Colourful Solutions > The rate of chemical change > The rate constant, k

The Mad Science Lab

Higher-level only

The rate constant relates the reaction rate to the concentrations of the reaction components.

Syllabus ref: R2.2.11

Reactivity 2.2.11 - The rate constant, k, is temperature dependent and its units are determined from the overall order of the reaction. (HL)

  • Solve problems involving the rate equation, including the units of k.

Guidance

Tools and links

  • Reactivity 3.4 - What are the rate equations and units of k for the reactions of primary and tertiary halogenoalkanes with aqueous alkali?

Significance of the rate constant

The rate constant, k, gives a direct measure of the relative reaction rate.

A very small value for the rate constant equates to a very slow reaction in general. Equally, a large value for the rate constant means a large value for the rate and that the reaction is rapid.

Rate = k[A]x[B]y

Example:

  • When k = 5 x 109 then the reaction (excluding situations where the concentrations are extremely small) is very fast.
  • When k = 1 x 10-10 the reaction is always very slow

Clearly, the actual rate is dependent on the concentrations and qualified by the orders, but it is still possible to get an idea about the relative rate of a reaction just by looking at the rate constant.

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Units of the rate constant

The units of the rate constant depend on the overall order of the reaction. As the rate equation can take different forms, i.e. the orders with respect to the individual reactants depend on the specific reaction under study, then the units for the rate constant change appropriately.

To find the units you must substitute the units of all of the other parts of the equation and then cancel down.

Units of rate = mol dm-3 s-1
Units of concentration = mol dm-3

When the concentration is raised to a specific order then the same thing must be done to the units of concentration. i.e.:

[A]1 has units of (mol dm-3)1 = mol dm-3

but [A]2 has units of (mol dm-3)2 = mol2 dm-6

So, the rate constant k for a first order reaction

rate = k [A]1

k = rate/[A]1

therefore units of k = mol dm-3 s-1/mol dm-3 = s-1

For a second order reaction

rate = k [A]2

k = rate/[A]2

therefore units of k = mol dm-3 s-1/mol2 dm-6 = dm3 mol-1 s-1

Note that the above refers to the overall order of the rate equation, not the specific individual orders. i.e the rate equation:

Rate = [A][B]

is second order overall, therefore:

Rate constant (k) units = dm3 mol-1 s-1

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Factors affecting the rate constant

The rate constant is just that, constant, unless the temperature conditions change.

However, catalysts also change the rate constant, but the reaction is now a different process with a different mechanism.

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Worked examples

Q632-01 A small increase in temperature often causes a large increase in the rate of a chemical reaction. This effect is best attributed to
  1. a decrease in the activation energy of the reaction
  2. more frequent collisions at the higher temperature
  3. the occurrence of more collisions with the needed energy
  4. different reaction pathways at the higher temperature
Answer

(A) a decrease in the activation energy of the reaction
Activation energy cannot be affected except by providing an alternative pathway (mechanism) using a catalyst

(B) more frequent collisions at the higher temperature
There are more collisions as the temperature rises but this is not as important as the success of the collisions.

(C) the occurrence of more collisions with the needed energy
The increase in temperature means that more particles will have the required energy to make their collisions successful

(D) different reaction pathways at the higher temperature
There is no reason to suppose that higher temperatures lead to different mechanisms
Q632-02 The dissociation of HI molecules, as shown below, occurs at a temperature of 629 K. The rate constant, k = 3.02 x 10-5 dm3 mol-1 s-1.

2HI(g) → H2(g) + I2(g)

What is the reaction order in the rate expression?

Answer

At first reading there appears to be insufficient information to answer the question - however the units of the rate constant provide the clue.

mol dm-3 s-1 are the units of the rate constant in a second order rate expression.

Rate = k[A]2

Therefore the reaction is second order.
Q632-03 The rate of the reaction 2NO + Cl2 → 2NOCl is given by the rate equation rate = k[NO]2[Cl2]. The value of the rate constant can be increased by:
  1. increasing the concentration of the NO.
  2. increasing the concentration of the Cl2.
  3. increasing the temperature.
  4. doing all of these.
Answer

The rate constant is only changed by variation in the temperature
Q632-04 For a certain second order decomposition reaction, the rate is 0.30 mol dm-3 s-1, when the concentration of the reactant is 0.20 M. What is the rate constant (dm3 mol-1 s-1) for this reaction?
  1. 1.5
  2. 2.2
  3. 3.0
  4. 7.5
Answer

Rate = k[A]2

[A] = 0.20 M, therefore 0.30 mol dm-3 s-1 = k x (0.20)2

therefore k = 0.30/0.04

Therefore the value of the rate constant = 7.5 dm3 mol-1 s-1
Q632-05 For the following reaction: H2(g) + I2(g) → 2 HI(g); the experimental rate law is: Rate = k[H2][I2]. When time is given in seconds and the concentration is in mol dm-3, the units for the rate constant are:
  1. mol dm-3 s-1
  2. mol-1 dm3 s-1
  3. s-1
  4. mol-1 dm3 s
Answer

The rate equation can be rearranged from rate = k[H2][I2] to give k = rate/[H2][I2]

Substituting the units into rate and the concentrations

k = mol dm-3 s-1 /mol dm-3 x mol dm-3

Therefore units of k = dm3 mol-1 s-1 (these are the units of the rate constant for a second order reaction)
Q632-06 At 800ºC, the instantaneous rate of the reaction 2NO + 2H2 → N2 + 2H2O is 3.5 mol dm-3 s-1 when the concentrations of nitrogen(II) oxide and hydrogen are each 0.30 mol dm-3. If the rate expression for this reaction is: Rate = k[NO]2[H2], the numerical value of the specific rate constant is:
  1. 1.2
  2. 3.5
  3. 3.90 x 101
  4. 1.30 x 102
Answer

Substituting values into the rate expression: Rate = k[NO]2[H2]

3.5 = k [0.30]2[0.30]

∴ k = 3.5/0.027 = 1.3 x 102 (three significant figures)
Q632-07 The reaction A + B → AB is 1st order with respect to A and zero order with respect to B. The reaction is begun with the initial concentration of both reactants at 0.100 mol dm-3. After 1.5 hours the concentration of B has dropped to 0.060 mol dm-3. What is the approximate value of the specific rate (reaction rate) constant for this reaction?
  1. 0.15 hr-1
  2. 0.33 hr-1
  3. 0.61 hr-1
  4. the specific rate constant cannot be determined unless the mechanism of the reaction known.
Answer

The rate expression for this reaction is: Rate = k[A][B]0

This can be simplified to Rate = k[A] as anything raised to the power of zero = 1

Initial concentration of B = 0.100 mol dm-3 and final concentration B = 0.0600 mol dm-3

Therefore the [B] changes by 0.100-0.060 = 0.040 mol dm-3 in 1.5 hours = 0.040 x 2/3 mol dm-3 hr-1 = 0.027 mol dm-3 hr-1

[A] changes in the same rate as the two reactants react in a 1:1 ratio.

Using the average value for the [A] over this time = 0.08 mol dm-3 and substituting values into the rate expression:

0.027 = k [0.08], ∴ k = 0.027/0.08 = 0.33 hr-1
Q632-08 The following data were obtained for the reaction between A and B:

Experiment Initial concentration of reactants
(mol dm-3)
A                       B		 Initial rate of reaction
(mol dm-3 h-1)
1 0.200 0.200 0.50
2 0.400 0.200 2.00
3 0.400 0.800 8.00

i) Give the order with respect to A
ii) Give the order with respect to B
iii) Write the rate expression for this reaction
iv) Using the data from the first experiment, calculate the value of the rate constant giving the units

Answer

Inspection of the data from experiments 1 & 2 (constant [B]) shows that the rate quadruples as [A] doubles, thus order with respect to [A] = 2.

Inspection of the data from experiments 2 & 3 (constant [A]) shows that the rate quadruples as [B] quadruples, thus order with respect to [B] = 1.

∴ Rate = k[A]2[B]

Value of the rate constant k is obtained by rearranging the rate equation from Rate = k[A]2[B] to k = rate/[A]2[B]

∴ using data from the first experiment k = 0.5/(0.04 x 0.2)

∴ k = 62.5 dm6 mol-2 hr-1 (units of a third order reaction)
Q632-09 Nitrogen(II) oxide reacts with hydrogen as shown by the following equation: 2NO(g) + 2H2(g) → N2(g) + 2H2O(g). The table below shows how the rate of reaction varies as the reactant concentrations vary:

Experiment Initial concentration of reactants
(mol dm-3)
NO                   H2		 Initial rate
(mol N2 dm-3 s-1)
1 0.100 0.100 2.53 x 10-6
2 0.100 0.200 5.05 x 10-6
3 0.200 0.100 10.10 x 10-6
4 0.300 0.100 22.80 x 10-6

i) Determine the order with respect to NO and with respect to H2.
ii) Write the rate expression for the reaction.
iii) Calculate the value for the rate constant including units.

Answer

Inspection of the data from experiments 1 & 2 (constant [NO]) shows that the rate doubles as [H2] doubles, thus order with respect to [H2] = 1.

Inspection of the data from experiments 1 & 3 (constant [H2]) shows that the rate quadruples as [NO] doubles, thus order with respect to [NO] = 2.

Hence the rate expression is Rate = k[NO]2[H2]

Using values from experiment 1, 2.53 x 10-6 = k [0.1]2[0.1]

Therefore k = 2.53 x 10-6 /0.001 = 2.53 x 10-3 dm6 mol-2 s-1 (units of a third order reaction)
Q632-10 The following data were obtained for the reaction between gases A and B:

Experiment Initial concentration of reactants
(mol dm-3)
A                    B		 Initial rate
(mol dm-3 s-1)
1 1.0 x 10-3 2.0 x 10-3 3.0 x 10-4
2 2.0 x 10-3 2.0 x 10-3 3.0 x 10-4
3 1.0 x 10-3 4.0 x 10-3 1.2 x 10-3

i) Define the term overall order of reaction.
ii) Deduce the order of the reaction with respect to A and the order of the reaction with respect to B
iii) Write the rate expression for the reaction between A and B
iv) Use the data from experiment 1 to calculate the value of the rate constant for the reaction and state its units.

Answer

Inspection of the data from experiments 1 & 2 (constant [B]) shows that the rate does not change as [A] doubles, thus order with respect to [A] = 0.

Inspection of the data from experiments 1 & 3 (constant [A]) shows that the rate quadruples as [B] doubles, thus order with respect to [B] = 2.

∴ rate expression is Rate = k[B]2 ([A] does not appear as anything raised to the power of zero equals 1)

Using date from experiment 1, 3.0 x 10-4 = k [1.0 x 10-3]2

∴ k = 3.0 x 10-4 /1.0 x 10-6 = 300 dm3 mol-1 s-1 (units of a 2nd order reaction)

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