17.4 Reaction Rates, Rate Equations, Order of Reaction & Activation Energy

Reaction Rates, Rate Equations, Order of Reaction & Activation Energy

In the Intermediate Chemistry Part I syllabus and the University of Engineering and Technology (UET) ECAT entry test, chemical kinetics is among the most quantitatively intensive topics. While chemical thermodynamics determines the theoretical feasibility and equilibrium position of a reaction (whether $\Delta G < 0$), chemical kinetics governs the time dimension: how rapidly reactants transform into products and through what microscopic elementary pathway (reaction mechanism). ECAT questions rigorously test rate law derivations from initial rate tables, dimensional units of rate constants ($k$), half-life diagnostic relationships, graphical slope interpretations of the Arrhenius equation, and the catalytic lowering of activation energy barriers.

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1. Reaction Rates: Instantaneous vs. Average Rates

The rate of a chemical reaction is defined quantitatively as the change in molar concentration of any reactant or product per unit time:

$\text{Rate} = \frac{\text{Change in concentration}}{\text{Time interval}} = \frac{\Delta c}{\Delta t}$

Standard SI units of reaction rate are $\text{mol}\cdot\text{dm}^{-3}\cdot\text{s}^{-1}$ (or $\text{mol}\cdot\text{L}^{-1}\cdot\text{min}^{-1}$ for slower transformations).

Stoichiometric Rate Relationships

For a generalized homogeneous chemical reaction:

$aA + bB \rightarrow cC + dD$

Because reactants disappear while products appear, a negative sign is prefixed to reactant terms so that reaction rate remains an intrinsically positive quantity. Furthermore, to eliminate ambiguity caused by varying stoichiometric coefficients, the rate of concentration change of each species is divided by its respective coefficient:

$\text{Rate} = -\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = +\frac{1}{c}\frac{d[C]}{dt} = +\frac{1}{d}\frac{d[D]}{dt}$

For example, in the decomposition of dinitrogen pentoxide ($2\text{N}_2\text{O}_5(g) \rightarrow 4\text{NO}_2(g) + \text{O}_2(g)$):

$\text{Rate} = -\frac{1}{2}\frac{d[\text{N}_2\text{O}_5]}{dt} = +\frac{1}{4}\frac{d[\text{NO}_2]}{dt} = +\frac{d[\text{O}_2]}{dt}$

This shows that nitrogen dioxide appears four times faster than oxygen gas is evolved, and twice as fast as dinitrogen pentoxide decomposes.

Average Rate vs. Instantaneous Rate

1. Average Rate: The rate of reaction measured over a finite macroscopic time interval $\Delta t = t_2 - t_1$: $\text{Rate}_{\text{avg}} = -\frac{\Delta [A]}{\Delta t} = -\frac{[A]_2 - [A]_1}{t_2 - t_1}$ As the reaction proceeds, reactant concentrations decline continuously, causing the average rate to decrease steadily with time.
2. Instantaneous Rate: The actual rate of reaction at one specific, infinitesimally small instant of time ($t$), defined mathematically as the limit of the average rate as $\Delta t$ approaches zero: $\text{Rate}_{\text{inst}} = \lim_{\Delta t \rightarrow 0} \left(-\frac{\Delta [A]}{\Delta t}\right) = -\frac{d[A]}{dt}$ Graphically, the instantaneous rate at any time $t$ is obtained by drawing a tangent line to the concentration-time curve at that specific point; the negative slope ($-\tan\theta$) of the tangent equals the instantaneous rate.
3. Initial Rate: The instantaneous rate measured at the exact moment of mixing ($t = 0$), prior to any significant accumulation of products or reverse reactions.

Physical Experimental Methods for Measuring Rates

Depending on the physical properties of the reacting system, instantaneous rates are monitored continuously without disturbing the reaction mixture:

• Spectrophotometry / Colorimetry: Measures light absorbance via Beer-Lambert law when a reactant or product has a characteristic color (e.g., brown $\text{NO}_2$ or violet $\text{I}_2$).
• Electrical Conductance: Measures changes in conductivity when the concentration of ions varies during the reaction (e.g., ester saponification releasing acetate ions).
• Dilatometry: Measures minuscule volume changes using a calibrated capillary dilatometer when liquid reactants and products possess different molar densities.
• Refractometry: Measures changes in refractive index over time.
• Polarimetry: Measures the optical rotation of plane-polarized light during reactions involving chiral compounds (e.g., the inversion of sucrose into glucose and fructose).
• Pressure Manometry: Measures total gas pressure changes at constant volume when $\Delta n_g \neq 0$.

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2. Rate Law, Rate Constant ($k$) and Order of Reaction

The Rate Law (or Rate Equation) is an experimentally determined mathematical relationship connecting the reaction rate to the molar concentrations of the reacting chemical species:

$\text{Rate} = k [A]^m [B]^n$

• The exponents $m$ and $n$ represent the partial orders of reaction with respect to reactants $A$ and $B$, respectively.
• The sum of exponents $n_{\text{overall}} = m + n$ represents the overall order of reaction.
• The proportionality constant $k$ is the specific rate constant (or velocity constant).

ECAT Golden Rule: The reaction order exponents ($m, n$) cannot be deduced from the stoichiometric coefficients of a balanced chemical equation! They can be determined solely through physical experimentation. For instance, while $\text{H}_2(g) + \text{I}_2(g) \rightarrow 2\text{HI}(g)$ is second order overall ($\text{Rate} = k[\text{H}_2][\text{I}_2]$), the identically balanced reaction $\text{H}_2(g) + \text{Br}_2(g) \rightarrow 2\text{HBr}(g)$ exhibits a complex fractional rate law: $\text{Rate} = \frac{k[\text{H}_2][\text{Br}_2]^{1/2}}{1 + k'[\text{HBr}]/[\text{Br}_2]}$.

The Specific Rate Constant ($k$)

If all reactant concentrations are set to unity ($[A] = [B] = 1.0\text{ mol}\cdot\text{dm}^{-3}$):

$\text{Rate} = k(1)^m(1)^n = k$

Thus, the specific rate constant $k$ is defined as the reaction rate when the concentrations of all reacting species are unity. Characteristics of $k$:

• $k$ is completely independent of reactant concentrations.
• $k$ is constant for a given reaction at a fixed temperature, but varies exponentially with temperature.
• $k$ increases substantially in the presence of a positive catalyst.

Dimensional Units of the Rate Constant ($k$)

Dimensional analysis of $k$ for an overall reaction order $n$:

$\text{Rate} = k [C]^n \implies k = \frac{\text{Rate}}{[C]^n} = \frac{\text{mol}\cdot\text{dm}^{-3}\cdot\text{s}^{-1}}{(\text{mol}\cdot\text{dm}^{-3})^n} = (\text{mol}\cdot\text{dm}^{-3})^{1-n}\cdot\text{s}^{-1}$

Expressing units across orders:

• Zero Order ($n = 0$): $(\text{mol}\cdot\text{dm}^{-3})^{1-0}\cdot\text{s}^{-1} = \mathbf{\text{mol}\cdot\text{dm}^{-3}\cdot\text{s}^{-1}}$
• First Order ($n = 1$): $(\text{mol}\cdot\text{dm}^{-3})^{1-1}\cdot\text{s}^{-1} = \mathbf{\text{s}^{-1}}$ (or $\text{min}^{-1}$)
• Second Order ($n = 2$): $(\text{mol}\cdot\text{dm}^{-3})^{1-2}\cdot\text{s}^{-1} = \mathbf{\text{dm}^3\cdot\text{mol}^{-1}\cdot\text{s}^{-1}}$
• Third Order ($n = 3$): $(\text{mol}\cdot\text{dm}^{-3})^{1-3}\cdot\text{s}^{-1} = \mathbf{\text{dm}^6\cdot\text{mol}^{-2}\cdot\text{s}^{-1}}$

ECAT Exam Shortcut: If an exam question provides a rate constant value without stating the reaction order, look immediately at its units! If units are $\text{s}^{-1}$, the reaction is unambiguously first order; if $\text{dm}^3\cdot\text{mol}^{-1}\cdot\text{s}^{-1}$, it is second order.

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3. Pseudo-Order Reactions & Molecularity vs. Order

Molecularity vs. Order of Reaction

Simultaneous collisions involving more than three molecules have an infinitesimally small statistical probability. Hence, reactions with molecularity $> 3$ are non-existent; all complex reactions with large stoichiometric coefficients proceed through a sequence of bimolecular and unimolecular steps.

Rate-Determining Step (RDS)

In a multi-step reaction mechanism, the individual elementary steps proceed at vastly different velocities. The overall rate of the chemical reaction is strictly governed and limited by the slowest elementary step in the reaction pathway, termed the Rate-Determining Step (RDS):

$\text{Step 1 (Slow, RDS): } \text{NO}_2 + \text{NO}_2 \rightarrow \text{NO}_3 + \text{NO}$ $\text{Step 2 (Fast): } \text{NO}_3 + \text{CO} \rightarrow \text{NO}_2 + \text{CO}_2$ $\text{Overall: } \text{NO}_2 + \text{CO} \rightarrow \text{NO} + \text{CO}_2$

The kinetic rate law reflects solely the molecular composition of the transition state in the rate-determining step. For this reaction (below about $500\text{ K}$) the observed rate law is $\text{Rate} = k[\text{NO}_2]^2$: second order in $\text{NO}_2$ and zero order in $\text{CO}$, even though $\text{CO}$ appears in the balanced equation, because $\text{CO}$ enters only after the slow step.

Pseudo-First-Order Reactions

A pseudo-first-order reaction is a higher-order reaction (typically bimolecular) that behaves kinetically as a first-order process because one of the reactants is present in vast stoichiometric excess relative to the other:

1. Acid-Catalyzed Hydrolysis of Ethyl Acetate: $\text{CH}_3\text{COOC}_2\text{H}_5 + \text{H}_2\text{O} \xrightarrow{\text{H}^+} \text{CH}_3\text{COOH} + \text{C}_2\text{H}_5\text{OH}$ The true differential rate law is bimolecular: $\text{Rate} = k'[\text{CH}_3\text{COOC}_2\text{H}_5][\text{H}_2\text{O}]$. However, because water acts as the solvent (concentration $\approx 55.5\text{ M}$), its concentration remains effectively constant throughout the reaction ($[\text{H}_2\text{O}] = \text{constant}$). Defining an apparent rate constant $k = k'[\text{H}_2\text{O}]$: $\text{Rate} = k [\text{CH}_3\text{COOC}_2\text{H}_5]$ The reaction exhibits pseudo-first-order kinetics with $k$ having units of $\text{s}^{-1}$.
2. Inversion of Cane Sugar (Sucrose): $\text{C}_{12}\text{H}_{22}\text{O}_{11} + \text{H}_2\text{O} \xrightarrow{\text{H}^+} \text{C}_6\text{H}_{12}\text{O}_6 (\text{glucose}) + \text{C}_6\text{H}_{12}\text{O}_6 (\text{fructose})$ Similarly pseudo-first-order with respect to sucrose.

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4. Mathematical Kinetics: Zero, First & Second Order Reactions

1. Zero-Order Reactions

A reaction is zero order when its rate is completely independent of the concentration of the reactants:

$\text{Rate} = -\frac{d[A]}{dt} = k [A]^0 = k$

• Integrated Rate Equation: $[A]_t = [A]_0 - kt$
• Linear Plot: A plot of reactant concentration $[A]_t$ versus time $t$ yields a straight line with slope $= -k$ and y-intercept $= [A]_0$.
• Half-Life ($t_{1/2}$): Time required for concentration to drop to half its initial value ($[A]_{t_{1/2}} = \frac{[A]_0}{2}$): $\frac{[A]_0}{2} = [A]_0 - k t_{1/2} \implies t_{1/2} = \frac{[A]_0}{2k}$ For zero-order kinetics, the half-life is directly proportional to initial concentration: $t_{1/2} \propto [A]_0$.
• Examples: Photochemical reactions ($\text{H}_2(g) + \text{Cl}_2(g) \xrightarrow{h\nu} 2\text{HCl}(g)$ over water), enzyme-catalyzed reactions at substrate saturation, and catalytic decomposition of ammonia on hot tungsten or platinum surfaces at high pressure.

2. First-Order Reactions

The reaction rate is directly proportional to the first power of reactant concentration:

$\text{Rate} = -\frac{d[A]}{dt} = k [A]$

• Integrated Rate Equation: Integrating $\int_{[A]_0}^{[A]_t} \frac{d[A]}{[A]} = -k \int_0^t dt$ yields: $\ln\left(\frac{[A]_0}{[A]_t}\right) = kt \implies [A]_t = [A]_0 e^{-kt}$ In common base-10 logarithms: $k = \frac{2.303}{t}\log_{10}\left(\frac{[A]_0}{[A]_t}\right)$
• Linear Plot: A plot of $\ln[A]_t$ versus $t$ gives a straight line with slope $= -k$. Alternatively, a plot of $\log_{10}[A]_t$ versus $t$ has slope $= -\frac{k}{2.303}$.
• Half-Life ($t_{1/2}$): Substituting $[A]_t = \frac{[A]_0}{2}$: $k t_{1/2} = \ln\left(\frac{[A]_0}{[A]_0 / 2}\right) = \ln(2) = 0.693$ $t_{1/2} = \frac{\ln 2}{k} = \frac{0.693}{k}$

  Crucial Diagnostic: For every first-order reaction, the half-life is completely independent of initial reactant concentration ($t_{1/2} \propto [A]_0^0$). Every radioactive decay follows first-order kinetics. After $n$ successive half-lives, the remaining reactant concentration is: $[A] = [A]_0 \left(\frac{1}{2}\right)^n$

• Examples: Decomposition of dinitrogen pentoxide ($2\text{N}_2\text{O}_5 \rightarrow 4\text{NO}_2 + \text{O}_2$), decomposition of sulfuryl chloride ($\text{SO}_2\text{Cl}_2 \rightarrow \text{SO}_2 + \text{Cl}_2$), and decomposition of hydrogen peroxide ($2\text{H}_2\text{O}_2 \rightarrow 2\text{H}_2\text{O} + \text{O}_2$).

3. Second-Order Reactions

The rate is proportional to the square of a single reactant concentration or the product of two reactant concentrations:

$\text{Rate} = -\frac{d[A]}{dt} = k [A]^2$

• Integrated Rate Equation: $\frac{1}{[A]_t} - \frac{1}{[A]_0} = kt$
• Linear Plot: A plot of $\frac{1}{[A]_t}$ versus $t$ yields a straight line with positive slope $= +k$ and y-intercept $= \frac{1}{[A]_0}$.
• Half-Life ($t_{1/2}$): Substituting $[A]_t = \frac{[A]_0}{2}$: $\frac{2}{[A]_0} - \frac{1}{[A]_0} = k t_{1/2} \implies t_{1/2} = \frac{1}{k [A]_0}$ For second-order reactions, the half-life is inversely proportional to initial concentration: $t_{1/2} \propto \frac{1}{[A]_0}$. Doubling initial concentration halves the half-life.

General Half-Life Proportionality

For any reaction of overall order $n$:

$t_{1/2} \propto \frac{1}{[A]_0^{n-1}}$

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5. Experimental Methods for Determining Reaction Order

1. Initial Rates Method: Reaction trials are conducted where the initial concentration of one reactant is varied systematically while keeping all other reactant concentrations constant. By comparing ratios of initial rates: $\frac{R_1}{R_2} = \left(\frac{[A]_1}{[A]_2}\right)^m \implies m = \frac{\log(R_1 / R_2)}{\log([A]_1 / [A]_2)}$
2. Half-Life Method: Half-lives ($t_{1/2}^{(1)}$ and $t_{1/2}^{(2)}$) are measured for two distinct initial concentrations ($[A]_{0,1}$ and $[A]_{0,2}$): $\frac{t_{1/2}^{(1)}}{t_{1/2}^{(2)}} = \left(\frac{[A]_{0,2}}{[A]_{0,1}}\right)^{n-1} \implies n = 1 + \frac{\log\left(t_{1/2}^{(1)} / t_{1/2}^{(2)}\right)}{\log\left([A]_{0,2} / [A]_{0,1}\right)}$
3. Graphical Integration Method: Experimental concentration-time data are plotted across different mathematical functions: if $[A]$ vs $t$ is linear $\rightarrow$ zero order; if $\ln[A]$ vs $t$ is linear $\rightarrow$ first order; if $1/[A]$ vs $t$ is linear $\rightarrow$ second order.
4. Ostwald's Isolation Method: In multi-reactant systems, all reactants except one are added in massive excess so their concentrations remain constant, isolating the individual partial order of the single limiting reactant.

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6. Collision Theory, Transition State & Activation Energy ($E_a$)

Collision Theory of Reaction Rates

Developed by Max Trautz (1916) and William Lewis (1918), Collision Theory assumes reacting molecules behave as hard, elastic spheres. For a chemical reaction to occur, reactant molecules must collide. However, at room temperature and atmospheric pressure, gas molecules undergo of the order of $10^{32}$ collisions per cubic decimeter per second (each molecule collides roughly $10^{9}$ to $10^{10}$ times per second), yet reactions proceed at measurable, finite rates. Hence, only a tiny fraction of collisions are effective collisions that lead to chemical transformation.

Two energetic and geometrical criteria govern effective collisions:

1. Energy Barrier (Threshold Energy): Colliding molecules must possess a minimum kinetic energy, termed the threshold energy ($E_{\text{threshold}}$). The extra kinetic energy above the average reactant kinetic energy required to cross the potential energy barrier is the Activation Energy ($E_a$): $E_a = E_{\text{threshold}} - E_{\text{reactants}}$ According to the Maxwell-Boltzmann energy distribution, the fraction of collisions possessing energy equal to or greater than $E_a$ is given by the exponential Boltzmann factor $f = e^{-E_a / (RT)}$.
2. Steric / Orientation Barrier ($P$): Colliding molecules must collide with proper spatial orientation so that reactive atoms come into direct contact and allow electronic rearrangement.

Combining these factors yields the collision rate expression:

$\text{Rate} = P \cdot Z_{AB} \cdot e^{-E_a / (RT)}$

where $Z_{AB}$ is collision frequency and $P$ is the steric (probability) orientation factor ($P < 1$).

Transition State / Activated Complex Theory

Henry Eyring and Michael Polanyi proposed that during an effective collision, reactant bonds stretch and weaken while new product bonds begin to form simultaneously, creating a transient, high-energy, unstable species known as the activated complex or transition state located at the apex of the potential energy curve:

$\text{Reactants} \rightleftharpoons [\text{Activated Complex}]^\ddagger \rightarrow \text{Products}$

• The activation energy of the forward reaction is $E_{a,f} = E_\ddagger - E_R$.
• The activation energy of the reverse reaction is $E_{a,r} = E_\ddagger - E_P$.
• The net reaction enthalpy is the difference between forward and reverse activation barriers: $\Delta H = E_{a,\text{forward}} - E_{a,\text{reverse}}$

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7. The Arrhenius Equation & Temperature Dependence

In 1889, Svante Arrhenius formulated the mathematical equation modeling the temperature dependence of the reaction rate constant:

$k = A \cdot e^{-E_a / (RT)}$

• $k$: Specific rate constant.
• $A$: Arrhenius pre-exponential factor (frequency factor), related to collision frequency and steric geometry; shares the identical units as $k$.
• $E_a$: Activation energy (in $\text{J}\cdot\text{mol}^{-1}$ or $\text{kJ}\cdot\text{mol}^{-1}$).
• $R$: Universal gas constant ($8.314\text{ J}\cdot\text{K}^{-1}\cdot\text{mol}^{-1}$).
• $T$: Absolute temperature in Kelvin.

Linear Logarithmic Forms

Taking natural logarithms of both sides:

$\ln k = \ln A - \frac{E_a}{R}\left(\frac{1}{T}\right)$

Converting to base-10 common logarithms:

$\log_{10} k = \log_{10} A - \frac{E_a}{2.303 R}\left(\frac{1}{T}\right)$

• Graphical Determination: Plotting $\ln k$ on the y-axis against $\frac{1}{T}$ on the x-axis yields a straight line with: $\text{Slope} = -\frac{E_a}{R} \implies E_a = -R \times \text{Slope}$ $\text{y-intercept} = \ln A$
• If plotted as $\log_{10} k$ vs $\frac{1}{T}$, the slope is $-\frac{E_a}{2.303 R}$.

Two-Temperature Ratio Formula

Measuring the rate constant at two distinct temperatures $T_1$ and $T_2$ allows direct algebraic calculation of $E_a$ without knowing the frequency factor $A$:

$\ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right) = \frac{E_a}{R}\left(\frac{T_2 - T_1}{T_1 T_2}\right)$

$\log_{10}\left(\frac{k_2}{k_1}\right) = \frac{E_a}{2.303 R}\left(\frac{T_2 - T_1}{T_1 T_2}\right)$

ECAT Temperature Coefficient Rule of Thumb: For most ordinary chemical reactions near room temperature ($25^\circ\text{C}$ to $35^\circ\text{C}$), the reaction rate approximately doubles or triples for every $10^\circ\text{C}$ rise in temperature: $\text{Temperature Coefficient } = \frac{k_{T + 10}}{k_T} \approx 2 \text{ to } 3$ This rapid acceleration is not due to an increase in molecular collision velocity (which increases by only $\sim 2\%$ over a $10^\circ\text{C}$ span), but rather due to the exponential increase in the fraction of molecules possessing kinetic energy $\ge E_a$ (the Boltzmann factor $e^{-E_a/RT}$). Reactions with higher activation energies are much more sensitive to temperature increases than reactions with lower activation energies.

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8. Catalysis and Reaction Energetics

A catalyst is a foreign substance that increases the velocity of a chemical reaction without being consumed or undergoing permanent chemical change during the process.

Classification of Catalysis

1. Homogeneous Catalysis: The catalyst and all reacting substances exist in the same physical phase (single uniform state):
   • Gas phase: Lead chamber process for $\text{H}_2\text{SO}_4$: $2\text{SO}_2(g) + \text{O}_2(g) \xrightarrow{\text{NO}(g)} 2\text{SO}_3(g)$.
   • Liquid phase: Hydrolysis of esters catalyzed by aqueous mineral acids: $\text{CH}_3\text{COOCH}_3 + \text{H}_2\text{O} \xrightarrow{\text{H}^+(aq)} \text{CH}_3\text{COOH} + \text{CH}_3\text{OH}$.
2. Heterogeneous Catalysis: The catalyst is in a different physical phase from the reactants (most commonly a solid transition metal catalyst operating on gaseous or liquid reactants):
   • Haber process: $\text{N}_2(g) + 3\text{H}_2(g) \xrightarrow{\text{Fe}(s)} 2\text{NH}_3(g)$.
   • Contact process: $2\text{SO}_2(g) + \text{O}_2(g) \xrightarrow{\text{V}_2\text{O}_5(s)} 2\text{SO}_3(g)$.
   • Catalytic hydrogenation of oils: $\text{Vegetable Oil}(l) + \text{H}_2(g) \xrightarrow{\text{Ni}(s)} \text{Vegetable Ghee}(s)$.
   • Mechanism: Adsorption of reactant molecules onto active surface metal d-orbitals, chemical bond weakening, formation of intermediate surface complexes, reaction between adsorbed radicals, and desorption of products.
3. Enzyme Catalysis: Highly specific biochemical protein catalysts featuring active site "lock and key" or "induced fit" geometry (e.g., urease decomposing urea, invertase inverting sucrose, pepsin digesting proteins).

Energetics of Catalysis

A catalyst accelerates chemical reactions by providing an alternative reaction pathway characterized by a lower activation energy ($E_a' < E_a$):

• Both the forward activation energy ($E_{a,f}$) and the reverse activation energy ($E_{a,r}$) are lowered by the exact same energy increment: $\Delta E_a$.
• Consequently, the forward rate constant $k_f$ and reverse rate constant $k_r$ increase by identical proportions.
• What a Catalyst Does NOT Change:
  • It does not change the initial energy of reactants ($H_R$) or final energy of products ($H_P$).
  • It does not alter the net reaction enthalpy: $\Delta H = E_{a,f}' - E_{a,r}' = E_{a,f} - E_{a,r}$.
  • It does not alter the Gibbs free energy change ($\Delta G$) or reaction spontaneity.
  • It does not alter the numerical value of the equilibrium constant ($K_c = k_f / k_r$) or the equilibrium yields. It merely allows dynamic equilibrium to be established substantially faster.

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9. Reaction Orders, Integrated Rate Laws & Half-Lives Reference Table