The Rate Constant (HL) (HL IB Chemistry)

How to calculate the rate constant 

• The rate constant, k, of a reaction can be calculated using the initial rate and the rate equation
• The reaction of sodium carbonate with chloride ions (from hydrochloric acid) to form sodium chloride 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:

Na₂CO₃ (aq) + 2Cl^- (aq) + 2H⁺ (aq) → 2NaCl (aq) + CO₂ (g) + H₂O (l)

rate = k[Na₂CO₃][Cl^-]

• The rate equation rearranges to give:

$\mathit{k}\mathbf{ }\mathbf{=}\mathbf{ }\frac{\mathbf{r}\mathbf{a}\mathbf{t}\mathbf{e}}{\mathbf{\left[}\mathbf{N}{\mathbf{a}}_{\mathbf{2}}\mathbf{C}{\mathbf{O}}_{\mathbf{3}}\mathbf{\right]}\mathbf{\left[}\mathbf{C}{\mathbf{l}}^{\mathbf{-}}\mathbf{\right]}}$

• 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

Measurement	[Na2CO3] / mol dm-3	[Cl–] / mol dm-3	[H+] / mol dm-3	Initial rate of reaction / mol dm-3 s-1
1	0.0250	0.0125	0.0125	4.38 x 10-6
2	0.0375	0.0125	0.0125	6.63 x 10-6
3	0.00625	0.0250	0.0250	2.19 x 10-6

• To find the rate constant, k:
  • Substitute the values of one of the experiments to find k (for example, measurement 1)

• • The values of measurement 2 or 3 could also have been used to find k
  • They all give the same result of 1.40 x 10^-2 

Calculating rate constant units

• When you are asked to calculate the rate constant, k, for a reaction you must also be able to deduce the units
• This is done by replacing the values in the rearranged rate equation with the units of that value
• The units can then be combined or cancelled as required
• For example, to calculate the units for the above reaction:

$\begin{array}{rcl}\mathbf{Units}\mathbf{ }\mathbf{of}\mathbf{ }\mathit{k}\mathbf{ }& \mathbf{=}& \mathbf{ }\frac{(\cancel{\mathrm{mol} {\mathrm{dm}}^{-3}} s^{-1})}{\mathbf{\left(}\cancel{\mathbf{mol}\mathbf{ }{\mathbf{dm}}^{\mathbf{-}\mathbf{3}}}\mathbf{\right)}\mathbf{(}\mathbf{mol}\mathbf{ }{\mathbf{dm}}^{\mathbf{-}\mathbf{3}}\mathbf{)}}\\ & & \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }& \mathbf{=}& \mathbf{ }\mathbf{ }\frac{ s^{-1}}{\mathbf{mol}\mathbf{ }{\mathbf{dm}}^{\mathbf{-}\mathbf{3}}}\\ & & \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }& \mathbf{=}& \mathbf{ }{\mathbf{mol}}^{\mathbf{-}\mathbf{1}}\mathbf{ }{\mathbf{dm}}^{\mathbf{3}}\mathbf{ }{\mathbf{s}}^{\mathbf{-}\mathbf{1}}\end{array}$

• The units of the rate constant, k, are dependent on the overall order of the reaction
• When the rate is measured per second:
  • Rate constant units of a zero order reaction are mol dm^-3 s^-1
  • Rate constant units of a first order reaction are s^-1
  • Rate constant units of a second order reaction are mol^-1 dm³ s^-1
  • Rate constant units of a third order reaction are mol^-2 dm⁶ s^-1

How does temperature affect the rate constant, k?

• Consider the following reaction:

A + B → C + D

Rate of reaction = k[A][B]

• The rate equation shows that the rate of reaction depends on the rate constant, k, and the concentration of the reactants, A and B
  • As the rate of reaction increases, the rate constant will increase
• Increasing the temperature of a reaction increases the rate of a chemical reaction
  • Remember: This does not necessarily increase the yield of a chemical reaction
  • The yield depends on whether a reaction is endothermic or exothermic according to Le Châtelier’s principle
• Therefore, increasing the temperature also increases the value of the rate constant, k, assuming that the concentration of the reactants remains unchanged

• An exponential relationship between the rate of reaction and temperature is observed when seen on a graph

A graph to show the rate constant against temperature

Relationship between temperature and rate constant, k

• This general relationship does not apply to all reactions
  • The graph shows that the rate of reaction roughly doubles with an increase of 10 °C
    • It is not necessarily every 10 °C, the rate may double every 9 °C or 11 °C
  • The number of degrees needed to double the rate also changes gradually as temperature increases