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2-78 does not apply to systems containing a monofunctional reactant unless it is present in addition to stoichiometric amounts of bifunctional reactants (either A B or A A B B) That is, for example, Eq 2-78 applies to a reaction system of 10 mol each of A and B plus 1 mol B but does not apply to a system of 10 mol A A B A, 9 mol B and 1 mol B In reality, the two reaction systems yield very nearly the same B, degree of polymerization at any particular extent of reaction The Carothers equation yields X n values of 210 and 200, respectively, for the two systems at complete conversion The use of Eq 2-78 for the second reaction system yields a gross distortion The value of r obtained from Eq.



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2-80 would be unity (the same as for a perfectly stoichiometric system of 10 mol each of A and B A B!) and this leads to a X n value of in nity at complete conversion This is correct for the stoichiometric system but clearly is incorrect for the reaction system containing the large amount of monofunctional B reactant The use of Eq 2-78 must be limited to situations in which there are stoichiometric amounts of bifunctional reactants (or A B) Also, Eq 2-78 is inapplicable to any systems containing a reactant with functionality greater than 2 The Carothers equation, without such limitations, has two general uses First, it can be used in the form of Eq 2-137 to calculate the degree of polymerization of a reaction system using the appropriate value of favg (whose calculation depends on whether stoichiometric or nonstoichiometric amounts are present).





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409. E. H. Lieb. The stability of matter. Reviews ofModem Physics, 48(4):553-569, October 1976. 410. M. Ligare and S. Becker. Simple soluble models of quantum damping applied to cavity quantum electrodynamics. American Journal of Physics, 63(9):788-796, September 1995.

If one obtains a negative value of X n in the calculation, it.

411. G. Lindblad. On the generators of quantum dynamical semigroups. Communi-

means that the system is past the gel point at that conversion. Second, the use of Eq. 2-139 allows one to calculate the critical extent of reaction at the gel point. Keep in mind that the extent of reaction calculated from Eq. 2-137 or 2-139 refers to the extent of reaction of the A functional groups (de ned as those groups not in excess). The extent of reaction of the B functional groups is rp or rpc. 2-10b Statistical Approach to Gelation: X w ! 1

cations in Mathematical Physics, 48: 119-130, 1976.

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Flory [1941, 1953] and Stockmayer [1943, 1952, 1953] used a statistical approach to derive an expression for predicting the extent of reaction at the gel point by calculating when X w approaches in nite size. The statistical approach in its simplest form assumes that the reactivity of all functional groups of the same type is the same and independent of molecular size. It is further assumed that there are no intramolecular reactions between functional groups on the same molecule. (These assumptions are also inherent in the use of Eqs. 2-137 through 2-143). For this derivation the branching coef cient a is de ned as the probability that a given functional group of a branch unit at the end of a polymer chain segment leads to another banch unit. For the polymerization of A with B and Af this corresponds to A B obtaining a chain segment of the type

412. M. Loffler, G. M. Meyer, and H. Walther. Spectral properties of the one-atom laser. Physical Review A, 55(5):3923-3930, May 1997. 413. F. London. Die Theorie von Weyl und die Quantenmechanik. Naturwissenschaften, 15:187, February 25,1927. 414. F. London. Quantenmechanische Deutung derTheorie von Weyl. Zeitschriftfiir Physik, 42:375-389, April 14, 1927. 415. H. A. Lorentz. The Theory of Electrons. Dover, New York, 1952. Originally published 1915.

A A + B B + Af A(f 1) A(B BA A)nB BA A(f 1)

416 R Loudon and P L Knight Squeezed light Journal ofModem Optics, 34:709759,1987 417 L A Lugiato, M O Scully, and H Walther Connection between microscopic andmacroscopicmasertheory PhysicaIReviewA,36(2):740-743,July 15,1987 418 J Luo, L-C Tu,Z-K Hu, and E-J Luan New experimental limit on the photon rest mass with a rotating torsion balance Physical Review Letters, 90(8):081801, February 28, 2003 This experiment sets an upper limit of 12x 10- 51 g to the photon's rest mass The previous record was 2 x 10-50 g due to R Lakes, Physical Review Letters, 80(9):1826-1829, March 2,1998 419 J Liitzen Heaviside's operational calculus and the attempts to rigorise it Archive for History of Exact Sciences, 21: 161-200, 1979 420 J Liitzen The Prehistory of the Theory of Distributions, volume 7 of Studies in the History of Mathematics and Physical Sciences Springer-Verlag, Berlin, 1982 421 A MacDonald and Q Niu.

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