# Chemical Kinetics, cont.

### So how do we determine the order of a reaction? cont.

Last time looked at an algebraic method to detrmine rate law. We can also determine the order graphically. If we plot ri = 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, t1/2, is independent of [A] and therefore of time.
• t1/2 = 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 t1/2 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 Zo[A][B] where Zo 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, Ea. 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-(Ea/RT). Note that
• as Ea increases the fraction decreases.
• as T increases, the fraction increases. Arrhenius Equation, k = Ae-(Ea/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-(Ea/RT)) Zo[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.

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

• Ea is given by the Arrhenius equation:

k = A e-(Ea/RT)

where A = pZo

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

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

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

public domain image via Wikipedia Creative Commons

• 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 = -Ea/R (1/T1 - 1/T2)

## Transition States and Reaction Progress (Reaction Coordinate) Diagrams

#### Reaction with -G:

How do we interpret this diagram?

• The x-axis can be thought of as a time axis for a single reaction, starting with the reacting molecules and ending with the product molecules.
• Thus as the reaction progresses the energy in the particles goes up (they become less stable) as they are "pushed together" in a collision.
• The upward slope represents the energy of repulsion and bond stretching.
• The downward slope represents the formation of the new bonds and the separation of the products.
• The top of the peak represents the "transition state" (TS) a high energy combination of atoms which can go to either reactants or products.
• The energy difference between the reactants and the TS is called the activation energy, Eact. This is the energy the reacting species must have (as kinetic energy etc.) in order to overcome the barriers to reaction (repulsion, bond energy). The greater the value of Eact the slower the reaction, because fewer molecules have enough energy to react. If Eact is very low (not much of a hill) then the reaction will proceed readily and rapidly.
• The energy difference between the reactants and the products (G) tell us how far the reaction will go (that is, the fractions of reactants and products in equilibrium with each other). If the change is negative (products have less energy than reactants as seen on the diagram above), then products are favored (the reaction will go to mostly product. If, on the other hand, the products are at a higher energy then G will be positive, and the reactants will be favored (the reaction will stay mostly reactants, and little product will be made - see the diagram below). If G is zero, then the reaction mixture will end up with equal amounts of reactant and product.

#### Reaction with +G:

Thus the size and sign of G tells us how far a reaction will proceed, while the size of Eact tells us how fast the reaction will be. Note on this plot that Eact goes from the lower blue line (reactant energy) to the top of the peak. Note that reactions can be thermodynamically very favorable (go nearly completely to product and release lots of energy), but kinetically unfavorable (they react very slowly). We say the reactants are thermodynamically unstable, but kinetically stable.

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