In the Bohr model of a `pi-mesic` atom , a `pi-mesic` of mass `m_(pi)` and of the same charge as the electron is in a circular orbit of ratio of radius `r` about the nucleus with an orbital angular momentum `h//2 pi`. If the radius of a nucleus of atomic number `Z` is given by `R = 1.6 xx 10^(-15) Z^((1)/(3)) m`, then the limit on `Z` for which `(epsilon_(0) h^(2)//pi me^(2) = 0.53 Å and m_(pi)//m_(e) = 264) pi-mesic` atoms might exist is

Options

A. `lt 105`
B. `gt 105`
C. `lt 37`
D. `gt 37`

Answer

C

Explanation

The angular momentum is mvr `=(nh )/(2 pi ) implies n=1`
centripetal force ,` (mv^2 )/(r ) = (Ze^2 )/( 4 pi epsi_0 r^2)`
` r= ( epsi_0 n^2 h^2)/( pi m_(pi) e^2 Z)= ((epsi_0 h^2)/( pi m_e e^2)) ((m_e )/(m_pi))1/Z`
`= (0.53 xx 10^(-10))/( 264 Z ) = ( 200 xx 10^(-15))/(Z) [ :. (m_(pi))/( m_e ) = 264]`
since r cannot be less than nuclear radius
` r gt 1.6 Z ^(1/3) xx 10^(-15)`m
`=200 xx 10^(-15)`
`or ( 200 xx 10^(-15))/(Z ) gt 1.6 xx 10^(-15) Z^(1/3) implies Z lt ((200 )/( 1.6 ))^(3/4) lt 37`