Question

If the de-Broglie wavelength of the electron in n^th Bohr orbit in a hydrogenic atom is equal to 1.5πa₀ (a₀ is Bohr radius), then the value of n/Z is
1. 1.0
2. 0.75
3. 0.40
4. 1.50

Answer 1

Correct Answer - Option 2 : 0.75

Calculation:

Number of waves \({\rm{}} = \frac{{{\rm{\;Circumference\;}}}}{{{\rm{\;Wavelength\;}}}}\) 

\(n = \frac{{2\pi r}}{\lambda }\)

∴ 2πr = nλ      ----(1)

Also, we know that radius (r) of an atom is given by

\(r = \frac{{{a_0}{n^2}}}{Z}\)

Thus, Eq. (1) becomes

\(2\pi {a_0}\frac{{{n^2}}}{Z} = n\lambda\)      ----(2)

\(\therefore {\rm{\;}}2\pi {a_0}\frac{{{n^2}}}{Z} = n\left( {1.5\pi {a_0}} \right)\left[ {{\rm{\;Given}},{\rm{\;}}\lambda = 1.5\pi {a_0}} \right]\)

\(\frac{n}{Z} = \frac{{15\pi {a_0}}}{{2\pi {a_0}}} = \frac{{1.5}}{2} = 0.75\)