Avogadro's number: Difference between revisions
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===Estimates from liquid solutions=== | ===Estimates from liquid solutions=== | ||
[[Albert Einstein|Einstein]] wrote in his 1905 Ph.D. thesis about the size of molecules and the closely related problem of the magnitude of ''N''<sub>A</sub>. He derived equations for [[diffusion coefficient]]s and [[viscosity|viscosities]]. From experimental values of the diffusion coefficients and viscosities of sugar solutions in water Einstein gave the estimate ''N''<sub>A</sub> = 2.1×10<sup>23</sup>. In a later paper | [[Albert Einstein|Einstein]] wrote in his 1905 Ph.D. thesis about the size of molecules and the closely related problem of the magnitude of ''N''<sub>A</sub>. He derived equations for [[diffusion coefficient]]s and [[viscosity|viscosities]] in which Avogadro's number appears. From experimental values of the diffusion coefficients and viscosities of sugar solutions in water Einstein gave the estimate ''N''<sub>A</sub> = 2.1×10<sup>23</sup>. In a later paper derived from his doctorate work<ref>A. Einstein, Ann. d. Physik, '''19''', 289, (1906)</ref> he gave a better estimate from improved experimental data: ''N''<sub>A</sub> = 4.15×10<sup>23</sup>, close to Maxwell's value of 1873. Later (1911) it was discovered that Einstein made an algebraic error in his thesis and the paper based on it. When this was corrected the very same experimental data gave ''N''<sub>A</sub> = 6.6×10<sup>23</sup>. | ||
The phenomenon of [[Brownian motion]] was first described by [[Robert Brown]] in 1828 as the "tremulous motion" of pollen grains (small solid particles of diameter on the order of a micrometer) suspended in water. Einstein's famous 1905 paper on the theory of Brownian motion<ref>A. Einstein, Ann. d. Physik, '''17''', 549, (1905)</ref> gives a method for determining ''N''<sub>A</sub>, but not yet a value. | The phenomenon of [[Brownian motion]] was first described by [[Robert Brown]] in 1828 as the "tremulous motion" of pollen grains (small solid particles of diameter on the order of a micrometer) suspended in water. Einstein's famous 1905 paper on the theory of Brownian motion<ref>A. Einstein, Ann. d. Physik, '''17''', 549, (1905)</ref> gives a method for determining ''N''<sub>A</sub>, but not yet a value. | ||
The first to give a value to ''N''<sub>A</sub> from Brownian motion was [[Jean Baptiste Perrin|Perrin]] in 1908. He considered the distribution of Brownian particles in a vertical column in the earths's gravitational field, and he used a similar mathematical approach to that which leads to the distribution of gas molecules in a vertical column of the atmosphere. This is a [[Boltzmann distribution]] that contains among the mass of the particle and the [[gravitation|gravitational acceleration]] ''g'' the [[Boltzmann constant]] ''k''. For suspended Brownian particles one has to make a correction for the buoyancy of the particles in the liquid (Archimedes principle), by using the expressions that contain the densities of the particles and the liquid. | The first to give a value to ''N''<sub>A</sub> from Brownian motion was [[Jean Baptiste Perrin|Perrin]] in 1908. He considered the distribution of Brownian particles in a vertical column in the earths's gravitational field, and he used a similar mathematical approach to that which leads to the distribution of gas molecules in a vertical column of the atmosphere. This is a [[Boltzmann distribution]] that contains among the mass of the particle and the [[gravitation|gravitational acceleration]] ''g'' the [[Boltzmann constant]] ''k''. For suspended Brownian particles one has to make a correction for the buoyancy of the particles in the liquid (Archimedes principle), by using the expressions that contain the densities of the particles and the liquid. | ||
Measuring the numbers of particles at two different heights allows one to determine Boltzmann’s constant, ''k'', and Avogadro’s constant through ''k'' = ''N''<sub>A</sub>/''R'' (the [[molar gas constant]] ''R'' already being known with high precision). Perrin in his first experiments prepared a monodisperse colloid of a gum called gamboge. The particle masses were determined by direct weighing of a specified number, and their radii (hence their volumes and densities) by using the Stokes-Einstein law for diffusion. Perrin’s first value for Avogadro’s number was ''N''<sub>A</sub> = 7.05× 10<sup>23</sup>. | Measuring the numbers of particles at two different heights allows one to determine Boltzmann’s constant, ''k'', and Avogadro’s constant through ''k'' = ''N''<sub>A</sub>/''R'' (the [[molar gas constant]] ''R'' already being known with high precision). Perrin in his first experiments prepared a monodisperse colloid of a gum called gamboge. The particle masses were determined by direct weighing of a specified number, and their radii (hence their volumes and densities) by using the Stokes-Einstein law for diffusion. Perrin’s first value for Avogadro’s number was ''N''<sub>A</sub> = 7.05×10<sup>23</sup>. In 1909 Perrin coined the name Avogadro's constant when he wrote: "The invariable number ''N'' is a universal constant, which may be appropriately designated Avogadro's Constant." | ||
Revision as of 07:49, 6 December 2007
Avogadro's number, NA, is defined as the number of atoms in 12 gram of carbon-12 atoms in their ground state at rest. By definition it is related to the atomic mass constant mu by the relation
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_\mathrm{A} m_\mathrm{u} = 1/1000 \quad \hbox{exact}. }
The exact factor 1/1000 appears here by the historic facts that the kilogram is the unit of mass and that in chemistry the mole is preferred over the Kmole. Recall that the atomic mass constant has the mass 1 u exactly (u is the unified atomic mass unit). Avogadro's number is indeed defined as number, a dimensionless quantity. Its latest numeric value[1] is NA = 6.022 141 79 1023.
The SI definition of Avogadro's constant (also designated by NA) is: the number of entities (such as atoms, ions, or molecules) per mole. (This definition requires, of course, a definition of mole that does not rely on NA, but one that is in terms of 12C atoms). In this definition NA has dimension mol−1. The numeric value of Avogadro's constant is NA = 6.022 141 79 1023 mol−1.
Because the mole and Avogadro's number are defined in terms of the atomic mass constant (one twelfth of the mass of a 12C atom), Avogadro's constant and Avogadro's number have by definition the same numerical value. In practice the two terms are used interchangeably.
History of Avogadro's number
Since 1811, when Amedeo Avogadro put forward his law stating that equal volumes of gas (we now know ideal gas) contain equal number of particles, increasingly sophisticated methods of determining Avogadro’s constant have been developed. These include the kinetic theory of gases, properties of liquid solutions, measurement of the electron charge, black-body radiation, alpha particle emission, and X-ray measurements of crystals.
Without the belief that a macroscopic substance consists of minute particles (initially called atoms, later also molecules), it does not make sense to speak of Avogadro's number. This belief—called atomism— was born in antiquity and grew further in importance with the developments of chemistry early in the 19th century. An important milestone was John Dalton’s law of multiple proportions published in 1804 that gave rise to the first table of the relative weights of atoms. In 1808 Joseph-Louis Gay-Lussac published his law for the combining volumes of gases, namely that gases combine among themselves in very simple proportions of their volumes, and if the products are gases, their volumes are also in simple proportions. Especially this latter law was of great influence on Avogadro's historical publication of 1811, in which he introduced the term "molecule" and enunciated his law.
In his 1811 paper Avogadro discusses Dalton’s atomic theory and calculates from gas densities that the molecular weight of nitrogen is nearly fourteen times the molecular weight of hydrogen. Avogadro was the first to propose that the gaseous elements, hydrogen, oxygen, and nitrogen, were diatomic molecules. He deduced that the molecule of water contains half a molecule of oxygen and one molecule (or two half molecules) of hydrogen. Dalton, who had assumed earlier that water is formed from a molecule each of oxygen and hydrogen, rejected Avogadro's and Gay-Lussac's laws. There are no testimonials that Avogadro ever speculated on the number of molecules in a given gas volume.
Avogadro's law went for a long time largely unnoticed, not in the least because it was not recognized that the law holds strictly only for ideal gases, which many dissociating and associating organic compounds are not. Four years after his death, at the historic (1860) chemistry conference in Karlsruhe, his countryman Stanislao Cannizaro explained why the exceptions to Avogadro's law happen and that it can determine molar masses.
Kinetic theory estimates
The first estimate of Avogadro’s constant is attributed to Johann Josef Loschmidt (1865). He gave a value for L, the number of molecules in 1 cm3 at standard temperature and pressure. The number L is called Loschmidt’s number; the Avogadro equivalent of this is
- NA = 0.410×1023.
Loschmidt estimated his number by applying the kinetic gas theory of James Clerk Maxwell and Rudolph Clausius, together with experimental data on gas viscosities and atomic volumes. Loschmidt’s work was the first to show that Avogadro’s constant very large. In 1873 Maxwell used his kinetic theory of the diffusion coefficient of a gas to obtain a ten times larger value: NA = 4.2×1023.
A simple method for getting the actual volume of molecules is to use the 1873 Van der Waals equation that contains a parameter b, which is the volume of a single molecule. From this and the volume of the total gas, an estimate of the number of molecules in the gas can be obtained. Much later (1923) Perrin measured b for mercury vapor, and combining this with results from viscosity measurements, he calculated Avogadro's number to be 6.2×1023. which is a very good value.
Estimates from liquid solutions
Einstein wrote in his 1905 Ph.D. thesis about the size of molecules and the closely related problem of the magnitude of NA. He derived equations for diffusion coefficients and viscosities in which Avogadro's number appears. From experimental values of the diffusion coefficients and viscosities of sugar solutions in water Einstein gave the estimate NA = 2.1×1023. In a later paper derived from his doctorate work[2] he gave a better estimate from improved experimental data: NA = 4.15×1023, close to Maxwell's value of 1873. Later (1911) it was discovered that Einstein made an algebraic error in his thesis and the paper based on it. When this was corrected the very same experimental data gave NA = 6.6×1023.
The phenomenon of Brownian motion was first described by Robert Brown in 1828 as the "tremulous motion" of pollen grains (small solid particles of diameter on the order of a micrometer) suspended in water. Einstein's famous 1905 paper on the theory of Brownian motion[3] gives a method for determining NA, but not yet a value.
The first to give a value to NA from Brownian motion was Perrin in 1908. He considered the distribution of Brownian particles in a vertical column in the earths's gravitational field, and he used a similar mathematical approach to that which leads to the distribution of gas molecules in a vertical column of the atmosphere. This is a Boltzmann distribution that contains among the mass of the particle and the gravitational acceleration g the Boltzmann constant k. For suspended Brownian particles one has to make a correction for the buoyancy of the particles in the liquid (Archimedes principle), by using the expressions that contain the densities of the particles and the liquid. Measuring the numbers of particles at two different heights allows one to determine Boltzmann’s constant, k, and Avogadro’s constant through k = NA/R (the molar gas constant R already being known with high precision). Perrin in his first experiments prepared a monodisperse colloid of a gum called gamboge. The particle masses were determined by direct weighing of a specified number, and their radii (hence their volumes and densities) by using the Stokes-Einstein law for diffusion. Perrin’s first value for Avogadro’s number was NA = 7.05×1023. In 1909 Perrin coined the name Avogadro's constant when he wrote: "The invariable number N is a universal constant, which may be appropriately designated Avogadro's Constant."
(To be continued)