Bohr Radius of Hydrogen Atom Formula
On the Bohr Radius Formula of Hydrogen Atoms
The Bohr radius of the hydrogen atom is very important in the field of atomic physics. The derivation of its formula contains profound physical concepts.
In the past, physicists based on classical electromagnetism and quantization assumptions constructed a model of the hydrogen atom. In this model, the electron makes a circular motion around the nucleus, and its centripetal force is derived from the Coulomb attractive force between the nucleus and the electron.
Let the electron mass be\ (m\), the charge be\ (-e\), the nucleus charge is\ (e\), and the electron orbital radius is\ (r\). According to Coulomb's law, the Coulomb force\ (F =\ frac {k e ^ 2} {r ^ 2}\) (where\ (k\) is the Coulomb constant) acts as the centripetal force of the circular motion of the electron\ (F = m\ frac {v ^ 2} {r}\).
According to the quantization assumption, the electron angular momentum\ (L = n\ hbar\) (\ (n = 1,2,3,\ cdots\),\ (\ hbar\) is the reduced Planck constant), and\ (L = mvr\).
From\ (m\ frac {v ^ 2} {r} =\ frac {k e ^ 2} {r ^ 2}\) we get\ (v ^ 2 =\ frac {k e ^ 2} {mr}\). And from\ (mvr = n\ hbar\) we get\ (v =\ frac {n\ hbar} {mr}\).
Substitute\ (v =\ frac {n\ hbar} {mr}\) into\ (v ^ 2 =\ frac {k e ^ 2} {mr}\), that is,\ ((\ frac {n\ hbar} {mr}) ^ 2 =\ frac {k e ^ 2} {mr}\).
Simplify and solve\ (r\) to get\ (r =\ frac {n ^ 2\ hbar ^ 2} {kme ^ 2}\). When\ (n = 1\), this is the Bohr radius of the hydrogen atom\ (a_0 =\ frac {\ hbar ^ 2} {kme ^ 2}\).
This formula reveals the minimum radius of possible orbits of electrons in hydrogen atoms, which is of great significance for understanding atomic structure, spectral properties, etc. Many atomic-related theories and experiments are based on this, and have been deeply explored and expanded, paving the way for the understanding of the microscopic world.
The Bohr radius of the hydrogen atom is very important in the field of atomic physics. The derivation of its formula contains profound physical concepts.
In the past, physicists based on classical electromagnetism and quantization assumptions constructed a model of the hydrogen atom. In this model, the electron makes a circular motion around the nucleus, and its centripetal force is derived from the Coulomb attractive force between the nucleus and the electron.
Let the electron mass be\ (m\), the charge be\ (-e\), the nucleus charge is\ (e\), and the electron orbital radius is\ (r\). According to Coulomb's law, the Coulomb force\ (F =\ frac {k e ^ 2} {r ^ 2}\) (where\ (k\) is the Coulomb constant) acts as the centripetal force of the circular motion of the electron\ (F = m\ frac {v ^ 2} {r}\).
According to the quantization assumption, the electron angular momentum\ (L = n\ hbar\) (\ (n = 1,2,3,\ cdots\),\ (\ hbar\) is the reduced Planck constant), and\ (L = mvr\).
From\ (m\ frac {v ^ 2} {r} =\ frac {k e ^ 2} {r ^ 2}\) we get\ (v ^ 2 =\ frac {k e ^ 2} {mr}\). And from\ (mvr = n\ hbar\) we get\ (v =\ frac {n\ hbar} {mr}\).
Substitute\ (v =\ frac {n\ hbar} {mr}\) into\ (v ^ 2 =\ frac {k e ^ 2} {mr}\), that is,\ ((\ frac {n\ hbar} {mr}) ^ 2 =\ frac {k e ^ 2} {mr}\).
Simplify and solve\ (r\) to get\ (r =\ frac {n ^ 2\ hbar ^ 2} {kme ^ 2}\). When\ (n = 1\), this is the Bohr radius of the hydrogen atom\ (a_0 =\ frac {\ hbar ^ 2} {kme ^ 2}\).
This formula reveals the minimum radius of possible orbits of electrons in hydrogen atoms, which is of great significance for understanding atomic structure, spectral properties, etc. Many atomic-related theories and experiments are based on this, and have been deeply explored and expanded, paving the way for the understanding of the microscopic world.

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