Rate-Concentration Graphs and Order Determination

Understanding how the rate of a reaction depends on the concentration of reactants is fundamental to chemical kinetics. For A-Level Chemistry, mastering the graphical methods to determine reaction order is a key skill that allows you to quantitatively analyze reaction mechanisms and predict how changes in conditions affect speed.

Understanding Reaction Order and the Rate Equation

The starting point for any kinetic analysis is the rate equation, which expresses the mathematical relationship between the reaction rate and the concentrations of reactants. For a reaction where reactant A is being consumed, the rate equation is typically expressed as: Rate = $k[A{]}^m$. Here, $k$ is the rate constant, a value that depends only on temperature, and $m$ is the order of the reaction with respect to A. The order ($m$) is not necessarily related to the stoichiometric coefficient; it must be determined experimentally and can be zero, one, two, or even a fraction.

The overall order of the reaction is the sum of the powers to which all concentration terms are raised in the rate equation. Determining this order is the primary goal of graphical analysis. A zero-order reaction means the rate is independent of concentration, a first-order reaction means rate is directly proportional to concentration, and a second-order reaction means rate is proportional to the square of the concentration. Each order produces a distinctive fingerprint on different types of graphs.

Graphical Method 1: Rate-Concentration Plots

The most direct way to find the order with respect to a single reactant is to plot a graph of initial rate against the initial concentration of that reactant, holding all other conditions constant. The shape of this graph reveals the order.

• Zero-Order Reaction: For a zero-order reaction, Rate = $k$. Plotting rate (y-axis) against $[A]$ (x-axis) yields a horizontal straight line. The rate does not change as the reactant is used up. The gradient of this line is zero.
• First-Order Reaction: For a first-order reaction, Rate = $k[A]$. Plotting rate against $[A]$ yields a straight line through the origin. The gradient of this line is equal to the rate constant, $k$.
• Second-Order Reaction: For a second-order reaction, Rate = $k[A{]}^2$. Plotting rate against $[A]$ yields a curve that gets steeper as concentration increases. If you plot rate against $[A{]}^2$, however, you would get a straight line through the origin.

Example: If you double the concentration of A and the initial rate also doubles, the graph of rate vs. $[A]$ is linear, indicating first-order behavior. If you double the concentration and the rate quadruples, the graph of rate vs. $[A]$ is curved, suggesting second-order behavior, which would be confirmed by a linear plot of rate vs. $[A{]}^2$.

Graphical Method 2: Concentration-Time Plots and Integrated Rate Laws

While rate-concentration plots are conceptually simple, they require multiple experiments. Often, you have data for how a single reaction mixture changes over time. Here, you use integrated rate laws, which relate concentration directly to time. You test which plot gives a straight line.

• Zero-Order: The integrated rate law is $[A{]}_t=-kt+[A{]}_0$. A plot of $[A]$ against time ($t$) yields a straight line with a negative gradient equal to $-k$.
• First-Order: The integrated rate law is $\ln [A{]}_t=-kt+\ln [A{]}_0$. A plot of $\ln [A]$ against time ($t$) yields a straight line with a gradient of $-k$. This is one of the most important diagnostic plots in kinetics.
• Second-Order (with respect to A): The integrated rate law is $\frac{1}{[A{]}_t}=kt+\frac{1}{[A{]}_0}$. A plot of $\frac{1}{[A]}$ against time ($t$) yields a straight line with a positive gradient equal to $+k$.

Worked Example (First-Order): For a reaction, you measure the concentration of reactant A over time. To test for first-order kinetics, calculate the natural log (ln) of each concentration value. If a plot of ln[A] vs. time is linear, the reaction is first order with respect to A. The gradient, calculated as (change in y)/(change in x), will be equal to $-k$. For instance, if the gradient is $-0.015{\text{s}}^{-1}$, then $k=0.015{\text{s}}^{-1}$.

The Half-Life Method for Order Determination

The half-life ($t_{1/2}$) of a reaction is the time taken for the concentration of a reactant to fall to half of its initial value. Its relationship to concentration is a powerful tool for determining order.

• Zero-Order: Half-life is dependent on initial concentration. $t_{1/2}=\frac{[A{]}_0}{2k}$. If you start with a higher concentration, the half-life is longer.
• First-Order: Half-life is independent of initial concentration. $t_{1/2}=\frac{\ln 2}{k}$. This is a defining characteristic. If you calculate the half-life at different starting points in a single reaction and it remains constant, the reaction is first order.
• Second-Order (with respect to A): Half-life is inversely proportional to initial concentration. $t_{1/2}=\frac{1}{k[A{]}_0}$. If you start with a higher concentration, the half-life is shorter.

To use this method experimentally, you would measure the time for $[A]$ to drop from its starting value to $[A{]}_0/2$, then from $[A{]}_0/2$ to $[A{]}_0/4$, and so on. Constant half-lives indicate first-order kinetics.

Calculating Rate Constants and Units

Once you have determined the order graphically, you can calculate the rate constant, $k$, from the gradient of the appropriate linear plot. The units of $k$ are crucial and change with the overall reaction order, as they must make the rate equation dimensionally consistent (Rate has units of mol dm${}^{-3}$ s${}^{-1}$).

• Zero-Order: Rate = $k$. Therefore, $k$ has the same units as rate: mol dm${}^{-3}$ s${}^{-1}$.
• First-Order: Rate = $k[A]$. Rearranging: $k=\text{Rate}/[A]$. Units: (mol dm${}^{-3}$ s${}^{-1}$) / (mol dm${}^{-3}$) = s${}^{-1}$.
• Second-Order (e.g., Rate = $k[A{]}^2$): $k=\text{Rate}/[A{]}^2$. Units: (mol dm${}^{-3}$ s${}^{-1}$) / (mol dm${}^{-3}$)${}^2$ = mol${}^{-1}$ dm${}^3$ s${}^{-1}$.

For a second-order reaction where Rate = $k[A][B]$, the units for $k$ would still be mol${}^{-1}$ dm${}^3$ s${}^{-1}$, as the overall concentration term is (mol dm${}^{-3}$)${}^2$.

Common Pitfalls

1. Confusing Graph Purpose: A common mistake is to try to determine order from the shape of a standard $[A]$ vs. time plot. This plot is curved for both first and second-order reactions. You must transform the data (plot ln[A] or 1/[A]) to see which yields a straight line. Only the zero-order plot is linear on a $[A]$ vs. $t$ graph.
2. Incorrect Units for k: Forgetting that the units of the rate constant depend on the overall reaction order is a frequent error in calculations. Always derive the units from the rate equation: substitute the units for rate (mol dm${}^{-3}$ s${}^{-1}$) and concentration (mol dm${}^{-3}$) to solve for the units of $k$.
3. Misapplying Half-Life Rules: Assuming a constant half-life without checking multiple half-life periods in a single experiment. For a first-order process, every successive half-life (e.g., from 100% to 50%, then 50% to 25%) must be the same, regardless of the starting concentration for that period.
4. Miscounting Overall Order: Remember that the overall order is the sum of the powers in the rate equation. For a reaction with rate equation Rate = $k[A][B{]}^{1/2}$, the order with respect to A is 1, with respect to B is 0.5, and the overall order is 1.5.

Summary

• The order of a reaction with respect to a reactant is determined experimentally and dictates how the reaction rate depends on that reactant's concentration.
• A plot of initial rate versus initial concentration directly reveals order: a horizontal line for zero order, a straight line through the origin for first order, and a curve for second order.
• For concentration-time data, integrated rate law plots are used: $[A]$ vs. $t$ (linear for zero order), $\ln [A]$ vs. $t$ (linear for first order), and $1/[A]$ vs. $t$ (linear for second order).
• Half-life provides a diagnostic test: it is constant for first-order reactions, proportional to $[A{]}_0$ for zero order, and inversely proportional to $[A{]}_0$ for second order.
• The rate constant, $k$, is obtained from the gradient of the appropriate linear plot, and its units vary with the overall reaction order (e.g., s${}^{-1}$ for first order, mol${}^{-1}$ dm${}^3$ s${}^{-1}$ for second order).