Avogadro constant

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Amedeo Avogadro

In chemistry and physics, the Avogadro constant, named after scientist Amedeo Avogadro, is the number of constituent particles, usually atoms or molecules, that are contained in the amount of substance given by one mole. Thus, it is the proportionality factor that relates the molar mass of a substance to the mass of a sample. The Avogadro constant, often designated with the symbol N_A or L, has the value 6.022140857(74)×10²³ mol⁻¹ in the International System of Units (SI). (The parentheses there represent the degree of uncertainty.)

Previous definitions of chemical quantity involved the Avogadro number, a historical term closely related to the Avogadro constant, but defined differently: the Avogadro number was initially defined by Jean Baptiste Perrin as the number of atoms in one gram-molecule of atomic hydrogen, meaning one gram of hydrogen. This number is also known as Loschmidt constant in German literature. The constant was later redefined as the number of atoms in 12 grams of the isotope carbon-12 (¹²C), and still later generalized to relate amounts of a substance to their molecular weight. For instance, to a first approximation, 1 gram of hydrogen element (H), having the atomic (mass) number 1, has 6.022×10²³ hydrogen atoms. Similarly, 12 grams of ¹²C, with the mass number 12 (atomic number 6), has the same number of carbon atoms, 6.022×10²³. The Avogadro number is a dimensionless quantity, and has the same numerical value of the Avogadro constant when given in base units. In contrast, the Avogadro constant has the dimension of reciprocal amount of substance. The Avogadro constant can also be expressed as 0.602214... mL⋅mol⁻¹⋅Å⁻³, which can be used to convert from volume per molecule in cubic ångströms to molar volume in millilitres per mole.

Pending revisions in the base set of SI units necessitated redefinitions of the concepts of chemical quantity. The Avogadro number, and its definition, was deprecated in favor of the Avogadro constant and its definition. Based on measurements made through the middle of 2017 which calculated a value for the Avogadro constant of N_A = 6.022140758(62)×10²³ mol⁻¹, the redefinition of SI units is planned to take effect on 20 May 2019. The value of the constant will be fixed to exactly 6.02214076×10²³ mol⁻¹.

History

The Avogadro constant is named after the early 19th-century Italian scientist Amedeo Avogadro, who, in 1811, first proposed that the volume of a gas (at a given pressure and temperature) is proportional to the number of atoms or molecules regardless of the nature of the gas. The French physicist Jean Perrin in 1909 proposed naming the constant in honor of Avogadro. Perrin won the 1926 Nobel Prize in Physics, largely for his work in determining the Avogadro constant by several different methods.

The value of the Avogadro constant was first indicated by Johann Josef Loschmidt, who in 1865 estimated the average diameter of the molecules in the air by a method that is equivalent to calculating the number of particles in a given volume of gas. This latter value, the number density n₀ of particles in an ideal gas, is now called the Loschmidt constant in his honor, and is related to the Avogadro constant, N_A, by

${\displaystyle n_{0}={\frac {p_{0}N_{\rm {A}}}{RT_{0}}},}$

where p₀ is the pressure, R is the gas constant and T₀ is the absolute temperature. The connection with Loschmidt is the origin of the symbol L sometimes used for the Avogadro constant, and German-language literature may refer to both constants by the same name, distinguished only by the units of measurement.

Accurate determinations of the Avogadro constant require the measurement of a single quantity on both the atomic and macroscopic scales using the same unit of measurement. This became possible for the first time when American physicist Robert Millikan measured the charge on an electron in 1910. The electric charge per mole of electrons is a constant called the Faraday constant and had been known since 1834 when Michael Faraday published his works on electrolysis. By dividing the charge on a mole of electrons by the charge on a single electron the value of the Avogadro number is obtained. Since 1910, newer calculations have more accurately determined the values for the Faraday constant and the elementary charge.

Perrin originally proposed the name Avogadro's number (N) to refer to the number of molecules in one gram-molecule of oxygen (exactly 32g of oxygen, according to the definitions of the period), and this term is still widely used, especially in introductory works. The change in name to Avogadro constant (N_A) came with the introduction of the mole as a base unit in the International System of Units (SI) in 1971, which regarded amount of substance as an independent dimension of measurement. With this recognition, the Avogadro constant was no longer a pure number, but had a unit of measurement, the reciprocal mole (mol⁻¹).

While it is rare to use units of amount of substance other than the mole, the Avogadro constant can also be expressed by pound-mole and ounce-mole.

General role in science

The Avogadro constant is a scaling factor between macroscopic and microscopic (atomic scale) observations of nature. As such, it provides the relationship between other physical constants and properties. For example, based on CODATA values, it establishes the following relationship between the gas constant R and the Boltzmann constant k_B,

${\displaystyle R=k_{\text{B}}N_{\text{A}}=8.314\,4598(48)\ {{\text{J}}{\cdot }{\text{mol}}^{-1}{\cdot }{\text{K}}^{-1}}}$

and the Faraday constant F and the elementary charge e,

${\displaystyle F=N_{\text{A}}e=96\,485.33289(59)\ {{\text{C}}{\cdot }{\text{mol}}^{-1}}.\,}$

The Avogadro constant also enters into the definition of the unified atomic mass unit, u,

${\displaystyle 1\ {\text{u}}={\frac {M_{\text{u}}}{N_{\text{A}}}}=1.660\,539\,040(20)\times 10^{-27}\ {\text{kg}}}$

where M_u is the molar mass constant.

Measurement

Coulometry

The earliest accurate method to measure the value of the Avogadro constant was based on coulometry. The principle is to measure the Faraday constant, F, which is the electric charge carried by one mole of electrons, and to divide by the elementary charge, e, to obtain the Avogadro constant.

${\displaystyle N_{\text{A}}={\frac {F}{e}}}$

The classic experiment is that of Bower and Davis at NIST, and relies on dissolving silver metal away from the anode of an electrolysis cell, while passing a constant electric current I for a known time t. If m is the mass of silver lost from the anode and A_r the atomic weight of silver, then the Faraday constant is given by:

${\displaystyle F={\frac {A_{\rm {r}}M_{\rm {u}}It}{m}}.}$

The NIST scientists devised a method to compensate for silver lost from the anode by mechanical causes, and conducted an isotope analysis of the silver used to determine its atomic weight. Their value for the conventional Faraday constant is F₉₀ = 96485.39(13) C/mol, which corresponds to a value for the Avogadro constant of 6.0221449(78)×10²³ mol⁻¹: both values have a relative standard uncertainty of 1.3×10⁻⁶.

Electron mass measurement

The Committee on Data for Science and Technology (CODATA) publishes values for physical constants for international use. It determines the Avogadro constant from the ratio of the molar mass of the electron A_r(e)M_u to the rest mass of the electron m_e:

${\displaystyle N_{\rm {A}}={\frac {A_{\rm {r}}({\rm {e}})M_{\rm {u}}}{m_{\rm {e}}}}.}$

The relative atomic mass of the electron, A_r(e), is a directly measured quantity, and the molar mass constant, M_u, is a defined constant in the SI. The electron rest mass, however, is calculated from other measured constants:

${\displaystyle m_{\rm {e}}={\frac {2R_{\infty }h}{c\alpha ^{2}}}.}$

As may be observed in the table below, the main limiting factor in the precision of the Avogadro constant is the uncertainty in the value of the Planck constant, as all the other constants that contribute to the calculation are known more precisely.

Constant	Symbol	2014 CODATA value	Relative standard uncertainty	Correlation coefficient with NA
Proton–electron mass ratio	mp/me	1836.152 673 89(17)	9.5×10–11	−0.0003
Molar mass constant	Mu	0.001 kg/mol = 1 g/mol	0 (defined)	—
Rydberg constant	R∞	10 973 731.568 508(65) m−1	5.9×10–12	−0.0002
Planck constant	h	6.626 070 040(81)×10–34 J s	1.2×10–8	−0.9993
Speed of light	c	299 792 458 m/s	0 (defined)	—
Fine structure constant	α	7.297 352 5664(17)×10–3	2.3×10–10	0.0193
Avogadro constant	NA	6.022 140 857(74)×1023 mol−1	1.2×10–8	1

X-ray crystal density (XRCD) methods

Ball-and-stick model of the unit cell of silicon. X-ray diffraction measures the cell parameter, a, which is used to calculate a value for the Avogadro constant.

A modern method to determine the Avogadro constant is the use of X-ray crystallography. Silicon single crystals may be produced today in commercial facilities with extremely high purity and with few lattice defects. This method defines the Avogadro constant as the ratio of the molar volume, V_m, to the atomic volume V_atom:

${\displaystyle N_{\rm {A}}={\frac {V_{\rm {m}}}{V_{\rm {atom}}}}}$, where ${\displaystyle V_{\rm {atom}}={\frac {V_{\rm {cell}}}{n}}}$ and n is the number of atoms per unit cell of volume V_cell.

The unit cell of silicon has a cubic packing arrangement of 8 atoms, and the unit cell volume may be measured by determining a single unit cell parameter, the length of one of the sides of the cube, a.

In practice, measurements are carried out on a distance known as d₂₂₀(Si), which is the distance between the planes denoted by the Miller indices {220}, and is equal to a/√8. The 2006 CODATA value for d₂₂₀(Si) is 192.0155762(50) pm, a relative standard uncertainty of 2.8×10⁻⁸, corresponding to a unit cell volume of 1.60193304(13)×10⁻²⁸ m³.

The isotope proportional composition of the sample used must be measured and taken into account. Silicon occurs in three stable isotopes (²⁸Si, ²⁹Si, ³⁰Si), and the natural variation in their proportions is greater than other uncertainties in the measurements. The atomic weight A_r for the sample crystal can be calculated, as the standard atomic weights of the three nuclides are known with great accuracy. This, together with the measured density ρ of the sample, allows the molar volume V_m to be determined:

${\displaystyle V_{\rm {m}}={\frac {A_{\rm {r}}M_{\rm {u}}}{\rho }}}$

where M_u is the molar mass constant. The 2006 CODATA value for the molar volume of silicon is 12.0588349(11) cm³⋅mol⁻¹, with a relative standard uncertainty of 9.1×10⁻⁸.

As of the 2006 CODATA recommended values, the relative uncertainty in determinations of the Avogadro constant by the X-ray crystal density method is 1.2×10⁻⁷, about two and a half times higher than that of the electron mass method.

International Avogadro Coordination

Achim Leistner at the Australian Centre for Precision Optics (ACPO) holding a one-kilogram single-crystal silicon sphere prepared for the International Avogadro Coordination.

The International Avogadro Coordination (IAC), often simply called the "Avogadro project", is a collaboration begun in the early 1990s between various national metrology institutes to measure the Avogadro constant by the X-ray crystal density method to a relative uncertainty of 2×10⁻⁸ or less. The project is part of the efforts to redefine the kilogram in terms of a universal physical constant, rather than the International Prototype Kilogram, and complements the measurements of the Planck constant using Kibble balances. Under the current definitions of the International System of Units (SI), a measurement of the Avogadro constant is an indirect measurement of the Planck constant:

${\displaystyle h={\frac {c\alpha ^{2}A_{\rm {r}}({\rm {e}})M_{\rm {u}}}{2R_{\infty }N_{\rm {A}}}}.}$

The measurements use highly polished spheres of silicon with a mass of one kilogram. Spheres are used to simplify the measurement of the size (and hence the density) and to minimize the effect of the oxide coating that inevitably forms on the surface. The first measurements used spheres of silicon with natural isotopic composition, and had a relative uncertainty of 3.1×10⁻⁷. These first results were also inconsistent with values of the Planck constant derived from Kibble balance measurements, although the source of the discrepancy is now believed to be known.

The main residual uncertainty in the early measurements was in the measurement of the isotopic composition of the silicon to calculate the atomic weight, so in 2007 a 4.8 kg single crystal of isotopically-enriched silicon (99.94% ²⁸Si) was grown, and two one-kilogram spheres cut from it. Diameter measurements on the spheres are repeatable to within 0.3 nm, and the uncertainty in the mass is 3 µg. Full results from these determinations were expected in late 2010. Their paper, published in January 2011, summarized the result of the International Avogadro Coordination and presented a measurement of the Avogadro constant to be 6.02214078(18)×10²³ mol⁻¹.