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A great simplification is possible when the conditions are such that only a pair of atomic levels is significantly involved in the interaction with the radiation field. This is the two-level atom approximation, and it is a very good approximation in many cavity QED experiments involving Rydberg atoms. Let us call I j) the upper atomic state and I !) the lower atomic state. The essence ofthe two-level atom approximation is that these two states span the whole of the relevant part of atomic space; that is, for all practical purposes, we can take

(A )



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1= I !)(! I + I i)(i I,





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In the above equations and are used to indicate all acid or alcohol species in the reaction mixture (ie, monomer, dimer, trimer, , n-mer) Species I and II are shown in the form of their associated ion pairs since polymerization often takes place in organic media of low polarity (A ) is the negative counterion derived from the acid HA Polyesteri cations, like many other step polymerizations, are equilibrium reactions However, from the practical viewpoint of obtaining high yields of high-molecular-weight product such polymerizations are run in a manner so as to continuously shift the equilibrium in the direction of the polymer In the case of a polysteri cation this is easily accomplished by removal of the water that is a by-product of the reaction species II (Eq 2-14) Under these conditions the kinetics of polymerization can be handled by considering the reactions in Eqs.

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You can write a formula to perform a calculation on data in your worksheet. In Excel, all formulas begin with an equals sign (=) and contain the cell references of the cells that contain the relevant data. (Note that in addition to referring to cells in the current worksheet, you can also build formulas that refer to cells in other worksheets.) For example, the formula for adding the contents of cells C3 and C4 together is =C3+C4. You create formulas in the Formula bar; formula results appear in the cell to which you assign a formula.

(5.63)

2-13 and 2-14 to be irreversible (A consideration of the kinetics of a reversible step polymerization is given in Sec 2-4c) The rate of a step polymerization is conveniently expressed in terms of the concentrations of the reacting functional groups Thus the polyesteri cation can be experimentally followed by titrating for the unreacted carboxyl groups with a base The rate of polymerization Rp can then be expressed as the rate of disappearance of carboxyl groups d[COOH]/dt For the usual polyesteri cation, the polymerization rate is synonomous with the rate of formation of species II; that is, k4 is vanishingly small (since the reaction is run under conditions that drive Eqs 2-13 and 2-14 to the right), and k1 , k2 , and k5 are large compared to k3 .

p = Ipl = (I !)(! I + I i)(i i)er{1 !)(! I + I i)(i = I i)(! Ix + I !)(i Ix*,

An expression for the reaction rate can be obtained following the general procedures for handling a reaction scheme with the characteristics described [Moore and Pearson, 1981] The rate of polymerization is given by.

(5.64)

5-113

methanol is added and then the AB block copolymer isolated. Various types of block copolymers (di, tri, multiblocks) are obtained by using the proper sequencing of additions

where X == (i Iii !) and we have assumed that (! If I !) = (i Irl i) = 0 (which is often the case for symmetry reasons). The atomic Hamiltonian also simplifies in this case. Setting the zero of energy halfway between the two atomic levels for convenience, the atomic Hamiltonian becomes (5.65) where w is the frequency of the atomic transition between I i) and I!). When adopting the two-level atom approximation, one often represents the atomic operators by Pauli matrices. These matTices are defined by

of different monomers provided that each propagating carbanion can initiate polymerization of the next monomer For example, polystyryl carbanion initiates polymerization of methyl methacrylate, but the reverse does not occur Crossover from one carbanion to another occurs only when the new carbanion is comparable or higher in stability than the original carbanion Poly(methyl methacrylate) carbanion is more stable than polystryrene carbanion Carbanion stability can be estimated from the pKa of the conjugate acids of model carbanions Toluene and ethyl acetate have pKa values of 43 and 30, in line with the higher stability of the PMMA carbanion Thus, the synthesis of a styrene-methyl methacrylate block polymer requires that styrene be the rst monomer Further, it is useful to decrease the nucleophilicity of polystyryl carbanions by adding a small amount of 1,1-diphenylethene to minimize attack at the ester function of MMA [Quirk et al, 2000].

a == I !)(i I, ax==a+a t ,

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