Number-average and weight-average molecular weights

Introduction

A simple compound has a fixed molecular weight; for example, Acetone has 58 molecular weights. Suppose this 58 molecular weight becomes 60; then the compound is no longer Acetone but may be acetic acid. Each molecule has the same molecular weight. It is valid for all low molecular-weight compounds. In contrast, a polymer comprises molecules of different molecular weights, and its molecular weight is expressed in terms of the ‘average’ value.

For example, ethylene has a fixed molecular weight 28. But, if we polymerize ethylene to make polyethylene with a different molecular weight nearer 14,000.

A polymer’s molecular weight uses either the number fraction or the weight fraction of the molecules present in the polymer to get either the number-average molecular weight or the weight average molecular weight.

Assume that there is n number of molecules in a polymer sample, and n₁ of them have M₁ molecular weight; n₁ has M₁ molecular weight and so on till we get n₁ having M₁ molecular weight.

A total number of molecules (n) given by,

\[\displaystyle n={{n}_{1}}+{{n}_{2}}+{{n}_{3}}+.....+{{n}_{i}}=\sum {{n}_{i}}\]

Number of molecules in fraction 1 = n₁

Number fraction of fraction 1 =n₁/n =n₁/n_i

Molecular weight contribution by fraction 1 = n₁M₁/n_i

Similarly, molecular weight contribution by other fractions will be as follows;

\[\displaystyle \frac{{{{n}_{2}}{{M}_{2}}}}{{\sum {{n}_{i}}}},\frac{{{{n}_{3}}{{M}_{3}}}}{{\sum {{n}_{i}}}},......,\frac{{{{n}_{i}}{{M}_{i}}}}{{\sum {{n}_{i}}}}\]

The number-average molecular weight of the whole polymer will then be given by,

\[\displaystyle \frac{{{{n}_{1}}{{M}_{1}}}}{{\sum {{n}_{i}}}}+\frac{{{{n}_{2}}{{M}_{2}}}}{{\sum {{n}_{i}}}}+\frac{{{{n}_{3}}{{M}_{3}}}}{{\sum {{n}_{i}}}}+......+\frac{{{{n}_{i}}{{M}_{i}}}}{{\sum {{n}_{i}}}}=\frac{{\sum {{n}_{i}}{{M}_{i}}}}{{\sum {{n}_{i}}}}={{{\bar{M}}}_{n}}---(1)\]

The above equation is used for calculating the number-average molecular weight.

Similarly, the total weight of the polymer= W= n₁M₁

Weight of fraction 1 = W₁ =n₁M₁

Weight fraction of fraction 1 = n₁M₁/W = n₁M₁/n_iM_i

Molecular weight contribution by fraction 1 is given by,

\[\displaystyle \frac{{{{n}_{1}}{{M}_{1}}{{M}_{1}}}}{{\sum {{n}_{i}}{{M}_{i}}}}=\frac{{{{n}_{1}}M_{1}^{2}}}{{\sum {{n}_{i}}{{M}_{i}}}}\]

Similarly, the molecular weight contribution by the other fractions will be,

\[\displaystyle \frac{{{{n}_{2}}M_{2}^{2}}}{{\sum {{n}_{i}}{{M}_{i}}}},\frac{{{{n}_{3}}M_{3}^{2}}}{{\sum {{n}_{i}}{{M}_{i}}}},.....,\frac{{{{n}_{i}}M_{i}^{2}}}{{\sum {{n}_{i}}{{M}_{i}}}}\]

The weight-average molecular weight of the whole polymer will be,

\[\displaystyle \frac{{{{n}_{1}}M_{1}^{2}}}{{\sum {{n}_{i}}{{M}_{i}}}}+\frac{{{{n}_{2}}M_{2}^{2}}}{{\sum {{n}_{i}}{{M}_{i}}}}+\frac{{{{n}_{3}}M_{3}^{2}}}{{\sum {{n}_{i}}{{M}_{i}}}}+.....+\frac{{{{n}_{i}}M_{i}^{2}}}{{\sum {{n}_{i}}{{M}_{i}}}}={{{\bar{M}}}_{w}}----(2)\]

The above equation is used for calculating the weight-average molecular weight.

For all synthetic polymers, the weight-average molecular weight is higher than the number-average molecular weight.