Answer:
[tex]1.06085\times 10^{-10}\ m[/tex]
Explanation:
h = Planck's constant = [tex]6.626\times 10^{-34}\ m^2kg/s[/tex]
m = Mass of electron = [tex]9.11\times 10^{-31}\ kg[/tex]
k = Coulomb constant = [tex]8.99\times 10^{9}\ Nm^2/C^2[/tex]
e = Charge of electron = [tex]1.6\times 10^{-19}\ C[/tex]
n = 1 (ground state)
Angular momentum is given by
[tex]L=mvr[/tex]
From Bohr's atomic model we have
[tex]L=\dfrac{nh}{2\pi}[/tex]
[tex]mvr=\dfrac{nh}{2\pi}\\\Rightarrow v=\dfrac{nh}{2\pi mr}[/tex]
The centripetal force will balance the electrostatic force
[tex]\dfrac{ke^2}{r^2}=\dfrac{mv^2}{r}\\\Rightarrow \dfrac{ke^2}{r}=mv^2\\\Rightarrow \dfrac{ke^2}{r}=m(\dfrac{nh}{2\pi mr})^2\\\Rightarrow r=\dfrac{n^2h^2}{4\pi^2mke^2}\\\Rightarrow r=\dfrac{1^2\times (6.626\times 10^{-34})^2}{4\pi^2 \times 9.11\times 10^{-31}\times 8.99\times 10^{9}\times (1.6\times 10^{-19})^2}\\\Rightarrow r=5.30426\times 10^{-11}\ m[/tex]
The diameter is [tex]2\times 5.30426\times 10^{-11}=1.06085\times 10^{-10}\ m[/tex]
