Chemical Kinetics

Study of rates and mechanisms of reactions. Experimentally, look at rates of reactions, use this information to guess mechanisms

• Rate is a measure of how fast the reaction goes. Can measure how fast a reactant is used up, or how fast a product appears. Use the stoichiometry of the reaction to relate a particular rate to the overall reaction rate.
• A mechanism is a detailed description of the steps leading from reactants to products.

Concentration Dependence of Reaction Rates: Concentrations are assumed to be in Molarity unless otherwise specified.

Consider the reaction:

A + 2B+ C D + E

• Let's assume that if [A] is doubled while all other concentrations are unchanged, that the rate doubles. We can then say that the rate is proportional to [A]; r [A]
• Let's now assume that if [B] is doubled the rate quadruples, while 3 x [B] gives 9 x rate (again, all other concentrations are unchanged). We can say that the rate is proportional to the square of [B]; r [B]².
• Finally let's say that if [C] is changed, no effect is seen on the rate; r [C]⁰ or r = constant.
• We can now combine these expressions to give

  r [A] [B]² [C]⁰,or

  r = k [A] [B]²

This expression is referred to as a Rate Lawwith the sum of various exponents referred to as the order of the reaction. The overall order of this reaction is thus 3rd order - it is first order in A, second order in B, and zero order in C.

Looking at the different reaction orders:

• First Order: For A C, or A + B C + D, etc.
  • r = k [A]; & r =-d [A]/dt = d[C]/dt.
  • Note that we could measure the rate by measuring the changes in concentrations of any of the species. That is, even though changing [B] won't affect the rate, we could measure the rate change occurring by changing [A] by measuring [B] since the stoichiometry says that for each A lost, one B is also lost!
  • Note: if double [A], double rate; triple [A], triple rate.

• Second order: For A C, or A + B C + D, etc. Two cases:

  r = k [A][B]

  • Note if double [A] or [B] will double rate; if double both [A] and [B] will quadruple rate

  r = k [A]², etc.

  • Note if double [A] will quadruple rate, if triple [A] will increase rate nine-fold

• Higher order reactions occur, but are uncommon.
• Zero order: r = k[A]⁰ = k: Only occurs with catalysts, important in enzyme catalysis. 0 order also only occurs above a minimum [A].

So how do we determine the order of a reaction? The experiment is to increase the concentration of a single reactant, and observe the rate. Sometimes the order will be obvious (i.e. double, double = directly proportional = 1st order). If not, then can take the results of two experiments and divide them and do some algebraic manipulations to find the correct order.

Example: Find the order of the reaction given the data below.

We can also determine the order graphically. If we plot r_i = d[P]/dt vs. [P] for 0 - 3rd order we see the plots below:

Its fairly easy to distinguish 0 order & first-order (linear), but the others can be difficult since they are all curves with only slightly different basic shapes (if we adjust the x-axis to fit them, they come fairly close for eye-balling). However, they are readily distinguished via linear plots.

Thus, the various orders can be linearized:

• First order reactions will give a linear plot with a negative slope when ln [A] is plotted against time.

• ln [A] = -kt + ln[A]_o
• Note that for first order only, the half life, t_1/2, is independent of [A] and therefore of time.
• t_1/2 = ln 2 / k

• Second order reactions give a linear plot when 1 / [A] is plotted against time
  • 1 / [A] = kt + 1 / [A]₀
  • in this case t_1/2 changes with [A], therefore it changes continuously as time goes on.

Collision Theory

We assume that particles must collide in order to react. Thus a first understanding of reaction rates is based on understanding what influences the frequency of collisions.

• Can have different molecularites. This refers to the numbers of molecules (or particles - can also be atoms or ions) involved in the collision.
  • unimolecular - only one molecule involved (there is no collision)
  • bimolecular - 2 body collision
  • termolecular - 3 body collision (this is very rare, essentially never get more than three body collisions)
• Since particles must collide to react, the rate will depend on concentration. Thus for the bimolecular reaction of A and B
  • rate Z_o[A][B] where Z_o is the collision frequency when [A] = [B] = 1 M
• But not all collisions will lead to reaction. At least two additional factors are important:
  • Activation energy, E_a. This is the energy needed to overcome repulsion, bond strength etc. At any particular temperature the number of particles with a given energy is specified by N = e^-(E_a/RT). Note that
    • as E_a increases the fraction decreases.
    • as T increases, the fraction increases. Arrhenius Equation, k = Ae^-(E_a/RT). Can plot fraction of particles with a given KE vs. KE. A crude relation, which holds approximately for many common reactions is the so called Q₁₀. This states that for every 10 °C increase in temperature we get a doubling of reaction rate.
  • Orientation of collision. The steric factor or probability factor (p) is the fraction of collisions which have orientations favorable for reaction. Can range from very small numbers (one atom or molecule must hit another at a very specific place) to about 1.
• Putting these factors together we can then write:

 

r = p (e^-(E_a/RT)) Z_o[A][B]

r = k [A][B]

• Note that for kinetics the stoichiometry of the reaction is not necessarily related to the rate law, rather the rate law is related to a single slow step in the reaction process.

Reaction Rates vs. Temperature

We know that reaction rates are influenced by temperature - wood reacts with oxygen much faster in a fire than at room temperature. From the expressions above it is also obvious that rate should be influenced by temperature.

Recall the distribution of KE in a population of molecules (the Maxwell-Boltzman Distribution). [overhead, Figure 12.12, p 588 in Zumdahl] Note that for this plot higher temperatures for Xe would look very similar to the plots for the smaller gases.

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At higher temperatures more molecule will have KE's exceeding E_a, thus the reaction rate will increase. So how can we determine the magnitude of this effect?

• E_a is given by the Arrhenius equation:

k = A e^-(E_a/RT)

where A = pZ_o

• By taking the log of both sides and rearranging we can get the equation of a straight line (y = ax + b):

ln k = (-E_a/R) (1/T) + ln A

• A plot of ln k vs. 1/T will then give a slope of -E_a/R, from which E_a is readily obtained.

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• Alternatively, k values can be obtained at two different temperatures. Subtracting the resultant equations (and recalling that ln a - ln b = ln a/b) then gives:

ln k₁/k₂ = -E_a/R (1/T₁ - 1/T₂)

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Last modified 17 April 2009