Rate Constant Calculations (Cambridge (CIE) A Level Chemistry): Revision Note

Rate Constant Calculations

• The rate constant (k) of a reaction can be calculated using:

  • The initial rates and the rate equation

  • The half-life

Calculating the rate constant from the initial rate

• The reaction of calcium carbonate (CaCO₃) with chloride (Cl^–) ions to form calcium chloride (CaCl₂) will be used as an example to calculate the rate constant from the initial rate and initial concentrations

• The reaction and rate equation are as follows:

CaCO₃ (s) + 2Cl^– (aq) + 2H⁺ (aq) → CaCl₂ (aq) + CO₂ (g) + H₂O (l)

Rate = k [CaCO₃] [Cl^–]

• The progress of the reaction can be followed by measuring the initial rates of the reaction using various initial concentrations of each reactant

Experimental results of concentrations & initial rates table

• To find the rate constant (k):

  • Rearrange the rate equation to find k:

  • Rate = k [CaCO₃] [Cl^–] → k = $\frac{\mathrm{rate}}{\left[{\mathrm{CaCO}}_3\right]\left[{\mathrm{Cl}}^{-}\right]}$

• Substitute the values of one of the experiments to find k:

  • For example, using the measurements from experiment 1

  • k = $\frac{4.38\times {10}^{-6}}{\left[0.0250\right]\left[0.0125\right]}$

  • k = 1.40 x 10^-2 dm³ mol^-1 s^-1

• The measurements from experiments 2 or 3 could also have been used to find k

  • They would also give the same result of 1.40 x 10^-2 dm³ mol^-1 s^-1 

Calculating the rate constant from the half-life

• The rate constant (k) can also be calculated from the half-life of a reaction

• You are only expected to deduce k from the half-life of a first-order reaction as the calculations for second and zero-order reactions are more complicated

• For a first-order reaction, the half-life is related to the rate constant by the following expression:

${\mathit{t}}_{\raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}$}\right.}\mathbf{=}\frac{\mathbf{0}\mathbf{.}\mathbf{693}}{\mathit{k}}$

• Rearranging the equation to find k gives:

$\mathit{k}\mathbf{=}\frac{\mathbf{0}\mathbf{.}\mathbf{693}}{{\mathbf{t}}_{\raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}$}\right.}}$

• So, for a first-order reaction such as the methyl (CH₃) rearrangement in ethanenitrile (CH₃CN) with a half-life of 10.0 minutes the rate constant is:

  • $k=\frac{0.693}{10.0\times 60}$ = 1.16 x 10^-3 dm³ mol^-1 s^-1