5.2.2 Deriving Rate Equations

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

Concentration-time graphs of a zero-order reaction

• In a first-order reaction the concentration of the reactant decreases with time
  • The graph is a curve going downwards and eventually plateaus

Concentration-time graphs of a first-order reaction

• 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

Concentration-time graphs of a second-order reaction

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 

Rate-time graph of a zero-order reaction

• 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]

Rate-time graph of a first-order reaction

• 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

Rate-time graphs of a second-order reaction

Order of reaction from half-life

• The order of a reaction can also be deduced from its half-life (t_1/2 )
• The half-life (t_1/2) 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

Half-lives of zero, first and second-order reactions

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 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