Question

Calculate the wavelength of electron in an orbit of $B{e}^{3+}$ ion having radius equal to the Bohr's radius ($a_0$). Also report the accelerating potential that must be imparted to an electron originally at rest to give an effective wavelength as calculated above.

• A. $9.66\times {10}^{-10}m$ and 27.2 eV
• B. $7.66\times {10}^{-12}m$ and 54.4 eV
• C. $1.66\times {10}^{-10}m$ and 54.4 eV
• D. Cannot be calculated

Answer 1

The correct option is C $1.66\times {10}^{-10}m$ and 54.4 eV

Radius in $n^{th}$ Bohr's orbit is given by
$r_n=\frac{n^2}{Z}{a}_0$
For $B{e}^{3+}$ ion Z=4
Thus from question $a_0=\frac{n^2}{4}{a}_0\therefore n=2$
$E_n=\frac{Z^2}{n^2}\times 13.6eV=\frac{4^2}{2^2}\times 13.6=-54.4eV$
$K.E=-E_n=54.4eV$
de Broglie wavelength is given by $\lambda =\frac{h}{\sqrt{2mK.E}}=\frac{6.625\times {10}^{-34}}{\sqrt{2\times 9.11\times {10}^{-31}\times 54.4\times 1.6\times {10}^{-19}}}=1.66\times {10}^{-10}m$
The K.E is 54.4 eV and hence accelerating potential must be 54.4 eV. Therefore, wavelength of electron in an orbit of $B{e}^{3+}$ is $1.66\times {10}^{-10}m$