Question

The acceleration of an electron in the first orbit of the hydrogen atom (n=1) is :
$\begin{aligned} & \text{A}\text{. }\dfrac{{{h}^{2}}}{\pi {}^{2}{{m}^{2}}{{r}^{3}}} \\\ & \text{B}\text{. }\dfrac{{{h}^{2}}}{8\pi {}^{2}{{m}^{2}}{{r}^{3}}} \\\ & \text{C}\text{. }\dfrac{{{h}^{2}}}{4\pi {}^{2}{{m}^{2}}{{r}^{3}}} \\\ & \text{D}\text{. }\dfrac{2{{\pi }^{2}}km{{e}^{2}}Z}{\eta h} \\\ \end{aligned}$

Answer 1

At first we need to write the formula given by scientist Bohr and known as Bohr’s quantization formula or principle. Then we have to know the formula for acceleration of the electron in the orbit. As the electron is in the first orbit n is considered as 1. Then replace the equation with the formula for acceleration.

Complete step-by-step answer:
We know, according to Bohr’s quantization principle
$mvr=\dfrac{nh}{2\pi }$,
$2\pi r=n\lambda$,
Where we can see that the angular momentum of an electron in an orbit that is stationary is quantized.
We know that acceleration is= $\dfrac{{{v}^{2}}}{r}$
Therefore we can say,
$mvr=\dfrac{nh}{2\pi }$,
$v=\dfrac{h}{2mr\pi }$, we are considering hydrogen in its first orbit.
So we can say that,
$a={{\left( \dfrac{h}{2\pi mr} \right)}^{2}}.\dfrac{1}{r}$ ,(as acceleration $a=\dfrac{{{v}^{2}}}{r}$)
$a=\dfrac{{{h}^{2}}}{4\pi {}^{2}{{m}^{2}}{{r}^{3}}}$

So, the correct answer is “Option C”.

Additional Information: According to Bohr, an electron can revolve only in certain orbit or path, we know that the non-radiating orbit for which total angular momentum of the revolving electron is an integral multiple of $h2\pi$, where h is the Planck’s constant.
The electron in an orbit can only revolve so when its acceleration increases it is known as angular acceleration.

Note: Students must know the Bohr’s quantization principle properly. In the formula $mvr=\dfrac{nh}{2\pi }$, n is the orbit in which the electron is, h is the Planck’s constant, m is the mass of the electron ‘v’ is the velocity of the electron and ‘r’ is the radius. ‘n’ can also be called a quantum number; it is always from (1, 2, 3, 4, 5…….) .