Bohr radius - wikipedia Bohr radius This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. ( July 2017 ) Bohr radius Unit of length Symbol a or r Named after Niels Bohr Unit conversions 1 a in...... is equal to... SI units 6989529000000000000 ♠ 5.29 × 10 m imperial / US units 2.08 × 10 in natural units 7015167272458542827 ♠ 2.68 × 10 / eV 7024327000000000000 ♠ 3.27 × 10 l The Bohr radius ( a or r ) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an atom. Its value is 6989529177210670000 ♠ 5.291 772 1067 ( 12 ) × 10 m. Contents 1 Definition and value 2 Use 3 Related units 4 Reduced Bohr radius 5 See also 6 Notes 7 References 8 External links Definition and value In SI units the Bohr radius is : a 0 = 4 π ε 0 ħ 2 m e e 2 = ħ m e c α ( displaystyle a_ ( 0 ) = ( \frac ( 4 pi \varepsilon _ ( 0 ) hbar ^ ( 2 ) ) ( m_ ( mathrm ( e ) ) e ^ ( 2 ) ) ) = ( \frac ( hbar ) ( m_ ( mathrm ( e ) ) , c , alpha ) ) ) where : a 0 ( displaystyle a_ ( 0 ) ) is the Bohr radius, ε 0 ( displaystyle \varepsilon _ ( 0 ) ) is the permittivity of free space, ħ ( displaystyle hbar ) is the reduced Planck 's constant, m e ( displaystyle m_ ( mathrm ( e ) ) ) is the electron rest mass, e ( displaystyle e ) is the elementary charge, c ( displaystyle c ) is the speed of light in vacuum, and α ( displaystyle alpha ) is the fine structure constant. In Gaussian units the Bohr radius is simply a 0 = ħ 2 m e e 2 ( displaystyle a_ ( 0 ) = ( \frac ( hbar ^ ( 2 ) ) ( m_ ( e ) e ^ ( 2 ) ) ) ) According to 2014 CODATA the Bohr radius has a value of 6989529177210670000 ♠ 5.291 772 1067 ( 12 ) × 10 m ( i.e., approximately 53 pm or 0.53 Å ). Use In the Bohr model of the structure of an atom, put forward by Niels Bohr in 1913, electrons orbit a central nucleus. The model says that the electrons orbit only at certain distances from the nucleus, depending on their energy. In the simplest atom, hydrogen, a single electron orbits the nucleus and its smallest possible orbit, with lowest energy, has an orbital radius almost equal to the Bohr radius. ( It is not exactly the Bohr radius due to the reduced mass effect. They differ by about 0.1 %. ) Although the Bohr model is no longer in use, the Bohr radius remains very useful in atomic physics calculations, due in part to its simple relationship with other fundamental constants. ( This is why it is defined using the true electron mass rather than the reduced mass, as mentioned above. ) For example, it is the unit of length in atomic units. An important distinction is that the Bohr radius gives the position of maximum probability density, not its expected radial distance. The expected radial distance is actually 1.5 times the Bohr radius, as a result of the long tail of the radial wave function. Related units The Bohr radius of the electron is one of a trio of related units of length, the other two being the Compton wavelength of the electron λ e ( displaystyle lambda _ ( mathrm ( e ) ) ) and the classical electron radius r e ( displaystyle r_ ( mathrm ( e ) ) ). The Bohr radius is built from the electron mass m e ( displaystyle m_ ( mathrm ( e ) ) ), Planck 's constant ħ ( displaystyle hbar ) and the electron charge e ( displaystyle e ). The Compton wavelength is built from m e ( displaystyle m_ ( mathrm ( e ) ) ), ħ ( displaystyle hbar ) and the speed of light c ( displaystyle c ). The classical electron radius is built from m e ( displaystyle m_ ( mathrm ( e ) ) ), c ( displaystyle c ) and e ( displaystyle e ). Any one of these three lengths can be written in terms of any other using the fine structure constant α ( displaystyle alpha ) : r e = α λ e 4 π = α 2 a 0. ( displaystyle r_ ( mathrm ( e ) ) = ( \frac ( alpha lambda _ ( mathrm ( e ) ) ) ( 4 pi ) ) = alpha ^ ( 2 ) a_ ( 0 ). ) The Compton wavelength is about 20 times smaller than the Bohr radius, and the classical electron radius is about 1000 times smaller than the Compton wavelength. Reduced Bohr radius The Bohr radius including the effect of reduced mass in the hydrogen atom can be given by the following equation : a 0 ∗ = λ p + λ e 2 π α, ( displaystyle a_ ( 0 ) ^ ( * ) = ( \frac ( lambda _ ( mathrm ( p ) ) + lambda _ ( mathrm ( e ) ) ) ( 2 pi alpha ) ), ) where : λ p ( displaystyle lambda _ ( mathrm ( p ) ) ) is the Compton wavelength of the proton. λ e ( displaystyle lambda _ ( mathrm ( e ) ) ) is the Compton wavelength of the electron. α ( displaystyle alpha ) is the fine structure constant. In the above equation, the effect of the reduced mass is achieved by using the increased Compton wavelength, which is just the Compton wavelengths of the electron and the proton added together. See also Angstrom Bohr magneton Rydberg energy Notes ^ Jump up to : The number in parenthesis denotes the uncertainty of the last digits. References ^ Jump up to : `` CODATA Value : Bohr radius ''. Fundamental Physical Constants. NIST. Retrieved 13 February 2016. Jump up ^ David J. Griffiths, Introduction to Quantum Mechanics, Prentice - Hall, 1995, p. 137. ISBN 0 - 13 - 124405 - 1 Jump up ^ Zettili, Nouredine ( 2009 ). Quantum Mechanics : Concepts and Applications ( 2nd ed. ). Chichester : Wiley. p. 376. ISBN 978 - 0 - 470 - 02678 - 6. External links Length Scales in Physics : the Bohr Radius Retrieved from `` https://en.wikipedia.org/w/index.php?title=Bohr_radius&oldid=811493334 '' Categories : Units of length Atomic physics Physical constants Hidden categories : Articles needing additional references from July 2017 All articles needing additional references Talk Contents About Wikipedia Asturianu Беларуская Bosanski Català Čeština Dansk Deutsch Español Esperanto فارسی Français 한국어 हिन्दी Hrvatski Italiano עברית Magyar Nederlands 日本 語 Norsk nynorsk Oʻzbekcha / ўзбекча Polski Português Русский Sicilianu Slovenčina Slovenščina Српски / srpski Suomi Türkçe Українська Tiếng Việt 中文 Edit links This page was last edited on 22 November 2017, at 00 : 08. 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Question: what is the radius of the hydrogen atom
Answer: 5.2917721067(12)×10−11 m