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A μ-meson particle of charge equal to that of an electron, e and mass 208 times the mass of the electron moves in a circular orbit around the nucleus of charge +4e. Assuming the Bohr’s model of the atom to be applicable to this system, the inverse of wavelength of photons emitted when an electron jumps in the 4th orbit of the atom from infinity in terms of Rydberg constant is given by αRH. Find the value of α/26.

Answer
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Hint: Use the mass-energy relation to determine the energy difference between the two orbits for meson. Since the meson has 208 times the mass of the electron, the energy difference will also be 208 times that of the electron. Use a formula for wavenumber from Bohr’s model to determine wave number for meson.

Formula used:
1λ=RHZ2(1n121n22)
Here, RH is Rydberg’s constant and Z is atomic number.

Complete step by step answer:When an electron jumps from higher orbit n2 to lower orbit n1, the difference in the energy of the orbit is given by Bohr’s atomic model as,
 ΔE=RHZ2(1n121n22) …… (1)

Here, RH is Rydberg’s constant and Z is atomic number.

We know that the difference in the energy is given as,
ΔE=hcλ

Therefore, the energy difference is inversely proportional to the wavelength of the electron. Therefore, the equation (1) is written as,
1λ=RHZ2(1n121n22)

The term 1λ is known as wave number.

According to the mass-energy relation, the difference in the energy is,
ΔE=mc2

Here, m is the mass of the particle and c is the speed of light.

We have given that the mass is 208 times the mass of the electron. Therefore, the energy difference will also be 208 times that of the electron. Therefore, we can write,
 1λ=208RHZ2(1n121n22)

Substitute 4 for Z, 4 for n1 and for n2 in the above equation.
 1λ=208RH(4)2(14212)
1λ=208RH

We have given that the wave-number for electrons is αRH. Since the given meson also follows Bohr’s model, we can write,
 αRH=208RH
α=208

Therefore,
α26=20826=8

Therefore, the value of α26 is 8.

Note:Rydberg constant RH consists of all the constants including mass of the electron m, charge e, speed of light c and Planck’s constant h. therefore, do not consider any other constants other than Z in the above formula. The mass-energy relation implies the energy possessed by the rest particle.
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