Question

The velocity of electrons in the fourth Bohr’s orbit of hydrogen is $u$. The velocity of the electron in the first orbit would be:
A.$4u$
B.$16u$
C.$\dfrac{u}{4}$
D.$\dfrac{u}{16}$

Answer 1

The Bohr’s model of hydrogen atom describes that the electron of the hydrogen moves in circular path orbit around the nucleus. In order to find the velocity of the electron in the fourth orbit of hydrogen,
We know that, angular momentum of electron,
$\Rightarrow mvr=\dfrac{nh}{2\pi }$
Now, the $r$ is the radius of the orbit and is equal to $\left( \dfrac{{{\varepsilon }_{\circ }}{{n}^{2}}{{h}^{2}}}{\pi m{{e}^{2}}} \right)$
Thus,
$\begin{aligned} & \Rightarrow mv\left( \dfrac{{{\varepsilon }_{\circ }}{{n}^{2}}{{h}^{2}}}{\pi m{{e}^{2}}} \right)=\dfrac{nh}{2\pi } \\\ & \Rightarrow v=\dfrac{nh}{2\pi }\dfrac{\pi {{e}^{2}}}{{{\varepsilon }_{\circ }}{{n}^{2}}{{h}^{2}}} \\\ & \Rightarrow v=\dfrac{{{e}^{2}}}{2{{\varepsilon }_{\circ }}nh}\text{ m}/\sec \end{aligned}$
Here,
$e=$ Charge of electron which has a constant value of $-1.6\times {{10}^{-19}}\text{C}$
${{\varepsilon }_{\circ }}=$ Constant of proportionality
$n=$ Number of hydrogen atom
$h=$ Plank’s constant

Formula Used:
The velocity of electron in nth orbit of an atom is,
$\Rightarrow v=\dfrac{z{{e}^{2}}}{2{{\varepsilon }_{\circ }}nh}\text{ m}/\sec$
Where, $z$ is the number of the orbit.

Complete Step by step solution:
Given that, $z=4$ and the velocity of the electron is $u$.
Thus, the equation becomes,
$\Rightarrow u=\dfrac{4{{e}^{2}}}{2{{\varepsilon }_{\circ }}nh}\text{ }$
Here, $n=1\text{ }$ as a hydrogen atom has only one electron.
$\Rightarrow u=\dfrac{2{{e}^{2}}}{{{\varepsilon }_{\circ }}h}$
Now, for the first orbit $z=1$, therefore the velocity of the electron is
$\Rightarrow v=\dfrac{{{e}^{2}}}{2{{\varepsilon }_{\circ }}h}$
Now, comparing the value of $u$ and $v$, we have
$\Rightarrow v = 4u$
Thus, the velocity of the electron in the first orbit is $4u$.

So, the correct answer is option A.

Note: Bohr’s model states that an atom consists of a small nucleus which is positively charged and is surrounded by the electrons which are negatively charged. These electrons revolve around the nucleus in a circular path known as orbit or shells. Each orbit and shell of an atom has a fixed amount of energy.