k$ (with units) that best fit these data are

Q14. For consecutive first-order reactions $A\xrightarrow{k_1}B\xrightarrow{k_2}C$ with $k_1\neq k_2$ and initial concentration $[A]_0$, the time $t$ at which the concentration of intermediate $B$ is maximum equals

Q15. Consider the mechanism: (1) $A+B \xrightleftharpoons[k_{-1}]{k_1} I$ (fast reversible), (2) $I+B \xrightarrow{k_2} P$ (slow). Using the steady-state approximation for intermediate $I$, the rate of formation of $P$ is