Polymerization

Kinetics of polymerization

i)Addition or chain polymerization

Let us derive the rate expression for polymer growth of chain polymerization.The mechanism for a addition reaction is given below:

Chain initiation:

I→2R°

R°+M→M°

Chain propagation:

M°+M→M₂^°

M2°+M→M₃^°

Chain termination:

M_n^°+M_m^°→M_n+m

As we see from the mechanism,the reaction involves many steps.THERe are some valid and simplifying assumptions makes in deriving the rate expressions for such a complicated mechanism,THe first assumption is that the radical concentration is determined only by the initiation and the termination step.In the intermediate step,as we see from the above scheme,one radical produce another and it is reasonable to assume that the propagation step neither increase nor decrease the radical concentration.Hence,the Steady State Concentration of the radical is given by,

\(\frac{d[M^°]{dt}\)=fK_i[I]-K_t[M^°]=0

=fK_i[I]=K_t[M^°]²

or,[M^°]²=fK_i/K_t

or,[M°]=\(\sqrt{fK_i[I]\)/K_{t............i)}

where M^°=any radical in the reaction

F=effectiveness of the initiator for producing radical

K_i=rate constant for the initiation step

K_t=rate constant for the termination step

Now,the rate of polymerization(R_p) is given by a relation,

R_p=K_p[M][M^°].....................ii)

Again,the rate of polymerization is independent of the size of the radical,so using equations i) and ii),then we get,

R_p=K_p[M].√fK_i/K_t

R_p=K_p[M].(fK_i/K_t)(I)^½

R_p=K[M][I]^½.................iii)

where K=K_p(fK_i/K_t)^½=constant

This equation predict that the rate of polymerization increases with the concentration of monomer and the square root of the initiator concentration.So,the rate of polymerization to be maximum at the start of the reaction when the concentration of the monomer is maximum and decreases with the time.

Kinetics of step growth polymerization

Sstep growth polymerization involves the condensation of monomer units and elimination of the small molecules usually water,carbondioxide,ammonia and alcohol,etc.For example:synthesis of polyesters

The monomer unit of polyesters are carboxylic acid and hydroxyl group.

Let us consider the reaction between ethylene glycol and terephthaleic acid.

The rate equation of this reactionis expressed as \(\frac{-d}{dt}\)[COOH]=K[COOH][OH].......i)

Here,we use the concentration of functional groupin the rate equation.If the reaction started with equal moles of functional group that is [COOH]=[OH]=c moles

So,above equation i) becomes

\(\frac{-dc}{dt}\)=Kc²...............ii)

This reaction is a second order reaction.So,theintegrated rate equation of this reaction,

k=\(\frac{x}{at(a-x)}\)

Here,x=concentration of the product formed at a time 't'

a=initial concentration of the reactants

Let a is equivalent to c₀and x is equivalent to c₀-c.Then,above relation can be written as,

kt=\(\frac{c_0-c}{c_0c}\)

or,c_o-c=ktc₀c

or,c_o=ktc_oc+c

or,c₀=(ktc₀+1)c

or,\(\frac{c_0}{c}\)=ktc₀+1........iii)

Let p be the fraction of functional group consumed then,

p=\(\frac{c_0-c}{c}\)

p=1-\(\frac{c}{c_0}\)

\(\frac{c}{c_0}\)=1-p

\(\frac{c_0}{c}\)=\(\frac{1}{1-p}\)........iv)

Compararing equations iii) and iv),we get

\(\frac{1}{1-p}\)=ktc₀+1.........v)

Equation v) is the required equation for step growth polymerization.

Degree of polymerization

The degree of polynerization is denoted by a symbol <η> and it is the ratio of the initial concentrationof the functional group (c₀) to the remaining concentration of the functional group at a time 't'.that is.

<η>=\(\frac{c_o}{c}\)

<η>=\(\frac{1}{1-p}\)

<η>=ktc_o+1

References

Paula, P.Atkins & J.de. Elements of Physical Chemistry. New Delhi: Oxford University Press, 2009.