Question

A beam of singly ionized atoms of carbon (each charge +e) all have the same speed enters a mass spectrometer, as shown in Fig. The ions strike the photographic plate in two different locations 5 cm apart. The ${}^{12}{C}_6$ isotope traces a path of smaller radius, 15.0 cm. What is the atomic mass number of other isotope?

Answer 1

Consider mass of proton =mass of neutron=m

Then mass of ${}^{12}{\text{C}}_6=12m$

and mass of isotope X=M

radius of path by ${}^{12}{\text{C}}_6=15cm=0.15m$

Using formula

$m=\frac{qBr}{\vartheta }$

for ${}^{12}{\text{C}}_6$

$12m=\frac{eB\ast 0.15}{\vartheta }\to \left(1\right)$

for X

as difference of diameter of two path=5cm

$\therefore$ radius of isotope X $=(15+\frac{5}{2})$

$=\frac{35}{2}\ast {10}^{-2}m$

$\therefore M=\frac{eB\ast \frac{35}{2}\ast {10}^{-2}}{\vartheta }\to \left(2\right)$

Dividing eqn (1) with eqn (2)

$\frac{12m}{M}=\frac{0.15}{\frac{35}{2}\ast {10}^{-2}}$

now mass of X must be integral multiple of m, say= $\lambda$m

then $\frac{12m}{\lambda m}=\frac{15}{100}\ast \frac{100\ast 2}{35}$

$\frac{12}{\lambda }=\frac{30}{35}$

$\lambda =\frac{12\ast 35}{30}=14$

Hence mass no. of isotopes $=14$