Kendrick Analysis

Kendrick Analysis helps classifying molecules into homologous groups using high resolution mass spectrometry. For example, homologous hydrocarbon molecules differ in the number of CH2 groups they contain. When measuring their masses with modular arithmetic using the mass m(CH2) as the modulus, all homologous molecules will have the same remainder mass.

Edward Kendrick achieved the same effect in 1963 by introducing a new mass scale, now called the Kendrick mass scale. In his mass scale the unit of mass corresponded to m(CH2)/14.This unit of mass, which has been called Kendrick mass unit or kendrick (Ke) in the literature, is very close to the dalton unit. All homologous molecules will have the same Kendrick Mass Excess Δm. This unit simplifies the interpretation of a hydrocarbon mass spectrum.

Methods
There are three different methods to do a Kendrick analysis:

Trend line method
The simplest way is to produce a graph where each molecule M is arranged according to its mass m(M) and its mass excess/defect ∆m(M) = m(M) - integer[m(M)]. When expressing the mass m in daltons, (in metrology terms: [m] = Da) this aligns all homologous molecules on trend lines whose slope depends on the "building block" of the molecules. The disadvantage of this method is that the offsets from sloped trend lines are difficult to read. The advantage is that no new units have to be introduced and that it works for all "building block" molecules.

Kendrick method
Kendrick realized that by using a mass scale with units equal to the mass of the building blocks, the trend lines would become horizontal for all homologous families of this "building block". This simplifies somewhat the reading of the mass defect. However, it requires a new mass scale and thereby a new unit for each building block as well as a conversion of each molecule mass into this unit.

For hydrocarbons, the Kendrick mass unit    is defined as:


 * m(CH2) := 14 Ke

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


 * 1 Ke = 14.01565/14.000 Da = 1.001118 Da = 1.001118 u

For other classes of molecules which are based on a different building block than CH2, another unit would have to be created.

How to correctly convert masses in Dalton units to masses in Kendrick units
Assume we have a molecule M with a known mass of m(M) = nD⋅Da. How is this mass m(M) expressed in Kendrick units Ke?


 * m(M) = nD⋅Da = nK⋅Ke
 * nK = nD⋅Da / Ke

We already know that the unit conversion factor Da/Ke = 14/14.01565, hence


 * nK = nD × 14/14.01565

Hence, the mass in Ke is:


 * m(M) = 14/14.01565⋅nD⋅Ke

Other terminology
In the literature it has been proposed that
 * the Kendrick mass scale is obtained by setting the mass of the 12CH2 radical to 14.00000. 

On the other hand, according to the rules of metrology outlined in the IUPAC green book, the IUPAP red book and the ISO 31 standards, a mass is not dimensionless and always needs a unit of dimension mass (like when indicating masses in Da).

The following conversion has also been suggested by the same authors:
 * Kendrick mass = SI mass · 14.00000 / 14.01565.

It is worth noting that
 * by "Kendrick mass" the authors mean the mass of a molecule measured in the Kendrick mass scale and not the Kendrick mass unit Ke.
 * by "SI mass" the authors mean the mass measured in dalton mass units Da even though the Da is explicitly referred to as "a unit outside the SI "
 * according to the rules of metrology outlined in the IUPAC green book, the IUPAP red book and the ISO 31 standards, above formula should be:
 * Kendrick mass = Dalton mass

The mass of a molecule is a global constant. Thereby it does not change depending on the units (or mass scale) used. According to the prior formula, a molecule would suddenly become lighter just by changing to Kendrick units (or mass scale) from Dalton units. The following equations further illustrate this, using the molecule 12CH2 as an example:
 * Kendrick mass of 12CH2 = mKe(12CH2) = 14 Ke
 * Dalton mass of 12CH2 = mDa(12CH2) = 14.01565 Da
 * mKe(12CH2) = mDa(12CH2)
 * 14 Ke = 14.01565 Da   → which is known to be true

It would be NPV to draw conclusions from these facts about the quality of this literature. Therefore, explicitly no conclusions are drawn.

Kendrick Mass Excess/Defect
Kendrick Mass Excess (or defect) Δm is defined as:


 * Δm = m - round(m)

or more rigorously


 * Δm = m - A·Ke

where:
 * Δm is the Kendrick mass excess
 * A is the mass number of the molecule
 * Ke is the mass unit kendrick
 * m is the mass of the molecule (or isotopologue) in kendricks.
 * round(m) and A·Ke are the integer masses of the molecule in kendricks.

Note:
 * the Kendrick mass excess Δm is defined different than the mass excess in nuclear physics

CH2 Modulo method
Math has a tool called modular arithmetic that can reveal the same homologous relation using the modulo operation.
 * 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."

The remainder mass of a molecule M, Δm(M), would be expressed as the remainder r:
 * Δm(M) = r = m(M) mod m(CH2)

If the modulo operation nor the remainder operation are defined
 * Δm(M) = m(M) - m(CH2)·round(m(M)/m(CH2))

This method has all advantages of the previous methods: it works with any building blocks, does not require new units, and it produces horizontal lines in a Kendrick plot.

Kendrick modulo method
Closer inspection reveals that above modulo method is not equivalent to the previous two methods. The modulo method above uses m(CH2) as the modulus whereas the Kendrick method effectively uses m(CH2)/14 as the modulus. The Kendrick mass defect of a molecule M, Δm(M), would be expressed as:
 * Δm(M) = m(M) mod m(CH2)/14

Thereby the range of the reminders shrinks from 14 Da to a range of 1 Da, effectively wrapping the reminders into a constrained space. Some groups have found that this leads to overcrowding with complex samples and therefore "unfolded" the effect of the Kendrick modulus by creating 14 versions of Kendrick plots, thereby recreating the properties of m(CH2) modulo method.