Exercises: Rate laws and order

For the gas-phase reaction $2\,\mathrm{NO}\,(g) + \mathrm{O_2}\,(g) \rightarrow 2\,\mathrm{NO_2}\,(g)$, it has been determined experimentally that its rate equation at $300\,\mathrm{K}$ is $v = k\,[\mathrm{NO}]^2\,[\mathrm{O_2}]$. Justify, with reasoning: a) Why the partial orders with respect to each reactant cannot be deduced directly from the stoichiometric coefficients of the overall chemical equation. b) What the overall order of the reaction is and what the units of the rate constant $k$ are, expressing time in seconds and concentrations in $\mathrm{mol\times L^{-1}}$.

The decomposition of the gaseous reactant $\mathrm{A}$ follows zero-order kinetics with respect to its concentration at a constant temperature of $500\,\mathrm{K}$, with a rate constant $k = 4.5\times 10^{-4}\,\mathrm{mol\times L^{-1}\times s^{-1}}$. a) Write the differential rate equation for this process and deduce the units of its rate constant. b) If the initial concentration of $\mathrm{A}$ is tripled, state, with reasoning, how the numerical value of the instantaneous reaction rate changes.

The kinetics of the reaction $\mathrm{CH_3Br}\,(aq) + \mathrm{OH^-}\,(aq) \rightarrow \mathrm{CH_3OH}\,(aq) + \mathrm{Br^-}\,(aq)$ at $298\,\mathrm{K}$ is studied. The process is known to be first order with respect to bromomethane and first order with respect to the hydroxide ion, with an initial rate of $1.2\times 10^{-5}\,\mathrm{mol\times L^{-1}\times s^{-1}}$ when $[\mathrm{CH_3Br}]_0 = 0.020\,\mathrm{mol\times L^{-1}}$ and $[\mathrm{OH^-}]_0 = 0.030\,\mathrm{mol\times L^{-1}}$. a) Write the rate law and calculate the numerical value of the rate constant $k$, stating its units. b) Calculate the rate if $[\mathrm{CH_3Br}]$ is doubled and $[\mathrm{OH^-}]$ is halved.

For the reaction $2\,\mathrm{A}\,(g) + \mathrm{B}\,(g) \rightarrow \mathrm{C}\,(g)$ at $350\,\mathrm{K}$, the initial rate is measured while keeping the concentration $[\mathrm{B}] = 0.40\,\mathrm{mol\times L^{-1}}$ constant and recording the response as $[\mathrm{A}]$ is varied. The graph of the experimental data obtained is shown below:

In the experiment, the pairs of values $([\mathrm{A}] = 0.10\,\mathrm{mol\times L^{-1}};\, v_0 = 0.0020\,\mathrm{mol\times L^{-1}\times s^{-1}})$, $([\mathrm{A}] = 0.20\,\mathrm{mol\times L^{-1}};\, v_0 = 0.0080\,\mathrm{mol\times L^{-1}\times s^{-1}})$ and $([\mathrm{A}] = 0.30\,\mathrm{mol\times L^{-1}};\, v_0 = 0.0180\,\mathrm{mol\times L^{-1}\times s^{-1}})$ were obtained. a) From the graph or the data provided, deduce the partial order with respect to $\mathrm{A}$. b) If an additional experiment at $[\mathrm{A}] = 0.10\,\mathrm{mol\times L^{-1}}$ and $[\mathrm{B}] = 0.80\,\mathrm{mol\times L^{-1}}$ gives $v_0 = 0.0040\,\mathrm{mol\times L^{-1}\times s^{-1}}$, obtain the complete rate equation and calculate the constant $k$.

For the reaction in aqueous solution $\mathrm{S_2O_8^{2-}} + 2\,\mathrm{I^-} \rightarrow 2\,\mathrm{SO_4^{2-}} + \mathrm{I_2}$, the following initial rate data were obtained at $298\,\mathrm{K}$:

• Experiment 1: $[\mathrm{S_2O_8^{2-}}]_0 = 0.080\,\mathrm{mol\times L^{-1}}$, $[\mathrm{I^-}]_0 = 0.034\,\mathrm{mol\times L^{-1}}$, $v_0 = 2.2\times 10^{-4}\,\mathrm{mol\times L^{-1}\times s^{-1}}$.

• Experiment 2: $[\mathrm{S_2O_8^{2-}}]_0 = 0.080\,\mathrm{mol\times L^{-1}}$, $[\mathrm{I^-}]_0 = 0.017\,\mathrm{mol\times L^{-1}}$, $v_0 = 1.1\times 10^{-4}\,\mathrm{mol\times L^{-1}\times s^{-1}}$.

• Experiment 3: $[\mathrm{S_2O_8^{2-}}]_0 = 0.160\,\mathrm{mol\times L^{-1}}$, $[\mathrm{I^-}]_0 = 0.017\,\mathrm{mol\times L^{-1}}$, $v_0 = 2.2\times 10^{-4}\,\mathrm{mol\times L^{-1}\times s^{-1}}$.

1. a) Determine the partial orders with respect to each reactant and the overall order. b) Calculate the value of the rate constant with its corresponding units and predict the initial rate if $[\mathrm{S_2O_8^{2-}}]_0 = 0.040\,\mathrm{mol\times L^{-1}}$ and $[\mathrm{I^-}]_0 = 0.050\,\mathrm{mol\times L^{-1}}$.

Consider the gas-phase synthesis reaction $\mathrm{X}\,(g) + 2\,\mathrm{Y}\,(g) \rightarrow \mathrm{Z}\,(g)$, whose empirical rate equation is $v = k\,[\mathrm{X}]\,[\mathrm{Y}]^2$. In a fixed-volume reactor $V_1$ at temperature $T$, the reaction proceeds with an initial rate $v_1$. a) Explain quantitatively how the initial rate changes if, keeping the temperature and volume constant, the amount of $\mathrm{X}$ is tripled and the amount of $\mathrm{Y}$ is halved. b) If the initial mixture is transferred to a vessel of volume $V_2 = V_1 / 3$ at the same temperature, calculate the factor by which the reaction rate is multiplied relative to the original rate in $V_1$.

The reaction $2\,\mathrm{NO}\,(g) + 2\,\mathrm{H_2}\,(g) \rightarrow \mathrm{N_2}\,(g) + 2\,\mathrm{H_2O}\,(g)$ at $1090\,\mathrm{K}$ is studied kinetically using the method of initial rates. The laboratory data, with the experimental measurement uncertainty, are given below:

• Experiment 1: $[\mathrm{NO}]_0 = (4.0 \pm 0.1)\times 10^{-3}\,\mathrm{mol\times L^{-1}}$, $[\mathrm{H_2}]_0 = (1.5 \pm 0.1)\times 10^{-3}\,\mathrm{mol\times L^{-1}}$, $v_0 = (3.0 \pm 0.1)\times 10^{-5}\,\mathrm{mol\times L^{-1}\times s^{-1}}$.

• Experiment 2: $[\mathrm{NO}]_0 = (4.0 \pm 0.1)\times 10^{-3}\,\mathrm{mol\times L^{-1}}$, $[\mathrm{H_2}]_0 = (3.0 \pm 0.1)\times 10^{-3}\,\mathrm{mol\times L^{-1}}$, $v_0 = (6.0 \pm 0.2)\times 10^{-5}\,\mathrm{mol\times L^{-1}\times s^{-1}}$.

• Experiment 3: $[\mathrm{NO}]_0 = (8.0 \pm 0.1)\times 10^{-3}\,\mathrm{mol\times L^{-1}}$, $[\mathrm{H_2}]_0 = (1.5 \pm 0.1)\times 10^{-3}\,\mathrm{mol\times L^{-1}}$, $v_0 = (1.2 \pm 0.1)\times 10^{-4}\,\mathrm{mol\times L^{-1}\times s^{-1}}$.

1. a) Deduce the rate law, the overall order and the mean value of $k$ with its units. b) The following mechanism is proposed:

Step 1 (fast, equilibrium): $2\,\mathrm{NO} \rightleftharpoons \mathrm{N_2O_2}$

Step 2 (slow): $\mathrm{N_2O_2} + \mathrm{H_2} \rightarrow \mathrm{N_2O} + \mathrm{H_2O}$

Step 3 (fast): $\mathrm{N_2O} + \mathrm{H_2} \rightarrow \mathrm{N_2} + \mathrm{H_2O}$

Justify whether this mechanism is kinetically consistent with the rate law deduced.

The gas-phase thermal decomposition reaction $2\,\mathrm{N_2O_5}\,(g) \rightarrow 4\,\mathrm{NO_2}\,(g) + \mathrm{O_2}\,(g)$ at $318\,\mathrm{K}$ is analysed. The plot of $\ln v_0$ against $\ln [\mathrm{N_2O_5}]_0$ for four independent experiments is observed to give a straight line of gradient $1.00 \pm 0.02$ with a y-intercept equal to $-6,215$. a) Determine, with justification, the order of reaction and the value of the rate constant $k$ with its corresponding units. b) Consider a proposed mechanism consisting of three elementary steps:

1) $\mathrm{N_2O_5} \rightleftharpoons \mathrm{NO_2} + \mathrm{NO_3}$ (fast equilibrium, constants $k_1$ and $k_{-1}$)

2) $\mathrm{NO_2} + \mathrm{NO_3} \rightarrow \mathrm{NO_2} + \mathrm{O_2} + \mathrm{NO}$ (slow step, constant $k_2$)

3) $\mathrm{NO} + \mathrm{NO_3} \rightarrow 2\,\mathrm{NO_2}$ (fast step, constant $k_3$)

Deduce analytically the expression for the rate law predicted by this mechanism and check whether it reproduces the experimental order. The logarithms are applied to the numerical values of the rates in $\mathrm{mol}\times\mathrm{L}^{-1}\times\mathrm{s}^{-1}$ and of the concentrations in $\mathrm{mol}\times\mathrm{L}^{-1}$, taking those units as references.

The kinetics of the reaction $A + B \rightarrow Productos$ at $300\,\mathrm{K}$ is studied under reactant-isolation (pseudo-order) conditions. In a first series with a large excess of $B$ ($[B]_0 = 2.0\,\mathrm{mol\times L^{-1}}$ constant), the initial rate is found to be proportional to $[A]_0^2$. In a second series with a large excess of $A$ ($[A]_0 = 1.5\,\mathrm{mol\times L^{-1}}$ constant), the initial rate does not change when $[B]_0$ is doubled or tripled. a) Formulate the complete differential rate law, justifying the partial orders and the overall order, and state the units of the rate constant. b) If in an experiment with $[A]_0 = 0.10\,\mathrm{mol\times L^{-1}}$ and $[B]_0 = 0.20\,\mathrm{mol\times L^{-1}}$ the initial rate is $v_0 = 3.5\times 10^{-3}\,\mathrm{mol\times L^{-1}\times s^{-1}}$, calculate the rate that would be recorded if the solution is diluted to twice its volume by adding pure solvent at the same temperature.