Tuesday, March 25, 2008

Wednesday, March 19, 2008

four experiences down but a final left

I have posted a final topic sheet and a final equation sheet. Not only does the latter have most of the important mathematical relations we studied in class but also makes a great wrapping paper. Solutions to hw.6 went up a couple of days ago...

A note on hw.6 problem 03 => answers for (a) and (b) were inadvertently switched: The faster rate has the larger k.

Thanks for all your hard work this quarter. Our final on Friday will likely be your last before spring break (sadly I have another one immediately afterwards) so make sure you get a good rest before returning for pchem III, perhaps the best of all the pchems.

To the three of you escaping to go on the boat, we'll miss you! [tear]

Wednesday, March 5, 2008

On the road to equilibrium

Equilibrium was an important theme last quarter, forming the basis of the ideas of reversibility and nearly all of the rest of thermodynamics. This quarter equilibrium has shown itself as the state towards which transport properties and chemical kinetics approach. As we solve the mechanism of opposing/reversible reactions, we are bridging the worlds of thermodynamics and kinetics.

For the reaction A ↔ B, we can readily write the differential equations for A and B:

dA/dt = - kfA + krB
dB/dt = + kfA - krB

Assuming that there is only A initially, we can state that Ao = A+B, or B = Ao - A, which, when plugged into equation 1 above, makes it integrable (two variables only). The solutions are:

A = Ao[ (kr + kfe-(kf+kr)t)/(kr + kf) ]
B = Ao[ 1 - (kr + kfe-(kf+kr)t)/(kr + kf) ]

To connect to equilibrium, we recognize that A(∞)=Aeq and that B(∞)=Beq:

Aeq = Ao[ kr/(kr + kf) ]
B = Ao[ kf/(kr + kf) ]

Cool, from just kinetics we can predict equilibrium quantities of A and B. But not only that, we can take their ratio and obtain the equilibrium constant:

K = kf/kr

Remarkably, this is the result for all reversible reactions ( 2A ↔ B, 3A ↔ 2B, etc) and is called the principle of microscopic reversibility.

More complicated mechanisms can be understood as combinations of branching, sequential and opposing reactions. When intermediates are reactive, we can often use the steady-state approximation, which says that dI/dt ≅ 0. This allows us to more easily obtain rate laws from mechanisms without solving elaborate series of differential equations that take lots of time and make your soul cry.

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].