Rates of Reaction

The Rate Equation: This mathematical expression relates the rate of a reaction to the concentrations of the reactants. For a reaction $A + B \rightarrow C$, the rate equation is: $Rate = k[A]^m[B]^n$

Rate Constant ($k$): A proportionality constant that is unique to every reaction at a specific temperature. If the temperature increases, the value of $k$ increases, leading to a faster rate.

Orders of Reaction: The powers $m$ and $n$ are the 'orders' with respect to each reactant. They indicate how sensitive the rate is to changes in concentration. The overall order is the sum of these individual powers ($m + n$).

Half-life ($t_{1/2}$): The time taken for the concentration of a reactant to decrease by half. For a first-order reaction, the half-life is constant regardless of the starting concentration.

Rate-Concentration Graphs: A zero-order reaction produces a horizontal line, a first-order reaction produces a straight line through the origin, and a second-order reaction produces a quadratic curve.

Elementary Steps: Most reactions occur via a series of simple steps rather than a single collision. These individual steps make up the reaction mechanism.

Rate-Determining Step (RDS): This is the slowest step in a multi-step reaction. The overall rate of the reaction is limited by this step, much like traffic is limited by the slowest vehicle on a single-lane road.

Predicting the Rate Equation: Only the species involved in the RDS (or steps preceding it) appear in the rate equation. The stoichiometry of the RDS matches the orders of the reactants in the rate equation.

Core Principle: The rate constant $k$ depends exponentially on temperature. As temperature increases, a larger fraction of molecules possess energy greater than the activation energy ($E_a$).

The Equation: $k = Ae^{-\frac{E_a}{RT}}$

$A$: Arrhenius constant (frequency factor)

$R$: Gas constant ($8.31 \, J \, K^{-1} \, mol^{-1}$)

$T$: Temperature in Kelvin.

Linear Form: To determine $E_a$ experimentally, the equation is rearranged into a linear form ($y = mx + c$): $\ln k = -\frac{E_a}{R} \left(\frac{1}{T}\right) + \ln A$ A plot of $\ln k$ against $1/T$ yields a straight line with a gradient of $-E_a/R$.

Units of $k$: Always derive the units of the rate constant based on the overall order. For example, in a first-order reaction, $k$ has units of $s^{-1}$, while in a second-order reaction, it is $dm^3 \, mol^{-1} \, s^{-1}$.

Graph Interpretation: When given a concentration-time graph, check if the half-life is constant. If it is, the reaction is first-order with respect to that reactant.

Initial Rates Data: When analyzing tables of data, look for experiments where only one reactant's concentration changes. If doubling $[A]$ quadruples the rate while $[B]$ is constant, the order with respect to $A$ is 2.

Temperature Conversion: Always ensure temperature is converted to Kelvin ($^\circ C + 273$) before using the Arrhenius equation.