A particle known as mu meson has a charge equal to that of no electron and mass `208`times the mass of the electron B moves in a circular orbit around a nucleus of charge `+3e` Take the mass of the nucleus to be infinite Assuming that the bohr's model is applicable to this system (a)drive an expression for the radius of the nth Bohr orbit (b) find the value of a for which the radius of the orbit it appropriately the same as that at the first bohr for a hydrogen atom (c) find the wavelength of the radiation emitted when the u - mean jump from the orbit to the first orbit

Explanation

a. we have redius of nth orbit of a hydrogen atom as ` r_(n) = (n^(2) h^(2))/(4 pi^(2) K ze^(2)m)`
If electron is replaced by a heavy partical of mass `208` times that of the electron , then the radius is given as
`r_(n) = (n^(2) h^(2))/(4 pi^(2) K (3) e^(2) (208m)) = (n^(2) h^(2))/(2496 pi^(2) K ze^(2)m)` ...(i)
Hence we have not used redused mass because it is given that mass of nucleus is assumed to be infinate and here we take `z = 3`
b. The radius of first Bohr orbit is given as
`r_(1) = ( h^(2))/(4pi^(2) K e^(2)m)`
From Eq. (i), we have
` = (n^(2) h^(2))/(2469 pi^(2) K e^(2) m) = ( h^(2))/(4 pi^(2) K ze^(2)m)`
or `n^(2) = 624`
or `n = 25`
c. Rydberg contant for hydrogen-like atom is
`R = (2 pi^(2) K ^(2) e^(4) m)/(ch^(3))`
or `= 208 R = 208 xx 1967800 m^(-1)`
Now , the wevelength `lambda` of the radition emitted when `mu`- meson makes a transition from `n_(2) = 3 to n = 1` is
`(1)/(lambda) = R'Z^(2) [(1)/(n_(1)^(2)) - (1)/(n_(2)^(2))]`
or `(1)/(lambda) = R' xx 9 [(1)/(1)^(2) - (1)/(3^(2))]` or `(1)/(lambda) = 8 R'`
or `lambda = (1)/(8 R')`
`lambda = (1)/(8 xx 208 xx 10967800) m^(-1)`
or `lambda = 0.548 Å`