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F Mark, N M Bikales, C G Overberger, and G Menges, eds, WileyInterscience, New York, 1985 Donnelly, E F, D S Johnston, D C Pepper, and D J Dunn, J Polym Sci Polym Lett Ed, 15, 399 (1977) Dotcheva, D T and C B Tsvetanov, Makromol Chem, 186, 2103 (1985) Drenth, W and R J M Nolte, Acc Chem Res, 12, 30 (1979) Dumas, S, V Marti, J Sledz, and F Schue, J Polym Sci Polym Lett Ed, 16, 81 (1978)..

The operator a (k) and its Hermitian conjugate obey the following commutation relations deduced from (5.116) and (5.117):

2-15



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(5.118) (5.119)

where [COOH], [OH], and [C OH 2 ] represent the concentrations of carboxyl, hydroxyl, and protonated carboxyl (I) groups, respectively. The concentration terms are in units of moles of the particular functional group per liter of solution. Equation 2-15 is inconvenient in that the concentration of protonated carboxylic groups is not easily determined experimentally. One can obtain a more convenient expression for Rp by substituting for C(OH)2 from the equilibrium expression

(5.120)

2-16

Constructing Arguments Arguments are enclosed in parentheses. When applying a function to individual cells in a worksheet, you can use a comma to separate the cell addresses, as in =SUM(A5,B5,C5). When applying a function to a range of cells, you can use a colon to designate the first and last cells in the range, as in =SUM(B5:E12). If your range has a name, you can insert the name, as in =SUM(Sales).





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d COOH k3 K COOH OH HA dt 2-17

= 0,

Two quite different kinetic situations arise from Eq. 2-17 depending on the identity of HA, that is, on whether a strong acid such as sulfuric acid or p-toluenesulfonic acid is added as an external catalyst. 2-2a Self-Catalyzed Polymerization

[a (k), a!, (k') J = 15 1() (k - k') .

In the absence of an externally added strong acid the diacid monomer acts as its own catalyst for the esteri cation reaction. For this case [HA] is replaced by [COOH] and Eq. 2-17 can be written in the usual form [Flory, 1953]

In terms of a (k) and a! (k), the expression for Htrans takes the familiar form (5.121)

d COOH k COOH 2 OH dt 2-18

9We notice that because the canonical variables of the field have only two independent components. the Poisson brackets associated with arbitrary Cartesian components of these variables do not have the form given by (5.113). In fact. a straightforward calculation of such Poisson brackets in terms of (5.113) yields [Ai (k, t), II; (k', t)] = {Oij - k i kj /k 2 } 0 (k - k').

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where K and k3 have been combined into the experimentally determined rate constant k. Equation 2-18 shows the important characteristic of the self-catalyzed polymerization the reaction is third-order overall with a second-order dependence on the carboxyl concentration. The second-order dependence on the carboxyl concentration is comprised of two rst-order dependencies one for the carboxyl as the reactant and one as the catalyst. For most polymerizations the concentrations of the two functional groups are very nearly stoichiometric, and Eq. 2-18 can be written as

There is an elegant and beautiful way of deriving the form of the coupling between matter and radiation: to use the basic principle of gauge invariance. This principle says that the physics cannot depend on our choice of gauge for the electromagnetic potentials; that is, when we make the gauge change A'(r, t) = A(r, t)

where [M] is the concentration of hydroxyl groups or carboxyl groups. Integration of Eq. 2-19b yields

+ \7x(r, t),

2kt 1 M 2 1 M 2 0 2-20

You use the Insert Function dialog box to look for a particular function from among Excel s 300-plus available functions. Functions are divided into 12 categories: Financial, Date & Time, Math & Trig, Statistical, Lookup & Reference, Database, Text, Logical, Information, Engineering, Cube and Compatibility. After you have selected your function, you build the formula using the Function Arguments dialog box. Functions use arguments to indicate the cells that contain the values you want to calculate.

cj/(r,t) = cjJ(r,t) -

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