Explanation:
According to Bohr's model, angular momentum is given by :
[tex]L=\dfrac{nh}{2\pi}[/tex]
Since, L = m v r
So, [tex]mvr=\dfrac{nh}{2\pi}[/tex]
[tex]v=\dfrac{nh}{2\pi mr}[/tex]...................(1)
The electrostatic force is balanced by the electrostatic force as :
[tex]\dfrac{ke^2}{r^2}=\dfrac{mv^2}{r}[/tex]
From equation (1),
[tex]r=\dfrac{n^2h^2}{4\pi^2mke^2}[/tex]
Where
r is the radius of ground state hydrogen atom
n is the orbit
h is Planck's constant
m is the mass of electron
k is the electrostatic constant
[tex]r=\dfrac{1^2(6.62\times 10^{-34})^2}{4\pi^2\times 9.1\times 10^{-31}\times 9\times 10^9\times (1.6\times 10^{-19})^2}[/tex]
[tex]r=5.29\times 10^{-11}\ m[/tex]
Diameter of hydrogen atom,
[tex]D=2r=2\times 5.29\times 10^{-11}[/tex]
[tex]D=1.058\times 10^{-10}\ m[/tex]
So, the diameter of a ground-state hydrogen atom is [tex]1.058\times 10^{-10}\ m[/tex]. Hence, this is the required solution.
