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

  8. 0 means the reaction is zero order with respect to that reactant

  9. 1 means first order

  10. 2 means second order

  
• The orders are not the same as the stoichiometric coefficients in the balanced equation. They are worked out experimentally.
• Total order for a reaction


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


  • The concentration of A has no effect on the rate of reaction
  • r = k[A]0 = k
• First order reaction


  • The rate of reaction is directly proportional to the concentration of A
  • r = k[A]1
• Second order reaction


  • The rate of reaction is proportional to the concentration of A squared
  • r = k[A]2
• Rate constant (k)


  • The units depend on the overall order of reaction
  • 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: s-1
  • For a 2nd order overall reaction: mol-1dm3s-1
  • For a 3rd order overall reaction: 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
  • 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
  • The rate drops as the reactants start to get used up and their concentration drops
  • The graph will eventual become horizontal and the gradient becomes zero which represents the reaction having stopped
• 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

  3. Typical Method:

  4. Measure 50 cm3 of the 1.0 mol dm–3 hydrochloric acid and add to conical flask

  5. Set up the gas syringe in the stand

  6. Weigh 0.20 g of magnesium

  7. Add the magnesium ribbon to the conical flask, place the bung firmly into the top of the flask and start the timer

  8. Record the volume of hydrogen gas collected every 15 seconds for 3 minutes
• 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
  • A measure of initial rate is preferable as we know the concentrations at the start of the reaction
• 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

  3. By repeating the experiment several times, varying the concentration of a reactant (keeping the other reactants at constant concentration) you can determine the order of reaction with respect to that reactant

  4. The initial rate of the reaction can be represented as (1/t )
• Example 3: Deduce the rate equation for the following reaction, A+ B+ 2C D + 2E, using the initial rate data in the table

  • To calculate the order for a particular reactant it is easiest to compare two experiments where only that reactant is being changed
  • For reactant A the order is first order
  • For reactant B the order is second order
  • For reactant C the order is zero order
  • The overall rate equation is r = k [A] [B]2
  • The reaction is 3rd order overall and the unit of the rate constant =mol-2dm6s-1
• Working out order graphically

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

  log rate = log k + n log [Y]
• Graph of log rate vs log [Y]

  • Yields a straight line
  • 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. Do a series of experiments where the initial concentrations of reactants are changed (one at a time)

  2. Measure the initial rate each time
• Log rate equation

  • y intercept = log K
  • 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, the effect of both individual changes on concentration are multiplied together to give on overall change on rate
• Calculating a value for k using initial rate data

  Choose one experiment and put the values into the rate equation rearranged to give k
• k is the same for all experiments done at the same temperature
• Increasing temperature

  Increases the value of the rate constant k
• Zero order reactants

  Rate stays constant as the reactant is used up, concentration has no effect on rate
• For zero order, rate = k
• Arrhenius equation

  • k = Ae-EA/RT
  • ln k = ln A - EA/(RT)
• Mechanism

  • Series of steps through which the reaction progresses, often forming intermediate compounds
  • Slowest step is the rate-determining step
  • Molecularity of the slow step determines the order of reaction