Chemical Kinetics: Rates of Reaction

Chemical kinetics is the study of the speed, or rate, of chemical reactions. While thermodynamics tells us if a reaction can happen, kinetics explains how fast it occurs, which is crucial for everything from designing life-saving pharmaceuticals to optimizing industrial chemical synthesis. Mastering kinetics allows you to predict how long a process will take and, more importantly, control it by altering conditions like concentration and temperature.

Determining the Rate Equation and Order of Reaction

The rate equation (or rate law) is the mathematical relationship that shows how the rate of reaction depends on the concentration of reactants. For a general reaction $aA+bB\to products$, the rate equation is expressed as:

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

Here, $k$ is the rate constant, a proportionality constant specific to a reaction at a given temperature. The exponents $m$ and $n$ are the orders of reaction with respect to reactants A and B, respectively. These orders are not necessarily the same as the stoichiometric coefficients $a$ and $b$; they must be determined experimentally.

You determine the order by analyzing how the initial rate changes when you vary the concentration of one reactant while keeping others constant. For example, if doubling $[A]$ doubles the rate, the reaction is first order ($m=1$) with respect to A. If doubling $[A]$ quadruples the rate, it is second order ($m=2$) with respect to A. A zero order ($m=0$) means the rate is independent of that reactant's concentration. The overall order is the sum of all individual orders ($m+n$).

Worked Example: Determining Order from Initial Rates Data

Consider data for the reaction $A+B\to C$:

1. Find order with respect to A: Compare Experiments 1 and 2, where $[B]$ is constant. $[A]$ doubles, and the rate doubles. This is a direct proportionality, so the reaction is first order with respect to A ($m=1$).
2. Find order with respect to B: Compare Experiments 1 and 3, where $[A]$ is constant. $[B]$ doubles, but the rate stays the same. Therefore, the reaction is zero order with respect to B ($n=0$).
3. Write the rate equation: $\text{Rate}=k[A{]}^1[B{]}^0=k[A]$.
4. Calculate the rate constant $k$: Use data from any experiment. From Experiment 1: $5.0\times 10^{-3}=k(0.10)$. Therefore, $k=0.05$ s${}^{-1}$. Remember the units of $k$ depend on the overall order; here it is s${}^{-1}$ for a first-order reaction.

Graphical Methods and Interpretation

Graphs provide a powerful visual method to confirm reaction order and determine $k$.

• Concentration-Time Graphs: Plot the concentration of a reactant against time.
• Zero Order: A straight-line decrease indicates zero order. The gradient (slope) is negative and equals $-k$.
• First Order: An exponential decay curve. A plot of $\ln [A]$ against time yields a straight line with gradient $-k$.
• Second Order (with respect to that reactant): A plot of $1/[A]$ against time yields a straight line with gradient $+k$.

• Rate-Concentration Graphs: Plot the initial rate against initial concentration.
• Zero Order: A horizontal line (rate is constant).
• First Order: A straight line through the origin (rate is directly proportional to concentration).
• Second Order: A curve (rate proportional to concentration squared). A plot of rate against $[A{]}^2$ would be a straight line.

The Arrhenius Equation: Linking Temperature, $k$, and $E_a$

Temperature has a dramatic effect on reaction rates, primarily by affecting the rate constant $k$. The Arrhenius equation quantifies this relationship:

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

Here:

• $k$ is the rate constant.
• $A$ is the Arrhenius constant or pre-exponential factor, related to the frequency of collisions with correct orientation.
• $E_a$ is the activation energy, the minimum kinetic energy particles must have to react.
• $R$ is the gas constant (8.31 J mol${}^{-1}$ K${}^{-1}$).
• $T$ is the temperature in Kelvin.
• $e^{-E_a/(RT)}$ is the fraction of particles with energy equal to or greater than $E_a$.

The equation shows that as $T$ increases, the value of $-E_a/(RT)$ becomes less negative, $e^{-E_a/(RT)}$ becomes larger, and therefore $k$ increases exponentially. This explains why a 10°C rise often doubles the rate of a reaction.

A more useful form for calculations is the linear form:

$$\ln k=\ln A-\frac{E_a}{R}\cdot \frac{1}{T}$$

By plotting $\ln k$ against $1/T$, you get a straight line with a gradient of $-E_a/R$. This allows you to determine the activation energy $E_a$ experimentally from rate measurements at different temperatures.

The Role of Catalysts

A catalyst is a substance that increases the rate of a reaction without being consumed. It works by providing an alternative reaction pathway with a lower activation energy ($E_a$).

• Effect on the Reaction Profile: On an energy level diagram, the catalyzed pathway has a lower "hump" (activation energy barrier) than the uncatalyzed pathway. The overall enthalpy change ($\Delta H$) for the reaction remains unchanged.
• Mechanism: Catalysts often work by adsorbing reactants onto their surface (heterogeneous catalysis, e.g., a platinum mesh in the Haber process) or by forming temporary, reactive intermediates (homogeneous catalysis, e.g., Fe${}^{2+}$ ions in the I${}^{-}$/S${}_2$O${}_8^{2-}$ reaction).
• Effect on the Arrhenius Equation: Lowering $E_a$ dramatically increases the value of $e^{-E_a/(RT)}$, and thus increases $k$. The catalyst does not affect the value of $A$ in the Arrhenius equation.

Think of $E_a$ as a high wall separating reactants from products. A catalyst doesn't remove the wall; it provides a door or a lower step to climb over.

Common Pitfalls

1. Confusing Order with Stoichiometry: A common exam trap is to assume the order is the same as the coefficient in the balanced equation. Always use experimental data, not the equation, to determine order. For example, in the reaction $2N_2{O}_5\to 4N{O}_2+O_2$, the rate equation is actually $\text{Rate}=k[N_2{O}_5]$ (first order), not based on the coefficient 2.
2. Misinterpreting Rate Constant ($k$) as Rate: The rate changes with concentration. The rate constant $k$ is only constant for a given reaction at a fixed temperature. A larger $k$ means a faster reaction potential, but the actual rate also depends on the concentrations plugged into the rate equation.
3. Incorrect Units for $k$: Forgetting that the units of $k$ vary with overall order is a frequent source of lost marks. For a reaction with overall order $n$, the units of $k$ are (mol dm${}^{-3}$)${}^{1-n}$ s${}^{-1}$. Always deduce the units from the rate equation: Rate (mol dm${}^{-3}$ s${}^{-1}$) = $k\times$ (concentration)${}^n$.
4. Muddling the Arrhenius Plot: When plotting $\ln k$ against $1/T$, students often plot against $T$ or forget to use Kelvin for temperature. Using Celsius will give an incorrect gradient and a wildly wrong value for $E_a$.

Summary

• The rate equation ($\text{Rate}=k[A{]}^m[B{]}^n$) defines the mathematical link between reaction rate and reactant concentrations, where the orders ($m,n$) are found experimentally.
• Graphical analysis of concentration-time or rate-concentration data is a key method for confirming reaction order and determining the rate constant $k$.
• The Arrhenius equation ($k=A{e}^{-E_a/(RT)}$) shows the exponential relationship between the rate constant $k$ and temperature $T$, governed by the activation energy $E_a$.
• Catalysts increase reaction rates by providing an alternative pathway with a lower $E_a$, which is visible on a reaction profile diagram. They do not alter the thermodynamic quantities ($\Delta H$) of the reaction.
• Always derive reaction orders from experimental data, not stoichiometry, and be meticulous with units and temperature scales in calculations.