 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

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 is 5.29177210903(80) ✕ 10-11 m.

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

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