Avogadro constant

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The Avogadro constant (symbols: L, NA), also called Avogadro's number, is the number of "elementary entities" (usually atoms or molecules) in one mole, that is (from the definition of the mole), the number of atoms in exactly 12 grams of carbon-12.12 The 2006 CODATA recommended value is 6.02214179(30)×1023 entities per mole.3

The Avogadro constant is named after the early nineteenth 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.4 The French physicist Jean Perrin in 1909 proposed naming the constant in honour of Avogadro.5 Perrin would win the 1926 Nobel Prize in Physics, in a large part for his work in determining the Avogadro constant by several different methods.6

The value of the Avogadro constant was first indicated by Johann Josef Loschmidt who, in 1865, estimated the average diameter of the molecules in air by a method that is equivalent to calculating the number of particles in a given volume of gas.7 This latter value, the number density of particles in an ideal gas, is now called the Loschmidt constant in his honour, and is approximately proportional to the Avogadro constant. The connection with Loschmidt is the root 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.8

Contents

In other units

While it is rare to use units of amount of substance other than the mole, the Avogadro constant can also be defined in units such as the pound mole (lb-mol.) and the ounce mole (oz-mol.).

NA = 2.731 597 57(14)×1026 lb-mol.–1 = 1.707 248 479(85)×1025 oz-mol.–1

Application

The Avogadro constant can be applied to any substance. It corresponds to the number of atoms or molecules needed to make up a mass equal to the substance's atomic or molecular mass, in grams. For example, the atomic mass of iron is 55.847 g/mol, so NA iron atoms (i.e., one mole of iron atoms) have a mass of 55.847 g. Conversely, 55.847 g of iron contains NA iron atoms. The Avogadro constant also enters into the definition of the unified atomic mass unit, u:

1 \ \mathrm{u} = \frac{1}{N_A} \ \mathrm{g} = (1.660 \, 538\, 86 \pm 0.000\, 000\, 28) 10^{-24} \ \mathrm{g}

Additional physical relations

Because of its role as a scaling factor, the Avogadro constant provides the link between a number of useful physical constants when moving between the atomic scale and the macroscopic scale. For example, it provides the relationship between:

R = k_BN_A = 8.314 \, 472 \, \pm \, 0.000 \, 015 \, \mbox{J}\cdot\mbox{mol}^{-1}\mbox{K}^{-1}\,
in J mol−1 K−1
F = N_Ae = 96 \, 485.3383 \, \pm \,0.0083 \,\, \mbox{C}\cdot\mbox{mol}^{-1} \,
in C mol−1

Measurement

Historical methods

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.

N_{\rm A} = \frac{F}{e}

The classic experiment is that of Bowers and Davis at NIST,9 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 Ar the atomic weight of silver, then the Faraday constant is given by:

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

The NIST workers devised an ingenious method to compensate for silver that was lost from the anode for mechanical reasons, and conducted an isotope analysis of their silver to determine the appropriate atomic weight. Their value for the conventional Faraday constant is F90 = 96 485.39(13) C/mol, which corresponds to a value for the Avogadro constant of 6.022 1449(78)×1023 mol–1: both values have a relative standard uncertainty of 1.3×10–6.

Electron mass method (CODATA)

The CODATA value for the Avogadro constant10 is determined from the ratio of the molar mass of the electron Ar(e)Mu to the rest mass of the electron me:

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

The "relative atomic mass" of the electron, Ar(e), is a directly-measured quantity, and the molar mass constant, Mu, is a defined constant in the SI system. The electron rest mass, however, is calculated from other measured constants:10

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

As can be seen from the table of 2006 CODATA values below,3 the main limiting factor in the accuracy to which the value of the Avogadro constant is known is the uncertainty in the value of the Planck constant, as all the other constants which contribute to the calculation are known much more accurately.

Constant Symbol 2006 CODATA value Relative standard uncertainty Correlation coefficient
with NA
Electron relative atomic mass Ar(e) 5.485 799 0943(23)×10–4 4.2×10–10 0.0082
Molar mass constant Mu 0.001 kg/mol defined —
Rydberg constant R∞ 10 973 731.568 527(73) m–1 6.6×10–12 0.0000
Planck constant h 6.626 068 96(33)×10–34 Js 5.0×10–8 –0.9996
Speed of light c 299 792 458 m/s defined —
Fine structure constant α 7.297 352 5376(50)×10–3 6.8×10–10 0.0269
Avogadro constant NA 6.022 141 79(30)×1023 mol–1 5.0×10–8 —

X-ray crystal density method

Ball-and-stick model of the unit cell of silicon. X-ray diffraction experiments can determine the cell parameter, a, which can in turn be used to calculate a value for Avogadro's constant

One modern method to calculate the Avogadro constant is to use ratio of the molar volume Vm to the unit cell volume Vcell for a single crystal of silicon:11

N_{\rm A} = \frac{8V_{\rm m}({\rm Si})}{V_{\rm cell}}

The factor of eight arises because there are eight silicon atoms in each unit cell.

The unit cell volume can be obtained by X-ray crystallography: as the unit cell is cubic, the volume is the cube of the length of one side (known as the unit cell parameter, 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 192.015 5762(50) pm, a relative uncertainty of 2.8×10–8, corresponding to a unit cell volume of 3.128 775 48(27)×10–31 m3.

The molar volume requires a series of measurements to be determined. Silicon occurs with three stable isotopes – 28Si, 29Si, 30Si – and the natural variation in the proportions of these isotopes is greater than the other uncertainties in the other measurements, so the proportions must be determined for each crystal which is used. With these values, the atomic weight Ar for that crystal can be calculated, as the relative atomic masses of the three nuclides are known with great accuracy. The crystal must also be weighed and measured to determine its density ρ. Once all these quantities are known, the molar volume Vm is given by:

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

where Mu is the molar mass constant. The 2006 CODATA value for the molar volume of silicon is 12.058 8349(11) cm3mol–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.


See also

References and notes

  1. ^ International Union of Pure and Applied Chemistry Commission on Physiochemical Symbols Terminology and Units (1993). Quantities, Units and Symbols in Physical Chemistry (2nd Edition). Oxford: Blackwell Scientific Publications. ISBN 0-632-03583-8, http://www.iupac.org/publications/books/gbook/green_book_2ed.pdf. Retrieved on 28 December 2006.  International Union of Pure and Applied Chemistry Commission on Quantities and Units in Clinical Chemistry; International Federation of Clinical Chemistry Committee on Quantities and Units (1996). "Glossary of Terms in Quantities and Units in Clinical Chemistry (IUPAC-IFCC Recommendations 1996)" (PDF). Pure Appl. Chem. 68: 957–1000. doi:10.1351/pac199668040957, http://www.iupac.org/publications/pac/1996/pdf/6804x0957.pdf. Retrieved on 28 December 2006. 
  2. ^ International Union of Pure and Applied Chemistry Commission on Atomic Weights and Isotopic Abundances (1992). "Atomic Weight: The Name, Its History, Definition and Units" (PDF). Pure Appl. Chem. 64: 1535–43. doi:10.1351/pac199264101535, http://www.iupac.org/publications/pac/1992/pdf/6410x1535.pdf. Retrieved on 28 December 2006. 
  3. ^ a b Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2008). "CODATA Recommended Values of the Fundamental Physical Constants: 2006". Rev. Mod. Phys. 80: 633–730. doi:10.1103/RevModPhys.80.633, http://physics.nist.gov/cuu/Constants/codata.pdf.  Direct link to value.
  4. ^ Avogadro, Amadeo (1811). "Essai d'une maniere de determiner les masses relatives des molecules elementaires des corps, et les proportions selon lesquelles elles entrent dans ces combinaisons". Journal de Physique 73: 58-76.  English translation.
  5. ^ Perrin, Jean (1909). "Mouvement brownien et réalité moléculaire". Annales de Chimie et de Physique, 8e Série 18: 1–114.  Extract in English, translation by Frederick Soddy.
  6. ^ Oseen, C.W. (December 10, 1926). Presentation Speech for the 1926 Nobel Prize in Physics.
  7. ^ Loschmidt, J. (1865). "Zur Grösse der Luftmoleküle". Sitzungsberichte der kaiserlichen Akademie der Wissenschaften Wien 52 (2): 395–413.  English translation.
  8. ^ Virgo, S.E. (1933). "Loschmidt's Number". Science Progress 27: 634–49, http://gemini.tntech.edu/~tfurtsch/scihist/loschmid.html. 
  9. ^ This account is based on the review in Mohr, Peter J.; Taylor, Barry N. (1999). "CODATA recommended values of the fundamental physical constants: 1998". J. Phys. Chem. Ref. Data 28 (6): 1713–1852. doi:0047-2689/99/28(6)/1713/140/$71.00. 
  10. ^ a b Mohr, Peter J.; Taylor, Barry N. (2005). "CODATA recommended values of the fundamental physical constants: 2002". Rev. Mod. Phys. 77 (1): 1–107. doi:0034-6861/2005/77(1)/1(107)/$50.00. 
  11. ^ Mineralogy Database (2000-2005). "Unit Cell Formula". Retrieved on 2007-12-09.

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