Question

Explain how you would determine the degree of polymerization using only NMR.

Answer 1

The degree of polymerization (DP) or number of repeating units of a polymer is defined as the average number of
structural units per polymer chain.. From ¹H NMR spectrum,by comparing the proton peak intensity of a known moiety (typically an end-group(s) with a known number of protons) to that of the repeating chain unit of interest can give DP. An equation is derived for calculating the value of DP, which is as follows.

We know that intensity or area of a proton signal in ¹H NMR is proportional to amount of that species present in sample (number of molecules of that species times its molecular weight.) From here, we have

$a_{i} = k \times n_{i}\times m_{i}$

Rearranging for k,

$k = \frac{a_{i}}{n_{i}\times m_{i}}$

Here, for a species i, a_i is the area or intensity of the ¹H NMR peak; n_i is the number of repeating units; m_i is the number of protons; and k is the constant, because number of molecules of that species is proportional to number of repeating units and molecular weight of that species is proportional to its number of protons.

Consider the ¹H NMR signals of two different moieties x and y,

$k_{x}= \frac{a_{x}}{n_{x}\times m_{x}}$

$k_{y}= \frac{a_{y}}{n_{y}\times m_{y}}$

But, in a given polymer k_x = k_y , so

$\frac{a_{x}}{n_{x}\times m_{x}} = \frac{a_{y}}{n_{y}\times m_{y}}$

where a_x is the area or intensity of the ¹H NMR peak of moiety x; n_x is the number of repeating units of moiety x; m_x is the number of protons of moiety x; a_y is the area or intensity of the ¹H NMR peak of moiety y; n_y is the number of repeating units of moiety y; and m_y is the number of protons of moiety y.

Rearranging  n_x,

$n_{x} = \frac{a_{x}\times m_{y}\times n_{y} }{a_{y}\times m_{x}}$

where n_x can be used to assess the polymer’s DP or number of repeating units.

Consequently, M_n can be calculated by substituting for n

$M_{n} =n\times M_{o} + M_{e}$

where n is the number of repeating units or DP; M_o is the molecular weight of one repeating unit, and M_e is the combined molecular weight of the end-groups.

Example :

Consider the spectrum of MPEG-NH_2. The protons of the methoxy (CH₃O?) and amino (?NH₂) end-groups resonated as a singlet and broad triplet, respectively, at 3.25 and 2.71 ppm (J = 5.6 Hz), whereas the prominent peak of the oxyethylene protons (?(OCH₂CH₂)n?) was centered at 3.52 ppm.

Determination of the Number of Repeating Units

To calculate the number of repeating units (n) in the MPEG-NH₂ chain, the peak areas of CH₃O– (OMe, ? 3.25) and ?(OCH₂CH₂)n– (OEt, ? 3.52) were obtained from the ¹H NMR spectrum

$n_{MPEG} = \frac{a_{OEt}\times m_{OMe}\times n_{OMe} }{a_{OMe}\times m_{OEt}}$

= (133.1*3*1)/(0.9*4)

= 110.92 $\approx$ 111.