The Mole and Avogadro’s Number

Measuring Chemical Quantities: The Mole introduces the mole as the chemist’s unit for the amount of substance and discusses its relationship with Avogadro’s number of chemical entities including atoms, ions, molecules and formula units. Calculations demonstrate the use of the mole to convert between mass and the number of chemical entities.

Macroscopic Limit of Quantum Systems

Classical physics is approached from quantum mechanics in the macroscopic limit. The technical device to achieve this goal is the quantum version of the central limit theorem, derived for an observable at a given time and for the time-dependent expectation value of the coordinate. The emergence of the classical trajectory can be followed for the average of an observable over a large set of independent microscopical systems, and the deterministic classical laws can be recovered in all practical purposes, owing to the largeness of Avogadro’s number. This result refers to the observed system without considering the measuring apparatus. The emergence of a classical trajectory is followed qualitatively in Wilson’s cloud chamber.

The Mole, Avogadro’s Number and Albert Einstein

The mole a concept and Avogadro’s number are discussed as sought by Albert Einstein in his PhD thesis of 1905. Einstein would probably have regarded the metric system of units based on centimetre-gram-second (cgs) preferable to today’s SI system and specifically he would have rejected a recent SI suggestion to redefine Avogadro’s constant as based on a nonatomistic continuum description of matter. He would probably also have preferred keeping a dualistic definition of mole able of bookkeeping both mass and number of particles: we advocate that here and call it the ‘Einstein Definition’ and as Avogadro’s number we shall adopt an integer, the cube of 84446888 as suggested by Fox and Hill, providing also a definition of the kilogram based on the atomic mass of the carbon 12 isotope. Einstein was the first to explain the microscopic movements of pollen grains reported by Robert Brown in 1828 and his explanation that the particles move as a result of an unequal number of water molecules bumping into them from opposite sides was what finally made the scientific world accept the atom theory in its modern shape. In a cosmic diffusion analogy, pollen or bacterial spores moving randomly in outer space driven by the solar winds between solar systems can be envisaged. Applying Einstein’s diffusion theory, one can argue that life might have emerged from far outside of our planet from billions of solar systems, though not from outside of our Milky Way galaxy. As a curiosity we note that the number of solar systems (stars) in the Universe has been estimated to be of the order of Avogadro’s number.

A relationship between the Avogadro’s number NA and the transition frequency of the caesium 133 atom ΔνCs

At the 26th meeting of the General Conference on Weights and Measures (CGPM) held on 13‐16 November 2018 at Versailles, France, the new International System of Units (SI) was established. Following the CGPM’s decision, the new SI units were established based upon a set of seven defining constants. This set of constants is the most fundamental feature in the definition of the entire system of units. What is truly remarkable about the new SI is the fact that all measurement units, except the amount of substance mole and Avogadro’s number NA , are defined based on the unperturbed ground-state hyperfine transition frequency of the caesium 133 atom <mml:math display="inline"> <mml:msub> <mml:mrow> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>ν</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">Cs</mml:mi> </mml:mrow> </mml:msub> </mml:math> equal to 9 192 631 770 Hz. This article, based on dimensional analysis, presents the possibility of connecting the Avogadro’s number NA and the mole, to the transition frequency <mml:math display="inline"> <mml:msub> <mml:mrow> <mml:mo>Δν</mml:mo> </mml:mrow> <mml:mrow> <mml:mtext>Cs</mml:mtext> </mml:mrow> </mml:msub> </mml:math> .

A Complementary Relation of Fluctuations between Energy and Temperature in a Small Specimen

Research on fluctuations in energy and temperature is presented for a small specimen. The small specimen in contact with a heat bath shows energy fluctuations, [Formula: see text], at the constant temperature. On the other hand, when this small specimen is isolated from the reservoir and adiabatic isolation is kept, it exhibits temperature fluctuations, [Formula: see text], at the constant energy. This means that the temperature is unsharp if a sharp energy is assigned. A complementary relation between [Formula: see text] and [Formula: see text] is proposed in a simple formula. The connection between [Formula: see text] and [Formula: see text] is mediated by the heat capacity [Formula: see text]. This complementary relation is valid in general and it does not depend on the amount of substance. If the constituent number[Formula: see text] of the system is of the order of Avogadro’s number, then the fluctuations have been masked by large [Formula: see text]and we cannot see the influence of the fluctuations. However, when the number [Formula: see text] decreases, the intrinsic features of fluctuations come out gradually. This paper presents the quantitative analyses of the fluctuations in the energy and temperature for several physical models. Typical characteristics in the fluctuations can be clearly seen only in a small specimen, which are shown in the graphical representations. It is stressed that the values of [Formula: see text] and [Formula: see text] are defined for the different prescribed conditions specified above.

Perrin’s Brownian Motion Experiments

Between 1908 and 1911 Perrin published values for Avogadro’s number—the number of molecules per mole of any substance—on the basis of theory-mediated measurements of the mean kinetic energies of granules in Brownian motion. The umbilical cord connecting these energies to Avogadro’s number was the assumption that they are the same as the mean kinetic energies of the molecules in the surrounding liquid. This, as van Fraassen has argued, seems to presuppose that molecules exist, thereby undercutting Perrin’s claim to be proving their existence. This chapter reviews Perrin’s four theory-mediated measurements, showing, on the one hand, that none of them in fact depended on molecular theory yet, on the other, that, by virtue of being exemplars of theory-mediated measurement at its best, they managed to establish several extraordinary landmark conclusions about Brownian motion in its own right.

Converging Values for Avogadro’s Number

Perrin’s values for Avogadro’s numbers presuppose that the mean kinetic energies in Brownian motion match those of molecules in the surrounding liquid. In support of these values and the presupposition, Perrin turns to values of Avogadro’s number and mean kinetic energies of molecules obtained by means of theory-mediated measurements by Planck (from blackbody radiation), Rutherford and his colleagues (from α‎-particle radiation), and Millikan (from ionization). While Perrin’s values differed from these others, they all collectively yielded values for the molecular constants within a theretofore unachievable window of ±5 percent. This chapter assesses first the evidence for Perrin’s values from his appeals to each of these “agreeing” complementary determinations, and then the evidence that molecules exist from all of them taken together. The conclusion is that the latter warranted taking molecules to exist in ongoing research into microphysics even though the referent of the word “molecule” remained seriously underspecified.