Question

The angular momentum of an electron in a Bohr orbit of H - atom is $\text{4}\text{.2178 }\times \text{ 1}{{\text{0}}^{-34}}\text{ kg}{{\text{m}}^{2}}/\sec $. Calculate the wavelength of the spectral line emitted when an electron falls from this level to the next lower level.

Answer 1

Hint: Bohr gave the expression for the angular momentum of an electron i.e.\[\text{mvr = }\dfrac{\text{nh}}{2\pi }\]. Here, m is the mass of an electron, v is the velocity, r is the radius of the orbit whereas n is the orbit in which the electron is present and h is the Planck's constant.

Complete step by step answer:
-It is given in the question that the angular momentum of an electron is $\text{4}\text{.2178 }\times \text{ 1}{{\text{0}}^{-34}}\text{ kg}{{\text{m}}^{2}}/\sec $ and the value of h is fixed and constant for every element i.e. $\text{6}\text{.626 }\times \text{ 1}{{\text{0}}^{-34}}$ and also the value of pi is 3.14 for every equation.
-Now, we have to find the value of n that is the orbit in which the electron is present.
-Now we know that the angular momentum is the product of the distance of the object from the rotational axis and the linear momentum of the electron.
-So, the formula of angular momentum becomes mvr.

-Now, we know that \[\text{mvr = }\dfrac{\text{nh}}{2\pi }\] so, the value of n will be:
\[4.2178\text{ }\times \text{ 1}{{\text{0}}^{-34}}\text{ = n }\times \text{ }\dfrac{6.626\text{ }\times \text{ 1}{{\text{0}}^{-34}}}{2\text{ }\times \text{ 3}\text{.14}}\]
$\text{n = 4}$

-Now, we have to calculate the wavelength of the spectral line when it travels from the 4 energy level to the 3 energy level.
-So, by using the Rydberg formula:
$\dfrac{1}{\lambda }\text{ = }{{\text{R}}_{\text{H}}}\left( \dfrac{1}{\text{n}_{1}^{2}}\text{ - }\dfrac{1}{\text{n}_{2}^{2}} \right)$
\[\dfrac{1}{\lambda }\text{ = 109678}\left( \dfrac{1}{{{3}^{2}}}\text{ - }\frac{1}{{{4}^{2}}} \right)\]
\[\lambda \text{ = 1}\text{.8 }\times \text{ 1}{{\text{0}}^{-4}}\text{cm}\]
Therefore, the wavelength of the spectral line is $\text{1}\text{.8 }\times \text{ 1}{{\text{0}}^{-4}}\text{cm}$.

Note: The Rydberg formula is the transition of the different energies that occur between the energy level. The electron emits a photon when they move from a high energy level to the low energy level. There are many series in the spectral line such as Lyman series, Balmer series, Paschen series, etc