Loschmidt constant

The Loschmidt constant or Loschmidt's number (symbol: n₀) is the number of particles (atoms or molecules) of an ideal gas in a given volume (the number density). It is usually quoted at standard temperature and pressure, the 2010 CODATA recommended value is 2.686 7805(24)×10²⁵ per cubic metre at 0 °C and 1 atm and the 2006 CODATA recommended value was 2.686 7774(47)×10²⁵ per cubic metre at 0 °C and 1 atm. It is named after the Austrian physicist Johann Josef Loschmidt, who was the first to estimate the physical size of molecules in 1865. The term "Loschmidt constant" is also sometimes used to refer to the Avogadro constant, particularly in German texts.

The Loschmidt constant is given by the relationship:

$n_0 = \frac{p_0}{k_{\rm B}T_0}$

where p₀ is the pressure, k_B is the Boltzmann constant and T₀ is the thermodynamic temperature. It is related to the Avogadro constant, N_A, by:

$n_0 = \frac{p_0N_{\rm A}}{RT_0}$

where R is the gas constant.

Being a measure of number density, the Loschmidt constant is used to define the amagat, a practical unit of number density for gases and other substances:

1 amagat = n₀ = 2.651 6467 x 1025 m-3(15)²⁵ m⁻³ ,

such that the Loschmidt constant is exactly 1 amagat.

Modern determinations

In the CODATA set of recommended values for physical constants, the Loschmidt constant is calculated from the gas constant and the Avogadro constant:

$n_0 = \frac{N_{\rm A}}{R}\frac{p_0}{T_0} = \frac{A_{\rm r}({\rm e})M_{\rm u}c\alpha^2}{2R_{\infty}hR}\frac{p_0}{T_0}$

where A_r(e) is the relative atomic mass of the electron, M_u is the molar mass constant, c is the speed of light, α is the fine structure constant, R_∞ is the Rydberg constant and h is the Planck constant. The pressure and temperature can be chosen freely, and must be quoted with values of the Loschmidt constant. The precision to which the Loschmidt constant is currently known is limited entirely by the uncertainty in the value of the gas constant.

First determinations

Loschmidt did not actually calculate a value for the constant which now bears his name, but it is a simple and logical manipulation of his published results. James Clerk Maxwell described the paper in these terms in a public lecture eight years later:

Loschmidt has deduced from the dynamical theory the following remarkable proportion:—As the volume of a gas is to the combined volume of all the molecules contained in it, so is the mean path of a molecule to one-eighth of the diameter of a molecule.

To derive this "remarkable proportion", Loschmidt started from Maxwell's own definition of the mean free path:

$\ell = \frac{3}{4n_0\pi d^2}$

where n₀ has the same sense as the Loschmidt constant, that is the number of molecules per unit volume, and d is the effective diameter of the molecules (assumed to be spherical). This rearranges to

$\frac{1}{n_0} = \frac{16}{3}\frac{\pi\ell d^2}{4}$

where 1/n₀ is the volume occupied by each molecule in the gas phase and πℓd²/4 is the volume of the cylinder made by the molecule in its trajectory between two collisions. However, the true volume of each molecule is given by πd³/6, and so n₀πd³/6 is the volume occupied by all the molecules not counting the empty space between them. Loschmidt equated this volume with the volume of the liquified gas. Dividing both sides of the equation by n₀πd³/6 has the effect of introducing a factor of V_liquid/V_gas, which Loschmidt called the "condensation coefficient" and which is experimentally measurable. The equation reduces to:

$d = 8\frac{V_{\rm l}}{V_{\rm g}}\ell$

relating the diameter of a gas molecule to measurable phenomena.

The number density, the constant which now bears Loschmidt's name, can be found by simply substituting the diameter of the molecule into the definition of the mean free path and rearranging:

$n_0 = \left (\frac{V_{\rm g}}{V_{\rm l}}\right )^2 \frac{3}{256\pi\ell^3}$

Instead of taking this step, Loschmidt decided to estimate the mean diameter of the molecules in air. This was no minor undertaking, as the condensation coefficient was unknown and had to be estimated–it would be another twelve years before Pictet and Cailletet would liquify nitrogen for the first time. The mean free path was also uncertain. Nevertheless, Loschmidt arrived at a diameter of about one nanometre, of the correct order of magnitude.

Loschmidt's estimated data for air give a value of n₀ = 1.81×10²⁴ m^-3. Eight years later, Maxwell was citing a figure of "about 19 million million million" per cm³, or 1.9×10²⁵ m^-3.