Question

A technique called mass spectrometry is used to segregate different isotopes of an element. In such a setup, hydrogen gas is sent which is ionized by the ion source and we get nuclei of protium ${(}_1^{1}H)$, deuterium ${(}_1^{2}H)$, and tritium
${(}_1^{3}H)$.
A perpendicular magnetic field is used to deviate these in separate beams. Pick the correct option.

• A. a - protium; b - deuterium; c - tritium
• B. a - protium; b - tritium; c - deuterium
• C. a - tritium; b - deuterium; c - protium
• D. a - deuterium; b - protium; c - tritium.

Answer 1

The correct option is C a - tritium; b - deuterium; c - protium

The three isotopes, once ionized carry an equal amount of charge each: +1e. The magnetic
field should result in an equal force on each of the charges, equal to $\overrightarrow{F}=e(\overrightarrow{v}\times \overrightarrow{B}),where\overrightarrow{v}$s the velocity of the nuclei when they enter the magnetic field
$\overrightarrow{B},and\overrightarrow{B}\perp \overrightarrow{v}$. This force will cause a centripetal acceleration $\overrightarrow{a}$ (because it's always perpendicular to $\overrightarrow{v})$, which we can write using Newton's second law, as -
$\overrightarrow{F}=m\overrightarrow{a}=m\frac{v^2}{r}\uparrow =e(\overrightarrow{v}\times \overrightarrow{B})$,
Which in scalar form looks like -
$F=m\frac{v^2}{r}=evBsin{90}^{\circ }=evB$
$\Rightarrow \frac{1}{r}=\frac{eB}{mv}\Rightarrow r=\frac{mv}{eB}.$ (1)

Interesting right? Since v, e, and B are constants in this situation, equation (1) means higher the mass, smaller the radius of the circular path that a charged particle takes once it enters a magnetic field perpendicularly. Among the three isotopes $-{(}_1^1)H,{(}_1^2)H,and{(}_1^3)H-tritium,{(}_1^3)H,$ is the heaviest - why?
Isotopes are different elements with same number of protons but different number of neutrons - same Z but different A. The protium nucleus has just one proton ${(}_1^{1}H),deuterium$ nucleus has one proton and one neutron ${(}_1^{2}H),tritium$ nucleus has one proton plus two neutrons ${(}_1^{3}H)$. Comparing masses, naturally, $m_{tritium}>m_{deuterium}{m}_{protium}$.
So when the three different kinds of nuclei enter the perpendicular magnetic field, their radii of curvature will $follow{r}_{tritium}>r_{deuterium}{r}_{protium}$, since by equation(1), the radius of curvature depends just on the mass. Therefore, a - tritium, b - deuterium, and c - protium.