Sunday, March 2, 2008

Chemical Kinetics and You

The last time-dependent phenomenon we study this quarter is chemical kinetics or the change of chemical identity as a function of time. In many ways this is an extension of transport phenomena except that the physical space over which it flows has been replaced with a chemical space. It is also one of the subfields of physical chemistry that ties most strongly with organic chemistry and biochemistry.

We can track the extent of the general reaction aA( ) + bB( ) → cC( ) + dD( ) by tracking A, B, C or D with whichever experimental techniques are most convenient, as in optical activity, absorbance or nmr. Formally, the number of moles of species i at any time t will be tethered to the extent of reaction ξ by the following relation:

ni = no,i + νiξ

It is through the extent of reaction that we define the [extensive] rate of reaction: Rate = dξ/dt. To obtain the intensive rate, we divide by the volume to get R = Rate/V. This is why, incidentally, that mol L-1 is the natural unit of kinetics (instead of, say, activity which establishes the basis of most of solution thermodynamics).

Several properties affect the rate of a reaction, which include (a) concentration, (b) temperature, (c) physical state and (d) presence of catalysts. The concentration dependence is most important, most common and, unfortunately, most complicated -- we are decades away, if that, from being able to predict the behavior of even the simplest reactions from their stoichiometry.

Rates of reaction are dependent on the frequency of collisions, which is in turn dependent on the concentration of reactants, and we model this mathematically with a rate equation (aka rate law): R = kAαBβ where α and β, the reaction orders with respect to A and B, are basically sensitivity factors to changes in concentration. The higher the order, the more sensitive that reaction is to changes in that reactant. The overall order is just the sum of the exponents, which are usually integers [like 1 or 2] or some simple fraction [like 3/2 or 1/4]. Incidentally, it will be through k itself that dependence on temperature and catalysis will appear, in the familiar, but empirical, Arrhenius equation.

Finding the rate law for a given reaction is the holy grail... once we have obtained it, we can in principle know the concentration of all reactants and products at any time afterwards (do you see the similarity to certain transport properties like, say, diffusion?). We discussed four methods of obtaining rate orders (and, hence, the general form of the rate law):

  1. isolation method
  2. method of initial rates
  3. ntegrated rate laws, and
  4. method of half-lives

When obtaining the integrated rate laws for 1st, 2nd and, finally, nth order, we discovered that the half-life is concentration dependent for all orders except for n=1. It is this concentration dependence that allows us to manipulate half-life experiments to give us information about the order of reaction, one of the most accurate methods currently at our disposal.

Another important goal in kinetics is the elucidation of the underlying mechanism for a given reaction, in which we conjecture a series of individual collisonal events [with molecularity 3 or less] that adds up to the overall stoichiometric reaction. It is only within these elementary step processes that we can correlate molecularity to reaction order (since, again, these represent actual collisions, not just stoichiometric bookkeeping).

Nearly all complex mechanisms are constructed from the following three elementary fragments:

branching/parallel: C ← A → B
sequential/consecutive: A → B → C
reversible/opposing: A ↔ B

In a multistep mechanism, all species are changing with time and it is the goal of pchem to know their functional forms, if possible [we will see shortly that this is easier said than done, since many mechanisms become mathematically intractable with just a few fragments].

For any mechanism, however, we can readilty write down the differential equations that could, in principle, be solved [analytically, numerically or graphically]. For instance, for the branching process above, we have three entities changing with time so we have three equations:

  1. dA/dt = -k1A - k2A
  2. dB/dt = k1A
  3. dC/dt = k2A
For the consecutive reaction mechanism, we get a different set of three:
  1. dA/dt = -k1A
  2. dB/dt = k1A - k2B
  3. dC/dt = k2B
In our next lecture, we will look at exact solutions to these two sets of coupled differential equations as well as reversible reactions [which will take us back to, of all places, equilibrium].

2 comments:

Anonymous said...

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Aliya said...

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