Question

Show that the orbit nearest to the nucleus in the hydrogen atom accommodates one electron waves. Given that the energy of the electron in the orbit is 13.6 eV, h=6.625x10-34Js, m=9.11x10-31kg, Bohr radius r1 =5.292x10-11 m.

Answer 1

The angular momentum for the electron in the first Bohr orbit is

$$mvr_0=\frac{h}{2\pi}.$$

Rearrange and notice there is de Broglie wave:

$$2\pi r_0=\frac h{mv}=\lambda.$$

Substitute the values and get

$$\lambda=2\pi r_0=333\text{ pm}.$$

Thus, the orbit nearest to the nucleus in the hydrogen atom accommodates one electron wave.