5.2.4 Rate Equation Calculations

Units of Rate and the Rate Constant ($k$)

The rate of a reaction is measured in mol dm$⁻³$ s$⁻¹$, representing how concentration changes over time. The units of the rate constant k depend on the overall order of the reaction and can be derived by examining the units in the rate equation.

Method

Here's a general method for determining the units of k using an example:

Step 1: Set Up the Rate Equation:

For a reaction with the rate equation:

$$\text{Rate} = k[A]^2[B]$$

where the overall order is $2 + 1 = 3$

Step 2: Substitute Units into the Rate Equation:

• Rate units are mol dm$⁻³$ s$⁻¹$.
• Concentration units for each reactant, such as $[A]$ and $[B]$, are mol dm$⁻³$. Substitute these units:

$$\text{mol dm}^{-3} \text{s}^{-1} = k (\text{mol dm}^{-3})^3$$

Step 3: Simplify to Find Units of $k$:

$$\text{mol dm}^{-3} \text{s}^{-1} = k \cdot \text{mol}^3 \text{dm}^{-9}$$

Rearrange to isolate $k$:

$$k = \frac{\text{mol dm}^{-3} \text{s}^{-1}}{\text{mol}^3 \text{dm}^{-9}} = \text{mol}^{-2} \text{dm}^6 \text{s}^{-1}$$

The resulting units of $k$ for this third-order reaction are mol$⁻²$ dm$⁶$ s$⁻¹$.

Temperature Dependence of the Rate Constant

The rate constant $k$ varies with temperature, as shown by the Arrhenius equation:

$$k = A e^{-\frac{E_a}{RT}}$$

where:

• $A$ is the Arrhenius constant (or pre-exponential factor), representing the frequency of collisions with proper orientation.
• $E_a$ is the activation energy (J mol$⁻¹$), the minimum energy required for the reaction to occur.
• $R$ is the gas constant (8.31 J K$⁻¹$ mol$⁻¹$).
• $T$ is the temperature in Kelvin. As temperature increases, $k$ increases, typically accelerating the reaction rate.

Rearranged Arrhenius Equation and Graphical Analysis

The Arrhenius equation can be rearranged into a linear form for graphical analysis:

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

This form ($y = mx + c$) allows you to plot $\ln k$ versus $\frac{1}{T}$, yielding a straight line with:

• Slope = $-\frac{E_a}{R}$
• y-intercept = $\ln A$ By calculating the slope of the line, you can determine $E_a$ and understand how the reaction rate varies with temperature.