Loschmidt's constant

This article deals with the relationship between the number of particles and the volume of a gas. For the relationship between the number of particles and the amount of substance of any given substance - sometimes also called the Loschmidt number - see Avogadro's constant .

Physical constant
Surname	Loschmidt's constant
Formula symbol	${\displaystyle n_{0}}$
value
SI	2.686 780 111 ...e25th ${\displaystyle \textstyle {\frac {1}{\mathrm {m^{3}} }}}$
Uncertainty  (rel.)	(exactly)
Gauss	2.686 780 111 ...e19th ${\displaystyle \textstyle {\frac {1}{\mathrm {cm^{3}} }}}$
Relation to other constants
${\displaystyle n_{0}={\frac {p_{0}}{k_{\mathrm {B} }\cdot T_{0}}}}$ at normal pressure and normal temperature ${\displaystyle p_{0}=101\,325\,\mathrm {Pa} }$ ${\displaystyle T_{0}=273,15\,\mathrm {K} }$

The Loschmidt constant (sometimes also referred to as) is a physical constant named after Josef Loschmidt , which describes the number of molecules per volume of an ideal gas under normal conditions (T ₀  = 273.15 K = 0 ° C) and (p ₀  = 101.325 kPa). ${\displaystyle n_{0}}$ ${\displaystyle N_{\mathrm {L} }}$ ${\displaystyle N}$ ${\displaystyle V_{0}}$

${\displaystyle n_{0}={\frac {N}{V_{0}}}}$

Relation to other sizes and value

The Loschmidt constant is linked to the Boltzmann constant k _B :

${\displaystyle n_{0}={\frac {p_{0}}{k_{\mathrm {B} }\cdot T_{0}}}\;}$,

where p ₀  = 101 325  Pa is the normal pressure and T ₀  = 273.15  K is the normal temperature. The Boltzmann constant defines the temperature scale and is defined as an exact value. As a result, the Loschmidt constant also has an exact value:

${\displaystyle n_{0}={\frac {101\,325\;\mathrm {Pa} }{1{,}380\,649\cdot 10^{-23}\;\mathrm {J/K} \cdot 273{,}15\;\mathrm {K} }}\quad =\quad {\frac {1{,}01325}{1{,}380649\cdot 2{,}7315}}\cdot 10^{24}\mathrm {\frac {Pa\cdot K}{J\cdot K}} }$

${\displaystyle n_{0}={\frac {1{,}01325}{1{,}380649\cdot 2{,}7315}}\cdot 10^{24}\mathrm {{\frac {kg}{m\cdot s^{2}}}\cdot {\frac {s^{2}}{kg\,m^{2}}}} \quad =\quad 2{,}686\,780\,111\,\ldots \cdot 10^{25}\;\mathrm {m} ^{-3}}$.

Before the redefinition of the International System of Units in 2019, the Loschmidt constant had to be determined experimentally and was subject to a measurement error.

The Loschmidt constant related to the Avogadro's number N _A over the molar volume of an ideal gas under normal conditions, V _m0 , via

${\displaystyle n_{0}={\frac {N_{\mathrm {A} }}{V_{\mathrm {m} 0}}}}$

together. The relationship can also be expressed using the universal gas constant R :

${\displaystyle n_{0}=N_{\mathrm {A} }\cdot {\frac {p_{0}}{R\cdot T_{0}}}}$

History and designation of the constant

Josef Loschmidt

The Italian physicist Amedeo Avogadro postulated in 1811 that equal volumes of different ideal gases contain the same number of molecules ( Avogadro's law ).

The Austrian physicist and chemist Josef Loschmidt succeeded for the first time in 1865 (after Avogadro's death) in determining this number of molecules by order of magnitude (see “The size of air molecules” ). Loschmidt's student and later friend Ludwig Boltzmann named the particle number of the molecules of an ideal gas at normal pressure and temperature per volume derived from Loschmidt's results as the Loschmidt constant . The Loschmidt constant multiplied by the CGS unit cubic centimeter (cm ³ ) is called Loschmidt's number (in the Gaussian CGS system) : ${\displaystyle n_{0}}$ ${\displaystyle \left\{n_{0}\right\}_{\mathrm {CGS} }}$

${\displaystyle n_{0}=\left\{n_{0}\right\}_{\mathrm {CGS} }\,{\frac {1}{\mathrm {cm} ^{3}}}}$

In 1909 (after both Loschmidt and Avogadro had died), the French chemist Jean-Baptiste Perrin suggested that size should not be given as the number of particles per volume, but as the number of particles per mole under the name Avogadro number . The Avogadro number (in the International System of Units (SI) ) indicates how many particles a substance of 1 mol consists of. In German-speaking countries, the name Loschmidt's number or Loschmidt number and the symbol L continued to be used, but now with a different meaning, namely as a synonym for Avogadro number or Avogadro constant . ${\displaystyle \left\{N_{\mathrm {A} }\right\}_{\mathrm {SI} }}$

The Avogadro number in the SI multiplied by the SI unit mol ⁻¹ is the (physical quantity of the) Avogadro constant : ${\displaystyle \left\{N_{\mathrm {A} }\right\}_{\mathrm {SI} }}$ ${\displaystyle N_{\mathrm {A} }}$

${\displaystyle N_{\mathrm {A} }=\left\{N_{\mathrm {A} }\right\}_{\mathrm {SI} }{\frac {1}{\mathrm {mol} }}}$

Avogadro's constant (not Loschmidt's constant) is used to convert molecular to molar sizes. The Loschmidt constant has been included in the CODATA recommendations for physical constants since the CODATA 1986 publication.

Loschmidt's work "On the size of air molecules"

The work in which Loschmidt determined the Loschmidt number later named after him in 1865 was published in 1866 as an article "On the size of the air molecules". It built on the kinetic gas theory and the related results of Clausius , Maxwell and Oskar Emil Meyer . Loschmidt defined the number of air molecules contained in the volume unit there, but he did not give a numerical value for this. The aim of his work was a preliminary approximation of the size of the diameter of the air molecules under normal conditions, here called Loschmidt's molecular diameter s _{0 of} an ideal gas. s ₀ was calculated from a so-called “condensation coefficient” and from the then known value of the mean free path λ for air at 0 ° C. The Loschmidt constant n ₀ can be derived from this - again over the mean free path

${\displaystyle n_{0}={\frac {3}{4\,\pi \,\lambda \,s_{0}^{2}}}}$

be calculated. If the value of Loschmidt's constant or Avogadro's constant recommended today is expressed as Loschmidt's molecular diameter, then s ₀  = 0.361 nm. Loschmidt's result from 1865 was s ₀  = 0.970 nm, i.e. 2.7 times that the actual value. However, he also gave an indication of the statistical uncertainty of his result. Original quote: " This value is only to be taken as an approximate approximation, but it is certainly not ten times too large or too small ". That was true.

Loschmidt had two different values ​​available for the mean free path: the value of λ  = 62 nm determined by Maxwell and a more recent, much larger and - as it later turned out - more inaccurate value of λ  = 140 nm published by Oskar Emil Meyer Today the mean free path of an ideal gas under normal conditions is λ  = 68 nm. If Loschmidt had used Maxwell instead of Meyer's value of the mean free path for his calculation, the result would have been s ₀  = 0.429 nm. This alternative result has a surprisingly low inaccuracy of only 1.28 times the actual value.

swell

1. ↑ ^{a b} CODATA Recommended Values. National Institute of Standards and Technology, accessed July 20, 2019 .  The value for Loschmidt's constant is exact; H. with no measurement uncertainty. However, the numerical value as the quotient of exact numbers does not have a finite representation of the decimal places and must therefore be abbreviated with ...
2. ↑ Josef Loschmidt: “On the size of the air molecules” in reports from the meetings of the Imperial Academy of Sciences in Vienna , 52, Dept. II, pp. 395-413 (1866), online in the Google book search.

Web links

• Joseph Loschmidt, Physicist and Chemist in Physics Today Online, March 2001