Question

The acceleration of an electron in the first orbit of the hydrogen atom will be
$\begin{aligned} & A.\dfrac{4{{\pi }^{2}}m}{{{h}^{3}}} \\\ & B.\dfrac{{{h}^{2}}}{4{{\pi }^{2}}mr} \\\ & C.\dfrac{{{h}^{2}}}{4{{\pi }^{2}}{{m}^{2}}{{r}^{3}}} \\\ & D.\dfrac{{{m}^{2}}{{h}^{2}}}{4{{\pi }^{2}}{{r}^{2}}} \\\ \end{aligned}$

Answer 1

First of all we should know that the angular momentum of an electron will be always quantised. This can be used to solve the question by substitution and comparing.
The centripetal acceleration is given by the square of the velocity of an electron divided by the radius of orbit.

Complete step-by-step answer:
First of all let us assume that the centripetal force acting on an atom will be equivalent to the electric force due to the charged electrons. In otherwise, we can say that this electric force of attraction is providing the atom the centripetal force.
Therefore, mathematically we can write that,
$\dfrac{m{{v}^{2}}}{r}=\dfrac{{{e}^{2}}}{4\pi {{\varepsilon }_{0}}{{r}^{2}}}$
According to the principle which says that the angular momentum is quantised, we can write that,
$mvr=\dfrac{h}{2\pi }$
Rearranging this equation will give the velocity of electron which can be written as,
$v=\dfrac{h}{2\pi mr}$
Squaring the velocity will give,
${{v}^{2}}={{\left( \dfrac{h}{2\pi mr} \right)}^{2}}$
Expanding the equation will give,
${{v}^{2}}=\left( \dfrac{{{h}^{2}}}{4{{\pi }^{2}}{{m}^{2}}{{r}^{2}}} \right)$
We know that the centripetal acceleration can be given as the formula,
${{a}_{c}}=\dfrac{{{v}^{2}}}{r}$
Therefore, dividing the obtained equation of the square of the velocity by $r$, we will get the centripetal acceleration. We can write it like this,
${{a}_{c}}=\dfrac{{{v}^{2}}}{r}=\left( \dfrac{{{h}^{2}}}{4{{\pi }^{2}}{{m}^{2}}{{r}^{3}}} \right)$

So, the correct answer is “Option C”.

Note: Centripetal acceleration is defined as the characteristics of the motion of a body which is traversing a circular path. The acceleration has been directed radially towards the centre of the circle. Any resulting force causing uniform circular motion is known as a centripetal force. As the charge of the nucleus is positive and the charge of the electron is negative, there will be some kind of force of attraction persisting which will be directed towards the centre. Basically the electrostatic force of attraction is supplying the required centripetal force for revolving of electrons around the nucleus in an atom.