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Mass spectrometry in the early days focuses on determining the charge-to-mass ratio, e/m, of an electron. This involves a few steps, beginning with the electric field turned on and the magnetic field turned off. The diagram below shows the mechanics of the electrons as they pass through the electric field plates.

An electric force, F_e, acts on an electron with charge, e, passing through the electric field, E, in the x-direction where

and gives the electron with mass, m, an acceleration, a:

Combining eq1 and eq2,

As the electric force acts only in the y-direction, the velocity of the electron in the x-direction remains constant at v_H. The velocity of the electron in the y-direction, v_V, is zero at the point where it enters the electric field and increases to a maximum value at the point where the electron leaves the electric field. This maximum value of v_V remains constant after the electron exits the electric field and is given by:

where S is the displacement of the electron in the y-direction, u is the initial velocity of the electron in the y-direction before entering the electric field (i.e. u = 0) and t is the time during which the electric force acts on the electron.

In other words, t is the time taken for the electron to travel the length of the electric field plate, L. Therefore,

The trigonometric relationship between v_H and v_V after the electron leaves the electric field is:

The electric field strength is adjusted so that the angle of deflection is small. Therefore, tanθ ≈ θ and eq5 becomes,

The value of θ is recorded, leaving v_H  as the only unknown variable (in eq6) for the calculation of e/m. To determine v_H , the magnetic field, B, is turned on while the electric field is still on. Using Fleming’s left hand rule, the magnetic force, F_B, acts on the electron (at the point when the electron just enters the magnetic field from the left side) and deflects it in the negative y-direction:

The magnetic field is then adjusted to the extent that the angle of deflection of the cathode ray is zero, meaning F_e = F_B. Substituting eq1 and eq7 in F_e = F_B, we have:

Substitute eq8 in eq6

The values of all the variables on the right hand side of eq9 are now known and the charge-to-mass ratio e/m of the electron can be determined. The magnitude of e/m that Thomson obtained ranged from 0.7 x 10¹¹ C/kg to 2 x 10¹¹ C/kg. The currently accepted value is 1.758820024 x 10¹¹ C/kg.

Finally, in 1909, when Robert Millikan determined the charge of an electron via his famous oil drop experiment, the estimated mass of an electron was calculated.

 

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A mass spectrometer measures the mass of an isotope relative to that of carbon-12 by analysing the ratio of the ionised isotope’s deflection to the ionised carbon-12’s deflection.

From eq9, we have:

Assume both ions are univalent so that  and divide eq11 by eq10:

Therefore,

We can also represent eq13 in the form

where we have replaced e with the notation z. The output of eq14 is a value with unit of unified atomic mass unit, u, since the ratio is unit-less while .

Finally, we divide the output of eq14 by 1u to obtain the relative isotopic mass, which is a dimensionless quantity that is defined as the ratio of the mass of an isotope in unified atomic mass unit to one unified atomic mass unit.

 

 

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Mass spectrometry is an analytical technique that utilises an instrument called a mass spectrometer to identify and quantify ions based on their mass-to-charge ratios. J. J. Thomson, an English physicist, constructed one of the earliest mass spectrometers (see diagram below) and demonstrated in an experiment in 1897 that atoms are made of subatomic particles called electrons. Using the spectrometer, he determined the charge-to-mass ratio of an electron.

The ioniser on the left of the spectrometer is not a complete vacuum; it is filled with a trace amount of air molecules, which are always in equilibrium between their neutral and ionic forms due to natural occurring processes like photoionisation. Consequently, free electrons are present among the trace amount of air molecules in the ioniser.

Instead of producing electrons from the cathode via a heated filament, the ioniser operates on the principle of a ‘cold cathode’, where a high potential difference maintained between the cathode and anode accelerates the free electrons present in the trace amount of air. These electrons collide with other gas molecules and knock more electrons off them, creating a cascade of ions and electrons called a Townsend discharge. The electrons are attracted towards the anode, where they are focused into a beam called a cathode ray and continue their path to the electric field plates and electromagnet, where they are deflected.

When the electrons eventually strike the atoms in the glass at the right end of the spectrometer, they excite electrons intrinsic to the atoms to a higher energy state. When these excited electrons relax back to the ground state, they emit light in the form of fluorescence, and the angle of deflection, θ, can be read from the fluorescent spot on the scale.

By working out the mechanics of the trajectory of the cathode ray, Thomson managed to calculate the charge-to-mass ratio of an electron and concluded that the negatively charged ‘corpuscles’ (the term he used at the time for electrons) must have originated from the dissociation of trace amounts of gas molecules. This conclusion was based on the fact that the charge-to-mass ratio of a corpuscle is about 1,800 times greater than that of a charged hydrogen atom (the charge-to-mass ratio of a hydrogen ion had been investigated earlier).

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Perrin needed to prove that N_A is a constant in the following equation that he derived:

The challenge was that N’, N and m could not be measured directly. He circumvented this issue by replacing the gas in the cylinder with visible particles suspended in a liquid. Drawing from the principles of Brownian motion, Perrin assumed that collisions between liquid molecules and the visible particles would result in a distribution similar to that of gas molecules in the cylinder. He also assumed that the motion of the particles obeyed the ideal gas law.

Perrin used a monodisperse colloid of a gum called gamboge, consisting of thousands of gamboge spheres in a water cylinder. He studied the distribution of the spheres with a microscope and adjusted eq9 to account for the upthrust of water on the gamboge spheres. For a single gamboge sphere (“particle”),

upthrust = weight of liquid displaced = m_l g = dV_l g                (10)

where m_l is the mass of the liquid displaced, d is the density of the liquid and V_l is the volume of the liquid displaced.

Furthermore, the volume of liquid displaced is equal to the volume of the particle, V_p :

V_l = V_p                (11)

Substituting eq11 in eq10 yields

upthrust = dV_p g               (12)

Since V_p = m/D, where m is the mass of a particle and D is the density of the particle, eq 12 becomes:

The effective weight of the particle is therefore the difference between the weight of the particle and the upthrust on the particle:

Replacing the weight mg of a gas molecule in eq9 with the effective weight of a suspended particle gives

The suspended particles must be heavier than the liquid molecules for the particles to exert a downward pressure on the liquid, producing an upthrust that results in a lower effective weight for the particles. This means that d < D for eq14 to be valid. Hence, the choice of particle material is important.

Perrin meticulously prepared emulsions containing particles that were equal in size. He calculated the average mass of a particle by weighing a specified number of particles, determined its density using various methods (including the specific gravity bottle method), and counted the number of suspended particles per unit volume at different heights using a microscope.

After repeating the experiment with different particle materials (e.g. mastic), sizes, masses, liquids and temperatures, he found that the value of N_A remained fairly constant, reporting numbers ranging from 6.5 x 10²³ to 7.2 x 10²³. He further conducted experiments using methods based on radioactivity, blackbody radiation and the motion of ions in liquids, obtaining very similar results for the value of N_A. Perrin concluded that the results justified the hypotheses that had guided him, including Avogadro’s law, and named the constant the Avogadro constant, in honour of Avogadro.

The accuracy of the value of the Avogadro constant was subsequently improved by other scientists, one of whom was Robert Millikan.

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Perrin developed the concept to analyse the vertical distribution of a volume of molecules of an ideal gas, with total mass of m_c, in a cylinder with a cross-sectional area A and small height h at temperature T. Firstly, he considered an infinitesimal layer of the gas with mass dm_c and thickness dh in the cylinder.

This layer of gas experiences an upward force F_u = pA from the pressure exerted by the gas it, and a downward force F_d = gdm_c + (p+dp)A due to the gravitational force and the pressure exerted by the gas above it. At equilibrium, F_u = F_d and we get:

At this juncture, Perrin, with reference to the works of previous scientists (Avogadro, Dalton, etc.) on relative mass of molecules and introduced an important definition:

A gramme-molecule, M, is the mass of a gas that occupies the same volume as 2g of hydrogen gas at the same temperature and pressure*

* Perrin’s definition of the gramme-molecule is somewhat similar to the 1967 definition of the mole. This definition, together with Avogadro’s law, implies that a gramme-molecule of gas X and a gramme-molecule of gas Y have the same number of molecules.

With Perrin’s definition, the physical state of the gas in the cylinder can be described using the ideal gas law (eq2), where the amount of gas n is expressed in multiples of gramme-molecule M:

Substituting eq4 in eq3 by eliminating dm_c/A yields

The expression for the distribution of the gas in the entire cylinder can be obtained by integrating both sides of eq5 (see above diagram):

From eq2, we have p’ = n’RT/V’ and p = nRT/V. Substituting these two equations in eq6 gives

where N’ = n’/V’ and N = n/V; that is, the number densities of the gas at the upper and lower levels of the cylinder.

Furthermore, the gramme-molecule of the gas, M, is equal to the number of molecules in a gramme-molecule, N_A, multiplied by the mass of a molecule of the gas, m:

Substituting eq8 in eq7 results in

If Avogadro’s hypothesis – that ‘equal volumes of all gases at the same temperature and pressure have the same number of molecules’ – is true, N_A in eq9 must be constant for different ideal gases. To determine the value of N_A, Perrin would need to (i) count the number of molecules per unit volume at the upper and lower levels of the cylinder (of fixed height, h, at a constant temperature, T) for a particular gas, (ii) repeat the count for other gases with different m, and  (iii) calculate the average value of N_A. However, a problem arose.

 

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Jean Perrin, a French scientist, was one of the earlier researchers who attempted to determine the Avogadro constant.

Who inspired Perrin?

Prior to Perrin’s work in 1909, Amedeo Avogadro, an Italian scientist, published papers between 1811 and 1841 and suggested that

Equal volumes of all gases at the same temperature and pressure have the same number of molecules

This became known as Avogadro’s law. It implies that for a given mass of an ideal gas at constant temperature and pressure, the volume V of the ideal gas is directly proportional to its amount n:

V = kn                 (1)

where k is the proportionality constant.

When eq1 is incorporated into the combined gas law, which was developed many years earlier, we have the ideal gas law:

pV = nRT                 (2)

At that time, n was known as ‘amount of gas’ rather than ‘number of moles’, as the mole concept had not yet been developed. Since the amount of gas can be in measured in different ways, the gas constant R had different units back then. 

Avogadro also investigated the relative mass of different gases; for example, he deduced from gas density data that the relative molecular weight of nitrogen and hydrogen is in the ratio of 13.2 : 1 and that the ratio of oxygen molecules and hydrogen molecules in water is 0.5 : 1.

Lastly, the research of botanist Robert Brown in 1872, which involved the random motion of particles suspended in a liquid or a gas due to their collisions with the surrounding molecules, also played an important role in Perrin’s calculations.

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Mass is the quantity of matter that a physical body contains. In daily life, we measure almost everything using the concept of inertia mass, which is a measure of a body’s resistance to acceleration. Such measurements are made in relation to the kilogramme, which is now defined by the Planck constant. Prior to Nov 2018, the kilogramme was defined by the mass of a platinum alloy cylinder called the International Prototype Kilogramme (IPK), which is stored in France. The IPK contains octillions of atoms and has inevitably gained or lost mass over time through oxidation. Therefore, it cannot be used to accurately measure the mass of atoms.

To circumvent this problem, scientists decided to measure the mass of an atom or isotope relative to that of another atom or isotope, just as the mass of everyday objects was measured in relation to the IPK. The question then arises: which isotope should be chosen as the reference mass, and why?

 

next article: unified atomic mass unit

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The unified atomic mass unit, defined as one twelfth the mass of a carbon-12 atom, serves as a standard reference for quantifying the masses of atoms and molecules, facilitating comparisons across the diverse elements of the periodic table.

An atom is made up of protons, neutrons and electrons, with almost all of its mass attributed to the combined masses of the protons and neutrons. Since the relative masses of a proton and a neutron are very similar, it is logical to select a reference isotope whose nucleus has an equal number of protons and neutrons. A few candidates, such as ²H, ⁴He and ¹²C immediately come to mind.

Ultimately, ¹²C was eventually chosen in 1961 because it has an equal number of protons and neutrons – six each – making it a better average for the relative masses of protons and neutrons in the nucleus compared to the other two isotopes. Furthermore, it has a high relative isotopic abundance, which makes it easier to isolate for measurement. Another reason for the choice of ¹²C is that it was already used as a reference standard in mass spectrometry prior to its selection.

Thus, ¹²C was arbitrarily assigned a value of exactly twelve unified atomic mass units or 12 u. Everything seems in order after this definition, but how is the carbon-12 unified atomic mass unit scale relevant to a chemist who is more familiar with calculating and measuring the inertia mass of macroscopic amounts of chemicals in the laboratory?

 

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How do you weigh an atom?

As mentioned in the earlier sections, the mass of an atom is measured on the unified atomic mass unit scale. This is carried out in a mass spectrometer with carbon-12 as a standard reference. For example, mass spectrometric data for the ratio of the mass-to-charge ratio (u/z) of ²H to that of ¹²C is 0.167842. Thus, the mass of ²H on the carbon-12 unified atomic mass unit scale is:

Using eq2, the mass of one ²H is 2.014104 x 1.660539 x 10^-27 = 3.3450 x 10^-27 kg.

The conversion from u to kg is based on eq2, which is dependent on the uncertainty in the Avogadro constant prior to Nov 2018. However, the molar mass of ²H, which is equal to 2.0141 g, is not. This is shown by substituting eq1 in eq6,

Multiplying both sides by N_A,

where the RHS of eq7 is independent of N_A.

This is why the molar mass of another isotope, silicon-28, is used in determining the Avogadro constant in X-ray diffraction experiments. Furthermore, it is simpler to present the mass of an atom or isotope in the form of relative isotopic mass, which is a dimensionless quantity defined as the ratio of the mass of an isotope in unified atomic mass unit to one unified atomic mass unit. The table below lists the relative masses of isotopes of the first few elements in the periodic table:

Atomic no. (Z)	Mass no. (A)	Symbol	Relative isotopic mass*
1	1	1H	1.007825
	2	2H	2.014104
2	3	3He	3.016029
	4	4He	4.002603
3	6	6Li	6.015122
4	9	9Be	9.012182
5	10	10B	10.012937
	11	11B	11.009305
6	12	12C	12.000000
8	16	16O	15.994915
	17	17O	16.999132
	18	18O	17.999160
9	19	19F	18.998403
10	20	20Ne	19.992440
	21	21Ne	20.993847
	22	22Ne	21.991386

*With the new definition of the Avogadro constant, the mass of an atom in kg is no longer subject to the uncertainty of the Avogadro constant, but is contingent on the uncertainty in the value of the molar mass constant, since . However, the relative masses of isotopes remain unchanged. 

 

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