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Perrin’s experiment: the concept

Perrin developed the concept to analyse the vertical distribution of a volume of molecules of an ideal gas, with total mass of mc, 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 dmc and thickness dh in the cylinder.

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

dp=-\frac{g}{A}dm_{c}\; \; \; \; \; \; \; \; (3)

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:

p(Adh)=\frac{dm_{c}}{M}RT\; \; \; \; \; \; \; (4)

Substituting eq4 in eq3 by eliminating dmc/A yields

\frac{1}{p}dp=-\frac{Mg}{RT}dh\; \; \; \; \; \; \; (5)

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

\int_{p}^{p'}\frac{1}{p}\: dp=-\frac{Mg}{RT}\int_{0}^{h}dh

\frac{p'}{p}=e^{-\frac{Mgh}{RT}}\; \; \; \; \; \; \; (6)

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

\frac{N'}{N}=e^{-\frac{Mgh}{RT}}\; \; \; \; \; \; \; (7)

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, NA, multiplied by the mass of a molecule of the gas, m:

M=N_{A}m\; \; \; \; \; \; \; (8)

Substituting eq8 in eq7 results in

\frac{N'}{N}=e^{-\frac{N_{A}mgh}{RT}}\; \; \; \; \; \; \; (9)

If Avogadro’s hypothesis – that ‘equal volumes of all gases at the same temperature and pressure have the same number of molecules’ – is true, NA in eq9 must be constant for different ideal gases. To determine the value of NA, 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 NA. However, a problem arose.

 

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Jean Perrin

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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What is mass?

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?

 

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Unified atomic mass unit

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.

12C was chosen in 1961 as the standard reference because 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 to ‘weigh’ an atom?

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:

one\; ^{2}H=\frac{\frac{u}{z}\; of\; one\; ^{2}H}{\frac{u}{z}\; of\; one\; ^{12}C}\times one\; ^{12}C\; \; \; \; \; \; \; \; (6)

one\; ^{2}H=0.167842\times 12\: u=2.014104\: u

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,

one\; ^{2}H=\frac{\frac{u}{z}\; of\; one\; ^{2}H}{\frac{u}{z}\; of\; one\; ^{12}C}\times \frac{0.012\: kg}{N_{A}}

Multiplying both sides by NA,

N_{A}\times one\; of\; ^{2}H=\frac{\frac{u}{z}\; of\; one\; ^{2}H}{\frac{u}{z}\; of\; one\; ^{12}C}\times 0.012\: kg

one\;mole\; of\; ^{2}H\left ( molar\; mass\; of \; ^{2}H \right )=\frac{\frac{u}{z}\; of\; one\; ^{2}H}{\frac{u}{z}\; of\; one\; ^{12}C}\times 0.012\: kg\; \; \; \; \; \; \; \; (7)

where the RHS of eq7 is independent of NA.

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 1u=\frac{M_u}{N_A}g. However, the relative masses of isotopes remain unchanged. 

 

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