Question

If the radius of the $3^{rd}$ Bohr orbit of hydrogen is r, then the wavelength of an electron in the $6^{th}$ orbit of a hydrogen atom is equal to

• A. $6\pi r$
• B. $2\pi r$
• C. $\frac{4\pi r}{3}$
• D. $12\pi r$

Answer 1

The correct option is C $\frac{4\pi r}{3}$

According to de-Broglie, circumference of the circular orbit must be an integral number of wavelength.
Thus we can write,
$2\pi r_n=n{\lambda }_n$
$\Rightarrow {\lambda }_6=\frac{2\pi r_6}{6}$
$=\frac{2\pi \times 36r_0}{6}$ ( $r_n=\frac{n^2{a}_0}{z}$)
$=\frac{2\pi \times (4\times 9)r_0}{(2\times 3)}$
$=\frac{4\pi \times 3^2{r}_0}{3}$
$=\frac{4\pi r}{3}$ ($r_3=9r$)