# Functional group equivalent weight

Last update 7 April 2017

The FGEW is used to determine if the RFGs in a polymer are substantially diluted by polymeric material to allow the polymer to be a PLC.

The FGEW of a polymer is defined as the ratio of the Number Average Molecular Weight (NAMW) to the number of functional groups in the polymer. It is the weight of a polymer that contains one formula weight of the functional group.

The level of low concern RFGs in the polymer is not restricted. Low concentrations of RFGs are permissible in polymer molecules, but the quantity is restricted by the reactivity of the functional group/s in question.

**How to calculate functional group equivalent weight**

Unless the FGEW can be determined empirically by recognised, scientific methodology (typically titration), a worst-case estimate must be made for the FGEW.

All moderate and high concern functional groups must be taken into account when calculating FGEW.

Guidance for estimating FGEW using specific methods is in D2.1 and D2.2, including illustrative examples for each (end-group analysis and per cent charged method).

### Method 1—End-group analysis

#### Equation 1: Calculating FGEW by simply counting the number of RFGs and dividing into the NAMW

where n = the number of RFGs^{[1]} in the monomer^{[2]}

^{[1] Reactive functional groups} ^{[2] Atom or small molecule that may bind chemically to other monomers to form a polymer}

##### Linear polymers

Linear polymers have the simplest polymer architecture: a *linear* chain: a single backbone with no branches.

For linear polymers, such as some condensational polymers (for example, polyesters and polyamides) the only RFGs are at the end of the chain because the others are used up in the condensation reaction. For linear polymers, where there are two RFGs per monomer, the FGEW is half the NAMW.

For example, for a polyamide of NAMW 1500 made from ethylenediamine and adipic acid, an amine group would be expected at each end of the polymer chain. Therefore, FGEW = 1500/2 = 750.

##### Branched polymers

For polymers where branching occurs, the RFGs at the end of each branched chain must be counted.

For example, consider the polymerisation of pentaerythritol (4 reactive groups) with polypropylene glycol (2 reactive groups) and an excess of isophorone diisocyanate (2 reactive groups) to give a polymer of NAMW 3000.

Due to the branching of pentaerythritol and excess diisocyanate, the resultant polymer will theoretically have six isocyanate end groups. Therefore, FGEW = 3000/6 = 500.

### Method 2—Per cent charged method

Some condensation and addition reactions create polymers where not all RFGs along the backbone of the polymer are consumed during the reaction, so an accurate FGEW cannot be determined through a simple end-group analysis.

In many of these cases, calculating the FGEW may be more complex. For example, in some condensation and addition reactions, some RFGs along the polymer backbone are unchanged during polymerisation. Also, in some cases, the structural formula of the final polymer is not accurately known.

In these cases, FGEW can be calculated according to Equation 2 or 3, using the weight percentage monomer in the polymer (W), the formula weight of the monomer (FW) and the number of RFGs on the monomer (n).

#### Equation 2

For example, for an acrylic polymer containing 7.5% acryloyl chloride monomer (FW 90.5), the FGEW of acid chloride groups in the polymer is:

#### Equation 3

If the various RFGs in a polymer arise from multiple monomers, the following equation can be used:

Where FGEW_{n} is the FGEW for each functional group in the polymer.

#### Examples

- Consider the reaction between ethanediamine (MW 60) (charged at 30%) and diglycidyl ether (MW 130) (70%) to give a polymer of NAMW 5000. The epoxides in the backbone are reacted to give an aliphatic alcohol (low concern). The amine groups remain intact, with their FGEW proportional to the charged amount of ethanediamine. As the diglycidyl ether is in excess, it can be assumed that the polymer is epoxide-terminated.

Using equation 2, the FGEW for the amine group is (100 x 60)/(30 x 2) = 100. The FGEW for the epoxide group can be calculated using end group analysis (Equation 1), that is, 5000/2 = 2500.

Then, using equation 3, FGEWcomb = inverse of [1/100 + 1/2500] = 96.

- Consider a p-cresol-formaldehyde condensation polymer which is reacted with 1.5% epichlorhydrin to give an epoxide-capped resin. As a worst-case scenario, it is assumed that the polymer is phenol-terminated and that epoxy rings from the epichlorhydrin (MW 92.5, 1 epoxy group) are also present. A NAMW of 10 000 is assumed.

Using equation 2, the FGEW for the epoxide group is (100 x 92.5)/(1.5 x 1) = 6167. The FGEW for the phenol group can be calculated using end group analysis (Equation 1), that is, 10 000/2 = 5000.

Then, using equation 3, FGEWcomb = inverse of [1/6167 + 1/5000] = 2762.

- Consider the addition reaction involving the polymerisation of three acrylates, glycidyl methacrylate (10%, MW 142, 1 RFG), hydroxymethyl acrylamide (2%, MW 101, 1 RFG) and acrylic acid (88%).

In this case, it can be assumed that each monomer is completely incorporated into the polymer, with the RFGs of concern being the epoxide group from glycidyl methacrylate and the hydroxymethyl amide group from the acrylamide. The carboxylic acid moiety from acrylic acid is of low concern and need not be included in FGEW calculations.

Using equation 2, the FGEW for the epoxide group is (100 x 142)/(10 x 1) = 1420. Again using equation 2, the FGEW for the hydroxymethyl amide group is (100 x 101)/(2 x 1) = 5050.

Then, using equation 3, FGEW_{comb} = inverse of [1/1420 + 1/5050] = 1108.