The **Bohr Radius** is a physical constant, that was formulated by *Neils Bohr* during the Bohr model, thus named after him. It is denoted by \(a_0\).

It is approximately equal to the most probable distance from the nucleus and the electron in a hydrogen atom in its ground state.

Index

**Bohr Radius Definition**

The physical constant \(a_0\) is given as,

\(\large{a_0 = \frac{4 \pi \epsilon_0 \hbar^2}{m_e e^2} = \frac{\epsilon_0 h^2}{\pi m_e e^2} = \frac{\hbar}{m_e c \alpha}}\)

Where,

\(\epsilon_0\) – Permittivity of Free Space

\(\hbar\) – Reduced Planck Constant

\(e\) – Elementary Charge

\(m_e\) – Mass of an Electron

\(c\) – Speed of Light in Vacuum

\(\alpha\) – Fine-structure Constant

**Bohr Radius Value**

Bohr radius value is 5.29177210903(80) ✕ 10^{-11} m.

The table below presents the Bohr model radius for a hydrogen atom in different units.

Units | Bohr Radii |

SI Units | 5.29 ✕ 10^{-11} m |

Imperial Units | 2.08 ✕ 10^{-9} in |

Natural Units | 2.68 ✕ 10^{-4} /eV3.27 ✕ 10 ^{24} lp |

**Derivation of Bohr Radius**

As we know,

Centripetal Force = \(\frac{mv^2}{r}\)

Electrostatic Force = \(\frac{1}{4 \pi \epsilon_0} . \frac{ze^2}{r^2}\)

In a hydrogen atom, Electrostatic force = Centrifugal force

\(\Rightarrow \frac{1}{4 \pi \epsilon_0} . \frac{ze^2}{r^2} = \frac{mv^2}{r}\)

\(\Rightarrow v^2 = \frac{1}{4 \pi \epsilon_0} . \frac{ze^2}{mr} … (1)\)

From Bohr’s quantum equation,

\(L = mvr = n \hbar … (2)\)

Where, \(\hbar = \frac{h}{2 \pi}\)

By solving \((2)\),

\(v = \frac{n \hbar}{mr}\)

Substituting \(v\) in \((1)\),

\((\frac{n \hbar}{mr})^2 = \frac{1}{4 \pi \epsilon_0} . \frac{ze^2}{mr}\)

\(\Rightarrow r = \frac{4 \pi \epsilon_0 (n \hbar)^2}{mze^2} … (3)\)

As we know, \(4 \pi \epsilon_0 = 1/9 ✕ 10^9\)

And, \(\hbar = \frac{h}{2 \pi} = \frac{6.625 ✕ 10^{-34}}{2 \pi}\)

Radius of a hydrogen atom with n = 1, m = 9.11 ✕ 10^{-31} kg, z = 1 and e = 1.6 ✕ 10^{-19} C can be calculated by substituting these values in \((3)\).

\(\Rightarrow r = \frac{(\frac{1}{9} \times 10^9)(1)^2(\frac{6.625 \times 10^{-34}}{2 \pi})^2}{(9.11 \times 10^{-31})(1)(1.6 \times 10^{-19})^2}\)

Simplifying this we get,

\(r = 5.2917721067 \times 10^{-11} m\).

**Applications**

As Bohrs radius is applicable only to Hydrogen and Hydrogen like atoms, it is not much used in modern physics. But it is still used for the following applications:

- Atomic unit
- Fine structure constant

**FAQs**

**What is the Bohr radius?**It is a physical constant, approximately equal to the most probable distance from the nucleus and the electron in a hydrogen atom in its ground state.

**What is the Bohr radius value?**The value of the physical constant is 5.29177210903(80)✕10^{-11}m.

**Related Topics:**