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Chapter 3 Chemical Kinetics
Rate Of A Chemical Reaction
Chemical kinetics studies the rates of chemical reactions and the factors influencing them. Reaction rates are measured by the change in concentration of reactants or products per unit time. The average rate ($r_{avg}$) is calculated over a time interval ($Δt$): $r_{avg} = - \frac{\Delta[Reactant]}{\Delta t}$ or $r_{avg} = + \frac{\Delta[Product]}{\Delta t}$. The instantaneous rate ($r_{inst}$) is the rate at a specific moment, determined by the slope of the tangent to the concentration-time curve: $r_{inst} = - \frac{d[Reactant]}{dt}$ or $r_{inst} = + \frac{d[Product]}{dt}$. For reactions with different stoichiometric coefficients (e.g., $aA + bB \rightarrow cC + dD$), the rate is expressed as $r = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = +\frac{1}{c}\frac{d[C]}{dt} = +\frac{1}{d}\frac{d[D]}{dt}$. Units of rate are typically concentration/time (e.g., mol L⁻¹ s⁻¹).
Factors Influencing Rate Of A Reaction
Dependence Of Rate On Concentration
The rate of most reactions increases with an increase in the concentration of reactants, as there are more molecules available to collide and react. This relationship is described by the rate law.
Rate Expression And Rate Constant
The rate law expresses the rate of a reaction as a function of reactant concentrations. For a general reaction $aA + bB \rightarrow cC + dD$, the rate law is typically written as $Rate = k[A]^x[B]^y$. Here, $k$ is the rate constant (or specific rate), a proportionality constant that is specific to the reaction and temperature. The exponents $x$ and $y$ represent the order of the reaction with respect to reactants A and B, respectively, and must be determined experimentally, not from stoichiometric coefficients.
Order Of A Reaction
The order of a reaction is the sum of the exponents ($x + y$) of the concentration terms in the rate law. It indicates how the rate changes with concentration and can be zero, a positive integer, a negative integer, or a fraction. The order must be determined experimentally.
Molecularity Of A Reaction
Molecularity applies only to elementary reactions (occurring in a single step). It is the total number of reactant molecules (atoms, ions, or molecules) that must collide simultaneously for the reaction to occur. Molecularity can only be a positive integer (1, 2, or rarely 3). Reactions involving more than three molecules colliding simultaneously are improbable.
Complex reactions occur in multiple steps. The overall rate of a complex reaction is determined by its slowest step, known as the rate-determining step. The molecularity of this slowest step usually corresponds to the experimental order of the reaction.
Key Differences: Order is experimental, can be fractional or zero, and applies to overall reactions. Molecularity is theoretical, must be a positive integer, and applies only to elementary steps.
Integrated Rate Equations
Integrated rate equations relate concentration directly to time, allowing calculation of rate constants from experimental data without needing instantaneous rates. They are derived by integrating the differential rate laws.
Zero Order Reactions
For a zero-order reaction ($Rate = k$), the rate is independent of reactant concentration. The integrated rate law is $[R]_t = [R]_0 - kt$, where $[R]_t$ is the concentration at time $t$, $[R]_0$ is the initial concentration, and $k$ is the rate constant. A plot of $[R]$ vs. $t$ yields a straight line with slope $-k$ and intercept $[R]_0$. The half-life ($t_{1/2}$) is $t_{1/2} = [R]_0 / (2k)$, meaning it is directly proportional to the initial concentration.
First Order Reactions
For a first-order reaction ($Rate = k[R]$), the rate is directly proportional to the reactant concentration. The integrated rate law is $ln[R]_t = -kt + ln[R]_0$, or $\frac{[R]_0}{[R]_t} = e^{kt}$, or $log\frac{[R]_0}{[R]_t} = \frac{kt}{2.303}$. Plots of $ln[R]$ vs. $t$ or $log\frac{[R]_0}{[R]_t}$ vs. $t$ yield straight lines. The half-life ($t_{1/2} = \frac{0.693}{k}$) is independent of initial concentration.
Half-Life Of A Reaction
The half-life ($t_{1/2}$) is the time required for the concentration of a reactant to decrease to half its initial value. For zero-order reactions, $t_{1/2} \propto [R]_0$. For first-order reactions, $t_{1/2}$ is constant and independent of $[R]_0$. Many radioactive decay processes follow first-order kinetics.
Temperature Dependence Of The Rate Of A Reaction
Reaction rates generally increase with temperature. The Arrhenius equation quantifies this relationship: $k = Ae^{-E_a/RT}$, where $k$ is the rate constant, $A$ is the pre-exponential factor (frequency factor), $E_a$ is the activation energy, $R$ is the gas constant, and $T$ is the absolute temperature. A plot of $ln k$ vs. $1/T$ yields a straight line with slope $-E_a/R$. Lowering the activation energy or increasing the temperature increases the rate constant.
Effect Of Catalyst
A catalyst increases the rate of a reaction without being consumed chemically. It achieves this by providing an alternative reaction pathway with a lower activation energy ($E_a$), thus increasing the fraction of effective collisions. Catalysts do not affect the overall thermodynamics ($ΔG$) or equilibrium constant ($K$) but help the reaction reach equilibrium faster.
Collision Theory Of Chemical Reactions
Collision theory states that for a reaction to occur, reactant molecules must collide with sufficient kinetic energy (threshold energy, $E_a$) and proper orientation. The rate is proportional to the collision frequency ($Z_{AB}$) and the fraction of effective collisions ($e^{-E_a/RT}$). The Arrhenius equation can be modified to include a steric factor ($P$) to account for the effect of orientation: $Rate = P \cdot Z_{AB} \cdot e^{-E_a/RT}$. This factor accounts for the fact that not all collisions, even with sufficient energy, lead to product formation.
Intext Questions
Question 3.1. For the reaction $R → P$, the concentration of a reactant changes from 0.03M to 0.02M in 25 minutes. Calculate the average rate of reaction using units of time both in minutes and seconds.
Answer:
Question 3.2. In a reaction, 2A $→$ Products, the concentration of A decreases from 0.5 $mol L^{–1}$ to 0.4 $mol L^{–1}$ in 10 minutes. Calculate the rate during this interval?
Answer:
Question 3.3. For a reaction, A + B $→$ Product; the rate law is given by, $r = k[A]^{1/2}[B]^2$. What is the order of the reaction?
Answer:
Question 3.4. The conversion of molecules X to Y follows second order kinetics. If concentration of X is increased to three times how will it affect the rate of formation of Y ?
Answer:
Question 3.5. A first order reaction has a rate constant $1.15 \times 10^{-3} s^{-1}$. How long will 5 g of this reactant take to reduce to 3 g?
Answer:
Question 3.6. Time required to decompose $SO_2Cl_2$ to half of its initial amount is 60 minutes. If the decomposition is a first order reaction, calculate the rate constant of the reaction.
Answer:
Question 3.7. What will be the effect of temperature on rate constant ?
Answer:
Question 3.8. The rate of the chemical reaction doubles for an increase of 10K in absolute temperature from 298K. Calculate $E_a$.
Answer:
Question 3.9. The activation energy for the reaction
$2 \text{ HI}(g) \rightarrow H_2(g) + I_2(g)$
is 209.5 kJ mol$^{–1}$ at 581K. Calculate the fraction of molecules of reactants having energy equal to or greater than activation energy?
Answer:
Exercises
Question 3.1. From the rate expression for the following reactions, determine their order of reaction and the dimensions of the rate constants.
(i) $3NO(g) \rightarrow N_2O (g)$ Rate = $k[NO]^2$
(ii) $H_2O_2 (aq) + 3I^-(aq) + 2H^+ \rightarrow 2H_2O (l) + I_3^-$ Rate = $k[H_2O_2][I^-]$
(iii) $CH_3CHO(g) \rightarrow CH_4(g) + CO(g)$ Rate = $k[CH_3CHO]^{3/2}$
(iv) $C_2H_5Cl(g) \rightarrow C_2H_4(g) + HCl(g)$ Rate = $k[C_2H_5Cl]$
Answer:
Question 3.2. For the reaction:
$2A + B \rightarrow A_2B$
the rate = $k[A][B]^2$ with $k = 2.0 \times 10^{–6} mol^{–2} L^2 s^{–1}$. Calculate the initial rate of the reaction when $[A] = 0.1 \text{ mol L}^{–1}$, $[B] = 0.2 \text{ mol L}^{–1}$. Calculate the rate of reaction after [A] is reduced to $0.06 \text{ mol L}^{–1}$.
Answer:
Question 3.3. The decomposition of $NH_3$ on platinum surface is zero order reaction. What are the rates of production of $N_2$ and $H_2$ if $k = 2.5 \times 10^{–4} mol^{–1} L s^{–1}$?
Answer:
Question 3.4. The decomposition of dimethyl ether leads to the formation of $CH_4$, $H_2$ and CO and the reaction rate is given by
Rate = $k [CH_3OCH_3]^{3/2}$
The rate of reaction is followed by increase in pressure in a closed vessel, so the rate can also be expressed in terms of the partial pressure of dimethyl ether, i.e.,
Rate = $k(p_{CH_3OCH_3})^{3/2}$
If the pressure is measured in bar and time in minutes, then what are the units of rate and rate constants?
Answer:
Question 3.5. Mention the factors that affect the rate of a chemical reaction.
Answer:
Question 3.6. A reaction is second order with respect to a reactant. How is the rate of reaction affected if the concentration of the reactant is
(i) doubled
(ii) reduced to half ?
Answer:
Question 3.7. What is the effect of temperature on the rate constant of a reaction? How can this effect of temperature on rate constant be represented quantitatively?
Answer:
Question 3.8. In a pseudo first order reaction in water, the following results were obtained:
| t/s | [A]/ mol L$^{–1}$ |
|---|---|
| 0 | 0.55 |
| 30 | 0.31 |
| 60 | 0.17 |
| 90 | 0.085 |
Calculate the average rate of reaction between the time interval 30 to 60 seconds.
Answer:
Question 3.9. A reaction is first order in A and second order in B.
(i) Write the differential rate equation.
(ii) How is the rate affected on increasing the concentration of B three times?
(iii) How is the rate affected when the concentrations of both A and B are doubled?
Answer:
Question 3.10. In a reaction between A and B, the initial rate of reaction ($r_0$) was measured for different initial concentrations of A and B as given below:
| A/ mol L$^{–1}$ | B/ mol L$^{–1}$ | $r_0$/mol L$^{–1}$s$^{–1}$ |
|---|---|---|
| 0.20 | 0.30 | $5.07 \times 10^{–5}$ |
| 0.20 | 0.10 | $5.07 \times 10^{–5}$ |
| 0.40 | 0.05 | $1.43 \times 10^{–4}$ |
What is the order of the reaction with respect to A and B?
Answer:
Question 3.11. The following results have been obtained during the kinetic studies of the reaction:
$2A + B \rightarrow C + D$
| Experiment | [A]/mol L$^{–1}$ | [B]/mol L$^{–1}$ | Initial rate of formation of D/mol L$^{–1}$ min$^{–1}$ |
|---|---|---|---|
| I | 0.1 | 0.1 | $6.0 \times 10^{–3}$ |
| II | 0.3 | 0.2 | $7.2 \times 10^{–2}$ |
| III | 0.3 | 0.4 | $2.88 \times 10^{–1}$ |
| IV | 0.4 | 0.1 | $2.40 \times 10^{–2}$ |
Determine the rate law and the rate constant for the reaction.
Answer:
Question 3.12. The reaction between A and B is first order with respect to A and zero order with respect to B. Fill in the blanks in the following table:
| Experiment | [A]/ mol L$^{–1}$ | [B]/ mol L$^{–1}$ | Initial rate/mol L$^{–1}$ min$^{–1}$ |
|---|---|---|---|
| I | 0.1 | 0.1 | $2.0 \times 10^{–2}$ |
| II | – | 0.2 | $4.0 \times 10^{–2}$ |
| III | 0.4 | 0.4 | – |
| IV | – | 0.2 | $2.0 \times 10^{–2}$ |
Answer:
Question 3.13. Calculate the half-life of a first order reaction from their rate constants given below:
(i) 200 s$^{–1}$
(ii) 2 min$^{–1}$
(iii) 4 years$^{–1}$
Answer:
Question 3.14. The half-life for radioactive decay of $^{14}C$ is 5730 years. An archaeological artifact containing wood had only 80% of the $^{14}C$ found in a living tree. Estimate the age of the sample.
Answer:
Question 3.15. The experimental data for decomposition of $N_2O_5$
$[2N_2O_5 \rightarrow 4NO_2 + O_2]$
in gas phase at 318K are given below:
| t/s | $10^2 \times [N_2O_5] / mol L^{–1}$ |
|---|---|
| 0 | 1.63 |
| 400 | 1.36 |
| 800 | 1.14 |
| 1200 | 0.93 |
| 1600 | 0.78 |
| 2000 | 0.64 |
| 2400 | 0.53 |
| 2800 | 0.43 |
| 3200 | 0.35 |
(i) Plot $[N_2O_5]$ against t.
(ii) Find the half-life period for the reaction.
(iii) Draw a graph between $\log[N_2O_5]$ and t.
(iv) What is the rate law ?
(v) Calculate the rate constant.
(vi) Calculate the half-life period from k and compare it with (ii).
Answer:
Question 3.16. The rate constant for a first order reaction is 60 s$^{–1}$. How much time will it take to reduce the initial concentration of the reactant to its 1/16th value?
Answer:
Question 3.17. During nuclear explosion, one of the products is $^{90}Sr$ with half-life of 28.1 years. If 1mg of $^{90}Sr$ was absorbed in the bones of a newly born baby instead of calcium, how much of it will remain after 10 years and 60 years if it is not lost metabolically.
Answer:
Question 3.18. For a first order reaction, show that time required for 99% completion is twice the time required for the completion of 90% of reaction.
Answer:
Question 3.19. A first order reaction takes 40 min for 30% decomposition. Calculate $t_{1/2}$.
Answer:
Question 3.20. For the decomposition of azoisopropane to hexane and nitrogen at 543 K, the following data are obtained.
| t (sec) | P(mm of Hg) |
|---|---|
| 0 | 35.0 |
| 360 | 54.0 |
| 720 | 63.0 |
Calculate the rate constant.
Answer:
Question 3.21. The following data were obtained during the first order thermal decomposition of $SO_2Cl_2$ at a constant volume.
$SO_2Cl_2(g) \rightarrow SO_2(g) + Cl_2(g)$
| Experiment | Time/s$^{–1}$ | Total pressure/atm |
|---|---|---|
| 1 | 0 | 0.5 |
| 2 | 100 | 0.6 |
Calculate the rate of the reaction when total pressure is 0.65 atm.
Answer:
Question 3.22. The rate constant for the decomposition of $N_2O_5$ at various temperatures is given below:
| T/°C | $10^5 \times k/s^{-1}$ |
|---|---|
| 0 | 0.0787 |
| 20 | 1.70 |
| 40 | 25.7 |
| 60 | 178 |
| 80 | 2140 |
Draw a graph between $\ln k$ and $1/T$ and calculate the values of A and $E_a$. Predict the rate constant at 30° and 50°C.
Answer:
Question 3.23. The rate constant for the decomposition of hydrocarbons is $2.418 \times 10^{–5}s^{–1}$ at 546 K. If the energy of activation is 179.9 kJ/mol, what will be the value of pre-exponential factor.
Answer:
Question 3.24. Consider a certain reaction $A \rightarrow$ Products with $k = 2.0 \times 10^{–2}s^{–1}$. Calculate the concentration of A remaining after 100 s if the initial concentration of A is 1.0 mol L$^{–1}$.
Answer:
Question 3.25. Sucrose decomposes in acid solution into glucose and fructose according to the first order rate law, with $t_{1/2} = 3.00$ hours. What fraction of sample of sucrose remains after 8 hours ?
Answer:
Question 3.26. The decomposition of hydrocarbon follows the equation
$k = (4.5 \times 10^{11}s^{–1}) e^{-28000K/T}$
Calculate $E_a$.
Answer:
Question 3.27. The rate constant for the first order decomposition of $H_2O_2$ is given by the following equation:
$\log k = 14.34 – \frac{1.25 \times 10^4K}{T}$
Calculate $E_a$ for this reaction and at what temperature will its half-period be 256 minutes?
Answer:
Question 3.28. The decomposition of A into product has value of $k$ as $4.5 \times 10^3 s^{–1}$ at 10°C and energy of activation 60 kJ mol$^{–1}$. At what temperature would $k$ be $1.5 \times 10^4s^{–1}$?
Answer:
Question 3.29. The time required for 10% completion of a first order reaction at 298K is equal to that required for its 25% completion at 308K. If the value of A is $4 \times 10^{10}s^{–1}$. Calculate k at 318K and $E_a$.
Answer:
Question 3.30. The rate of a reaction quadruples when the temperature changes from 293 K to 313 K. Calculate the energy of activation of the reaction assuming that it does not change with temperature.
Answer: