In the Bohr atomic model, if the electron is replaced by a negative muon (μ ) which has samecharge as electron but mass ≈200 times that of electron, then, find – (Assuming no relative motionbetween the negative muon and nucleus)a. Relation between eV and μV (V = velocity of the species)b. Relation between radius of nth orbit of electron and negative muon
Question
In the Bohr atomic model, if the electron is replaced by a negative muon (μ ) which has samecharge as electron but mass ≈200 times that of electron, then, find – (Assuming no relative motionbetween the negative muon and nucleus)a. Relation between eV and μV (V = velocity of the species)b. Relation between radius of nth orbit of electron and negative muon
Solution
a. Relation between eV and μV (V = velocity of the species)
In the Bohr model, the kinetic energy of an electron in an atom is given by KE = 1/2 mv^2, where m is the mass of the electron and v is its velocity. The kinetic energy is also equal to the charge of the electron times the voltage, or KE = eV, where e is the charge of the electron and V is the voltage.
If we replace the electron with a muon, the mass changes but the charge remains the same. Therefore, the kinetic energy of the muon is KE = 1/2 m'v'^2 = eV', where m' is the mass of the muon, v' is its velocity, and V' is the voltage.
Since the charge e is the same for both the electron and the muon, we can equate the two expressions for the kinetic energy to find the relation between eV and μV:
1/2 mv^2 = 1/2 m'v'^2 => mv^2 = m'v'^2 => v^2/v'^2 = m'/m => V/V' = sqrt(m'/m) => V/V' = sqrt(200) => V = sqrt(200) * V'
So, the velocity of the electron is sqrt(200) times the velocity of the muon.
b. Relation between radius of nth orbit of electron and negative muon
In the Bohr model, the radius of the nth orbit is given by r = n^2h^2/4π^2mke^2, where h is Planck's constant, k is Coulomb's constant, and e is the charge of the electron.
If we replace the electron with a muon, the mass changes but the charge remains the same. Therefore, the radius of the nth orbit for the muon is r' = n^2h^2/4π^2m'ke^2.
We can find the relation between the radii of the nth orbits of the electron and the muon by dividing the two expressions:
r/r' = m'/m => r = m'/m * r' => r = 200 * r'
So, the radius of the nth orbit of the electron is 200 times the radius of the nth orbit of the muon.
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