Question:
The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is [$a_0$ is Bohr radius]
The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is [$a_0$ is Bohr radius]
Updated On: Jun 14, 2022
- $\frac{h^2}{4 \pi^2 ma_0^2}$
- $\frac{h^2}{16 \pi^2 ma_0^2}$
- $\frac{h^2}{32 \pi^2 ma_0^2}$
- $\frac{h^2}{64 \pi^2 ma_0^2}$
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The Correct Option is C
Solution and Explanation
According to Bohr's model, $\hspace15mm mvr=\frac{nh}{2 \pi} \, \, \, \Rightarrow \, \, (mv)^2=\frac{n^2 h^2}{4 \pi^2 r^2}$
$\hspace15mm KE=\frac{1}{2} mv^2 =\frac{n^2 h^2}{8\pi^2 r^2 m} \hspace25mm ...(i)$
Also, Bohr's radius for H-atom is, r =$n^2 \, a_0$ Substituting 'r' in E (i) gives
$ \, \, \, \, \, \, KE =\frac{h^2}{8 \pi^2 \, n^2 \, a_0^2 m} \, \, when \, n=2, KE=\frac{h^2}{32 \, \pi^2 \, a_0^2 \, m}$
$\hspace15mm KE=\frac{1}{2} mv^2 =\frac{n^2 h^2}{8\pi^2 r^2 m} \hspace25mm ...(i)$
Also, Bohr's radius for H-atom is, r =$n^2 \, a_0$ Substituting 'r' in E (i) gives
$ \, \, \, \, \, \, KE =\frac{h^2}{8 \pi^2 \, n^2 \, a_0^2 m} \, \, when \, n=2, KE=\frac{h^2}{32 \, \pi^2 \, a_0^2 \, m}$
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Concepts Used:
Bohr's Model of Hydrogen Atom
Niels Bohr introduced the atomic Hydrogen model in 1913. He described it as a positively charged nucleus, comprised of protons and neutrons, surrounded by a negatively charged electron cloud. In the model, electrons orbit the nucleus in atomic shells. The atom is held together by electrostatic forces between the positive nucleus and negative surroundings.
Read More: Bohr's Model of Hydrogen Atom
Bohr's Theory of Hydrogen Atom and Hydrogen-like Atoms
A hydrogen-like atom consists of a tiny positively-charged nucleus and an electron revolving around the nucleus in a stable circular orbit.
Bohr's Radius:
If 'e,' 'm,' and 'v' be the charge, mass, and velocity of the electron respectively, 'r' be the radius of the orbit, and Z be the atomic number, the equation for the radii of the permitted orbits is given by r = n2 ｘr1, where 'n' is the principal quantum number, and r1 is the least allowed radius for a hydrogen atom, known as Bohr's radius having a value of 0.53 Å.
Limitations of the Bohr Model
The Bohr Model was an important step in the development of atomic theory. However, it has several limitations.
- Bohr’s model of the atom failed to explain the Zeeman Effect (effect of magnetic field on the spectra of atoms).
- It failed to explain the Stark effect (effect of electric field on the spectra of atoms).
- The spectra obtained from larger atoms weren’t explained.
- It violates the Heisenberg Uncertainty Principle.