Equations Carothers

Carothers, the pioneer of step-growth reactions, proposed a simple equation relating to a quantity p describing the extent of the reaction for linear polycondensations or polyadditions. 

If Nq is the original number of molecules present in an A-B monomer system and N is the number of all molecules remaining after time t, then the total number of functional groups of either A or B that have reacted is (Nq- N). At that time t, the extent of reaction p is given by 

If we remember that c = NJN, a combination of expressions gives the Carothers equation. 

This equation is also vahd for an A-B + B-B reaction when one considers that, in this case, there are initially 2A/q molecules. 

The Carothers equation is particularly enlightening when we examine the numerical relation between x andp thns, for p = 0.95 (i.e., 95% conversion), x = 50 and when p = 0.99, then x = 100. In practical terms, it has been fonnd that for a fiberforming polymer such as nylon-6.6 NH(CH2)6NHCO(CH2)4CO the value of 

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This result is known as the Carothers equation. It is apparent that this expression reduces to Eq. (5.4) for the case of f = 2. Furthermore, when f exceeds 2, as in the AA/BB/Af mixture under consideration, then n is increased over the value obtained at the same p for 7= 2. A numerical example will help clarify these relationships ... 

The Carothers equation relates the number-average degree of polymerization to the extent of reaction and average functionality of a step-growth polymer. In the Carothers equation, the number-average degree of polymerization, X , relates to the extent of reaction, p, and average functionality, /avg, of the polymer system ... 

Carboxylic-sulfonic anhydrides, 80 Cardiovascular surgery, 27 Cardo diamines, 277, 278 Carothers, Wallace, 198 Carothers equation, 11, 59 Carothers group, 1 Carpet waste... 

The last equation commonly named the Carothers equation is of general use in step polymerisation. 

Crosslinking in step polymerisation-modified Carothers equation... 

The latter equation is a modification of the Carothers equation and can be used to approximate the critical conversion value at the gelation. Indeed, when Xn goes to infinity the above equation leads to... 

The relationship given in Equation 4.21 is called the Carothers equation because it was first found by Carothers while working with the synthesis of polyamides (nylons). For an essentially quantitative synthesis of polyamides where p is 0.9999, the DP is approximately 10,000, the value calculated using Equation 4.21 ... 

Thus, the Carothers equation allows calculation of maximum DP as a function of extent of polymerization, and the purity of reactants. This value of 10,000 Da is sufficient to produce polyesters that will give strong fibers. The high value of p decreases, as does DP, if impurities are present or if some competing reaction, such as cyclization, occurs. Since the values of k at any temperature can be determined from the slope (2kAl) when 1/(1 —p) is plotted against t, DPn at any time t can be determined from the expression... 

Since quenching the reaction or adding a stoichiometric excess of one reactant is seldom economical, the commercial practice is to add a specific amount of a monofunctional reactant in the synthesis of polyesters, nylons, and other similar polymers. In these cases, a functionality factor / is used that is equal to the average number of functional groups present per reactive molecule. While the value of / in the preceding examples has been 2.0, it may be reduced to lower values and used in the following modified Carothers equation ... 

The average DP for formation of linear condensation polymers can be calculated using Carothers equation, average DP = 1/(1 — p). 

This equation relating the degree of polymerization to the extent of reaction was originally set forth by Carothers [1936] and is often referred to as the Carothers equation. 

The/ in Eq. 2-145 is the functionality of the branch units, that is, of the monomer with functionality greater than 2. It is not the average functionality/avg from the Carothers equation. If more than one type of multifunctional branch unit is present an average / value of all the monomer molecules with functionality greater than 2 is used in Eq. 2-145. 

The two approaches to the problem of predicting the extent of reaction at the onset of gelation differ appreciably in their predictions of pc for the same system of reactants. The Carothers equation predicts the extent of reaction at which the number-average degree of polymerization becomes infinite. This must obviously yield a value of pc that is too large because polymer molecules larger than Xn are present and will reach the gel point earlier than those of size Xn. The statistical treatment theoretically overcomes this error, since it predicts the extent of reaction at which the polymer size distribution curve first extends into the region of infinite size. 

Compare the gel points calculated from the Carothers equation (and its modifications) with those using the statistical approach. Describe the effect of unequal functional groups reactivity (e.g., for the hydroxyl groups in glycerol) on the extent of reaction at the gel point. 

Clearly, the average degree of polymerization, X (number of repeat units), depends only on the stoichiometric imbalance ratio, r (r < 1), and the extent of conversion, p (expressed as fractional percentage of functional groups consumed). Considering a step polymerization that contains only one type of functional group, as is the case for metathesis polymerization of acyclic dienes, then stoichiometry is perfectly balanced and equation (17) reduces to the Carothers equation. 

A simple plot of the Carothers equation illustrates how functional group conversion governs step polymerization chemistry (Figure 2). Conversions of above 99% are required for high polymer to form. 

Figure 2 Plot of the Carothers equation in step chemistry...

Equation 10.16 for bifunctional monomers is a special case of the more general Carothers equation [18] that is applicable to monomers with any functionalities ... 

The Carothers equation becomes invalid if cyclization occurs to a significant extent. Cyclization reduces the number of functional groups, but leaves the number of molecules unchanged. This violates the underlying premise that there are always twice as many functional groups as there are molecules. 

Gel point. The Carothers equation can also be used to estimate the conversion needed to reach the so-called gel point in condensation polymerization involving monomers with functionalities higher than 2. The gel point is defined as the state of conversion at which gel formation caused by crosslinking begins to become apparent. With the assumption that this occurs when practically all molecules of the limiting monomer have reacted, the requisite fractional conversion of functional groups can be estimated with a rearranged form of the Carothers equation ... 

This section develops the relation between the number average size of polymers produced in a Step-growth polymerization and the fraction of functional groups which have been reacted at any point in the process. The basic equation which will be developed, Eq. (5-19), is called the Carothers equation. It illustrates the fundamental principles which underlie the operation of such polymerizations to produce good yields of polymers with the desired molecular weights. 

The basic assumptions involved in deriving the Carothers equation are reviewed in this section. 

This is the Carothers equation. Note that Xn. is the number average degree of polymerization of the reaction mixture and not just of the polymer which has been formed. Note also that Y is not necessarily equal to the degree of polymerization DPn, defined in Section 1.2 as the average number of repea ng units per polymer molecule. In the present context a single monomer has X = 1, a molecule containing two monomers has X = 2, and so on. Thus for structure 5-3, X = DPn = 1(X), but for 5-4, Xn = 200 while DP = 100, since each... 

The following paragraphs include sample calculations which illustrate the practical application of the Carothers equation to step-growth polymerizations which yield branched polymers. 

Consider the simple alkyd recipe shown in Table 5-1, part (i). Alkyds are polyesters produced from polyhydric alcohols and polybasic and monobasic acids. They are used primarily in surface coatings. The ingredients of these polymers contain polyfunctional monomers and it is possible that such polymerizations could produce a thermoset material during the actual alkyd synthesis. This is of course an unwanted outcome, and calculations based on the Carothers equation can be used to adjust the polymerization recipe to produce a finite molecular weight polymer in good yield. The recipe can also be adjusted to provide other desirable characteristics of the product, such as an absence of free acid groups which may react adversely with some pigments. 

Note that substitution of p = I into Eq. (5-19) with /av > 2 yields a negative value for X . The Carothers equation obviously does not hold beyond the degree of conversion at which X is infinite. 

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