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 `m v r = (n h)/(2 pi) implies n = 1`
centripetal force, `(m v^(2))/( r) = (Ze^(2))/(4 pi epsilon_(0) r^(2))`
`r = (epsilon_(0) n^(2) h^(2))/(pi m_(pi) e^(2) Z) = ((psilon_(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`
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`