A particle of mass m moves in circular orbits with potential energy V(r)= Fr, where F is a positive constant and r is its distance from the origin Its energies are calculated using the Bohr model If the radius of the particle's bit is denoted by R and its speed and energy are denoted by v and E, respectively, then for the \(n ^{\text {th }}\) orbit (here h is the Planck's constant)
- \(R \propto n ^{1 / 3}\) and \(v \propto n ^{2 / 3}\)
- \(R \propto n ^{2 / 3}\) and \(v \propto n ^{1 / 3}\)
- \(E=\frac{3}{2}\left(\frac{n^{2} h^{2} F^{2}}{4 \pi^{2} m}\right)^{1 / 3}\)
- \(E=2\left(\frac{n^{2} h^{2} F^{2}}{4 \pi^{2} m}\right)^{1 / 3}\)
The Correct Option is C
Solution and Explanation
The Correct Option is (C): \(E=\frac{3}{2}\left(\frac{n^{2} h^{2} F^{2}}{4 \pi^{2} m}\right)^{1 / 3}\)
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Concepts Used:
Atomic Spectra
The emission spectrum of a chemical element or chemical compound is the spectrum of frequencies of electromagnetic radiation emitted due to an electron making a transition from a high energy state to a lower energy state. The photon energy of the emitted photon is equal to the energy difference between the two states.
Read More: Atomic Spectra
Spectral Series of Hydrogen Atom
Rydberg Formula:
The Rydberg formula is the mathematical formula to compute the wavelength of light.
\[\frac{1}{\lambda} = RZ^2(\frac{1}{n_1^2}-\frac{1}{n_2^2})\]Where,
R is the Rydberg constant (1.09737*107 m-1)
Z is the atomic number
n is the upper energy level
n’ is the lower energy level
λ is the wavelength of light
Spectral series of single-electron atoms like hydrogen have Z = 1.
Uses of Atomic Spectroscopy:
- It is used for identifying the spectral lines of materials used in metallurgy.
- It is used in pharmaceutical industries to find the traces of materials used.
- It can be used to study multidimensional elements.