Concept explainers
To explain:
The size of Bohr orbits for a “muonic” hydrogen atom is much smaller than the corresponding Bohr orbits of an ordinary hydrogen atom.
To compare:
The energy levels of muonic hydrogen atom with the ordinary hydrogen atom.
To find:
The difference in
Answer to Problem 2C
Bohr orbits for a “muonic” hydrogen atom is much smaller than the corresponding Bohr orbits of an ordinary hydrogen atom.
For muonic hydrogen atom the energy levels are given by
The emission energy of the muonic hydrogen atom will increase by a factor of 200 as compared to the ordinary hydrogen atom
Explanation of Solution
Given:
The mass of muon is 200 times larger than the electron.
Formula used:
The radius of the Bohr orbit is given by
Calculation:
According to de Broglie’s wave hypothesis the wavelength of the orbiting electron can be given as
where p is the momentum of the electron.
Circumference of the orbit is given by
Substituting equation (1.1) in equation (1.2) we can write
So, from equation (1.3) we see that radius of the Bohr orbit is inversely proportional to the mass of the electron.
Since, the mass of the muon (
So, using equation (1.3) and (1.4) we can see that for muonic hydrogen atom the radius of Bohr orbit is smaller compared to ordinary hydrogen atom.
For muonic hydrogen atom the energy levels are given by
According to Bohr orbital theory the energy of an electron in nth orbit is given by
Where
So for muonic hydrogen atom as
For ordinary Hydrogen atom the emission energy is given by
where
Now for muonic hydrogen atom the emission energy can be given by
Conclusion:
Bohr orbits for a ‘muonic’ hydrogen atom is much smaller than the corresponding Bohr orbits of an ordinary hydrogen atom
For muonic hydrogen atom the energy levels are given by
The emission energy of the muonic hydrogen atom will increase by a factor of 200 as compared to the ordinary hydrogen atom.
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