5.2.2 Deriving Rate Equations

Deducing Orders

Order of reaction

• For the general reaction

A + B → C + D

• The order of reaction shows how the concentration of a reactant affects the rate of reaction

Rate = k [A]m [B]n

• When m or n is zero = the concentration of the reactants does not affect the rate
• When the order of reaction (m or n) of a reactant is 0, its concentration is ignored
• The overall order of reaction is the sum of the powers of the reactants in a rate equation
• For example, in the reaction below, the overall order of reaction is 2 (1 + 1)

Rate = k [NO2] [Cl2]

Order of reaction from concentration vs. time graphs

• In a zero-order the concentration of the reactant is inversely proportional to time
• This means that the concentration of the reactant decreases with increasing time
• The graph is a straight line going down
• In a first-order reaction the concentration of the reactant decreases with time
• The graph is a curve going downwards and eventually plateaus
• In a second-order reaction the concentration of the reactant decreases more steeply with time
• The concentration of reactant decreases more with increasing time compared to in a first-order reaction
• The graph is a steeper curve going downwards

Order of reaction from rate vs. time graphs

• The progress of the reaction can be followed by measuring the initial rates of the reaction using various initial concentrations of each reactant
• These rates can then be plotted against time in a rate-time graph
• In a zero-order reaction the rate doesn’t depend on the concentration of the reactant
• The rate of the reaction therefore remains constant throughout the reaction
• The graph is a horizontal line
• The rate equation for this one reactant is rate = k
• In a first-order reaction the rate is directly proportional to the concentration of a reactant
• This means that if you doubled the concentration of the reactant, the rate would also double
• If you increased the concentration of the reactant by a factor of 3, the rate would increase by this factor as well
• The graph is a straight line
• The rate equation for this one reactant is rate = k [A]
• In a second-order reaction, the rate is directly proportional to the square of concentration of a reactant
• This means that if you doubled the concentration of the reactant then the rate would increase by 4 (22)
• If you increase the concentration by a factor of 3, then the rate would increase by a factor of 9 (32)
• The graph is a curved line
• The rate equation for this one reactant is rate = k [A]2

Order of reaction from half-life

• The order of a reaction can also be deduced from its half-life (t1/2 )
• The half-life (t1/2) is the time taken for the concentration of a limiting reactant to become half of its initial value
• For a zero-order reaction the successive half-lives decrease with time
• This means that it would take less time for the concentration of reactant to halve as the reaction progresses
• The half-life of a first-order reaction remains constant throughout the reaction
• The amount of time required for the concentration of reactants to halve will be the same during the entire reaction
• For a second-order reaction, the half-life increases with time
• This means that as the reaction is taking place, it takes more time for the concentration of reactants to halve

Calculating the initial rate

• The initial rate can be calculated using the initial concentrations of the reactants in the rate equation
• For example, in the reaction of bromomethane (CH3Br) with hydroxide (OH) ions to form methanol (CH3OH) the reaction equation and rate are as follows:

CH3Br + OH → CH3OH + Br (aq)

Rate = k [CH3Br][OH]

Where k = 1.75 x 10-2 mol-1 dm3 s-1

• If the initial concentrations of CH3Br and OH are 0.0200 and 0.0100 mol dm-3 respectively, the initial rate of reaction is:

Rate = k [CH3Br] [OH]

Initial rate = (1.75 x 10-2) x (0.0200) x (0.0100)

Initial rate = 3.50 x 10-6 mol dm-3 s-1

Deriving Rate Equations

Deriving Rate Equations from data

• Let’s take the following reaction and derive the rate equation from experimental data

(CH3)3CBr  +  OH  →  (CH3)3COH  +  Br

Table to show the experimental data of the above reaction • To derive the rate equation for a reaction, you must first determine all of the orders with respect to each of the reactants
• This can be done using a graph, but it doesn’t have to be – you can use tabulated data provided
• Take the reactants one at a time and find the order with respect to each reactant individually
• Identify two experiments where the concentration of the reactant you are looking at first changes, but the concentrations of all other reactants remain constant
• Repeat this for all of the reactants, one at a time, until you have determined the order with respect to all reactants

Order with respect to [(CH3)3CBr]

• From the above table, that is experiments 1 and 2
• The [(CH3)3CBr] has doubled, but the [OH] has remained the same
• The rate of the reaction has also doubled
• Therefore, the order with respect to [(CH3)3CBr] is 1 (first order)

Order with respect to [OH–]

• From the above table, that is experiments 1 and 3
• The [OH] has doubled, but the [(CH3)3CBr] has remained the same
• The rate of reaction has increased by a factor of 4 (i.e. increased by 22)
• Therefore, the order with respect to [OH] is 2 (second order)

Putting the rate equation together

• Once you know the order with respect to all of the reactants, you put them together to form the rate equation
• If a reactant has an order of 0, then you do not include it in the rate equation
• If a reactant has an order of 1, then you do not need to include the number 1 as a power
• If a reactant has an order of 2, then you raise that reactant concentration to the power of 2
• For this reaction, the rate equation will be:

Rate = k [(CH3)3CBr] [OH]2

Exam Tip

Be careful when reading the values in standard form! It is easy to make a mistake.

Close Close