# Chemical Sciences: A Manual for CSIR-UGC National Eligibility Test for Lectureship and JRF/Kendrick mass

The Kendrick mass is a mass obtained by scaling the atomic mass unit (u), or dalton (Da) to simplify the display of peak patterns in hydrocarbon mass spectra.

## Definition

The Kendrick mass unit is defined as 

m(12CH2) = 14 Ke

In words: "the group 12CH2 has a mass of 14 Ke exactly, by definition."

1 Ke = 14.0156/14.000 Da = 1.00111429 Da = 1.00111429 u

## Kendrick mass defect

When expressing the masses of hydrocarbon molecules in Kendrick mass, all homologous molecules will have the same mass defect Δm defined as:

Δm = m - round(m)

or more rigorously

Δm = m - A·Ke

where

• Δm is the Kendrick mass defect
• A is the mass number of the molecule
• m is the mass of the molecule (or isotopologue) which is also referred to as exact mass
• round(m) and A·Ke are the integer masses of the molecule

## Equivalence relation

The Kendrick mass scale was introduced to find an equivalence relation for hydrocarbons. The same relation could be expressed with modular arithmetic using the modulo operation without introducing a new mass scale.

A ~ B (mod CH2)

The above statement is read: "A is modulo CH2 equivalent to B."

Or, when considering the mass of the molecules A and B:

m(A) ~ m(B) (mod m(CH2))

"A has the same modulo CH2 mass as B."

In a computing code the Kendrick mass defect of a molecule M, Δm(M), would be expressed as the remainder r:

Δm(M) = r = m(M) mod m(CH2)

or, if the modulo operation nor the remainder operation are defined

Δm(M) = m(M) - m(CH2)·round(m(M)/m(CH2))

Note that:

• most programming languages implement the modulo operation with trunc or floor instead of round
• this approach with modular arithmetic works independent of the mass units (or mass scale)
• this approach is more generalized and allows for other building blocks than CH2, e.g. in polymer chemistry
• the Kendrick mass defect Δm is defined different than the mass defect in nuclear physics