Question

An electron in hydrogen atom stays in 2nd orbit for ${{10}^{-8}}s$ . How many revolutions will it undergo till it jumps to the ground state?

Answer 1

Hint: Here the electron is in second orbit so that it can make possible transition to first orbit only. Then we can find out the frequency by using the equation,\[-13.6eV\]
$E=h\nu $
Where $h$ be the Planck’s constant and $E$ be the energy of the state of the hydrogen atom. These all will help you to solve this problem.

Complete answer:
As we can see the electron is currently in second orbit from which the transition take place to the first orbit of the hydrogen electron,
Therefore as per the equation of energy is concerned, we can write that,
$\Delta E=h\nu $
Rearranging this equation will give,
$\nu =\dfrac{\Delta E}{h}$
As the transition is from second orbit to the first one,
$\nu =\dfrac{{{E}_{2}}-{{E}_{1}}}{h}$
As we know the energy of the second orbit of the hydrogen atom is given as,
${{E}_{2}}=-3.4eV$
And the energy of the first orbit will be,
${{E}_{1}}=-13.6eV$
And the Planck’s constant is given as,
$h=6.63\times {{10}^{-34}}{{m}^{2}}kg{{s}^{-1}}$
Substituting these all in the equation,
$\nu =\dfrac{\left( -3.4 \right)-\left( -13.6 \right)}{6.63\times {{10}^{-34}}}$
As the energies are given in $eV$, we have to convert it into joules, for that multiply them with the value of charge of an electron in the equation, therefore we can write that,
\[\nu =\dfrac{\left( -3.4 \right)-\left( -13.6 \right)\times 1.6\times {{10}^{-19}}}{6.63\times {{10}^{-34}}}\]
Simplifying these will give,
\[\nu =2.46\times {{10}^{15}}Hz\]
Therefore we got the answer for one electron. This means that the electrons complete \[2.46\times {{10}^{15}}\] revolutions in a second. This is in the case of one electron. If there is \[{{10}^{-8}}\] electrons, then we can write that,
\[2.46\times {{10}^{15}}\times {{10}^{-8}}\] revolutions have been completed.
Simplifying this will give,
Total number of revolutions is \[2.46\times {{10}^{7}}\] in number. Therefore the correct answer is obtained.

Note:
The ground state defines the least possible energy that an atom can possess. Atoms may be in different energy states. This is the energy state that is considered to be normal for the atom. The value of energy of this state is \[-13.6eV\].