1.9 Physical Chemistry (Rate Equations)

• Rate Equation


  Relates mathematically the rate of reaction to the concentration of the reactants
• Rate Equation


  1. r = k[A]m[B]n

  2. r is used as symbol for rate

  3. Unit of r is usually mol dm-3 s-1

  4. [A] means the concentration of A (unit mol dm-3)

  5. k is called the rate constant

  6. m, n are called reaction orders

  7. Orders are usually integers 0,1,2
• Reaction Orders


  • 0 means the reaction is zero order with respect to that reactant
  • 1 means first order
  • 2 means second order
  • Orders are not the same as the stoichiometric coefficients in the balanced equation. They are worked out experimentally.
• Total Order


  Worked out by adding all the individual orders together (m+n)
• Zero Order


  1. The concentration of A has no effect on the rate of reaction

  2. r = k[A]0 = k
• First Order


  1. The rate of reaction is directly proportional to the concentration of A

  2. r = k[A]1
• Second Order


  1. The rate of reaction is proportional to the concentration of A squared

  2. r = k[A]2
• Rate Constant (k)


  • The units of k depend on the overall order of reaction
  • The value of k is independent of concentration and time. It is constant at a fixed temperature.
  • The value of k refers to a specific temperature and it increases if we increase temperature
• Units of k


  • For a 1st order overall reaction the unit of k is s-1
  • For a 2nd order overall reaction the unit of k is mol-1dm3s-1
  • For a 3rd order overall reaction the unit of k is mol-2dm6s-1
• Calculating units of k

  1. Rearrange rate equation to give k as subject

  2. Insert units and cancel
• Example 1 (first order overall)

  • Rate = k[A][B]0 m = 1 and n = 0
  • Reaction is first order in A and zero order in B
  • Overall order = 1 + 0 = 1
  • Usually written: Rate = k[A]
• Example 2


  • Write rate equation for reaction between A and B where A is 1st order and B is 2nd order
  • r = k[A][B]2
  • Overall order is 3
  • Calculate the unit of k: Unit of k = mol-2dm6s-1
• Continuous Monitoring


  • Following one experiment over time recording the change in concentration
  • The gradient represents the rate of reaction
  • The reaction is fastest at the start where the gradient is steepest
• Measurement of the change in volume of a gas

  1. Mg + 2HCl MgCl2 +H2

  2. Using a gas syringe is a common way of following this
• Typical Method for measuring gas volume
• Initial Rate


  • The rate at the start of the reaction, where it is fastest
  • Can be calculated from the gradient of a continuous monitoring conc vs time graph at time = zero
• Large Excess of Reactants

  If the concentration of one of the reactant is kept in a large excess then that reactant will appear not to affect rate and will be pseudo-zero order
• Comparing continuous rate curves


  • Different volumes of the same initial concentrations will have the same initial rate (if other conditions are the same) but will end at different amounts
  • The higher the concentration/temperature/surface area the faster the rate (steeper the gradient)
• Initial Rate Method


  • The initial rate can be calculated from taking the gradient of a continuous monitoring conc vs time graph at time = zero
  • Initial rate can also be calculated from clock reactions where the time taken to reach a fixed concentration is measured
• A Common Clock Reaction

  1. Hydrogen peroxide reacts with iodide ions to form iodine. The thiosulfate ion then immediately reacts with iodine formed in the second reaction

  2. When the I2 produced has reacted with all of the limited amount of thiosulfate ions present, excess I2 remains in solution. Reaction with the starch then suddenly forms a dark blue-black colour.
• Example 3: Deduce the rate equation for the following reaction, A+ B+ 2C D + 2E, using the initial rate data in the table

  • Working out order graphically
  • Working out orders from experimental initial rate data
  • Working out orders when two reactant concentrations are changed simultaneously
• Log rate


  log k + n log [Y]
• A graph of log rate vs log [Y] will yield a straight line where the gradient is equal to the order n
• Rate equation


  Y = c + m x
• High concentrations with quick times will have the biggest percentage errors
• Working out orders from experimental initial rate data


  1. Normally do a series of experiments where the initial concentrations of reactants are changed (one at a time) and measure the initial rate each time

  2. log (rate) y intercept = log K

  3. gradient = n = change in y/change in x
• Working out orders when two reactant concentrations are changed simultaneously

  1. Compare between two experiments where only one reactant has its initial concentration changed

  2. If both reactants are changed then the effect of both individual changes on concentration are multiplied together to give on overall change on rate
• Reaction where the rate equation is r = k [A] [B]2

  • If the [A] is x2 that rate would x2
  • If the [B] is x3 that rate would x32= x9
  • If these changes happened at the same time then the rate would x2x9= x 18
• Deducing the rate equation from initial rate data

  1. Compare between experiments to determine the order with respect to each reactant

  2. The overall rate equation is r = k [X] [Y]2

  3. The reaction is 3rd order overall and the unit of the rate constant =mol-2dm6s-1
• Calculating a value for k using initial rate data

  1. Rearrange the rate equation to solve for k

  2. k = r/([X][Y]2)

  3. Remember k is the same for all experiments done at the same temperature