Question

Chlorine has two stable isotopes, ³⁵Cl and ³⁷Cl. Chlorine gas which consists of singly ionized ions is to be separated into its isotopic components using a mass spectrometer. The magnetic field strength in the spectrometer is 1.2 T. What is the minimum value of the potential difference through which these ions must be accelerated so that the separation between them, after they complete their semicircular path, is 1.5 cm?

Answer 1

let potential difference be V volts.

let speed after acceleration be v m/s.

then balancing electrical energy with kinetic energy:

e*V=0.5*m*v^2

where m=mass of the ion

==>v=sqrt(2*e*V/m)

let radius of the path is r.

balancing centripetal force with magnetic force:

q*v*B=m*v^2/r

where B=applied magnetic field

==>r=m*v/(q*B)=m*sqrt(2*e*V/m)/(q*B)

=sqrt(2*e*m*V)/(q*B)

separation between the two ions =difference between the diameters

=2*(r2-r1)

=2*((sqrt(2*e*m2*V)/(q*B))-(sqrt(2*e*m1*V)/(q*B)))

=(2*sqrt(2*e*V)/(q*B))*(sqrt(m2)-sqrt(m1))

where m1=mass of 35Cl isotope=35*1.66*10^(-27) kg

m2=mass of 37Cl isotope=37*1.66*10^(-27) kg

minimum separation required is 1.5 cm

=>1.5*0.01=(2*sqrt(2*e*V)/(q*B))*(sqrt(m2)-sqrt(m1))

==>sqrt(2*e*V)=1.5*0.01*q*B/(2*((sqrt(m2))-(sqrt(m1))))=2.1204*10^(-7)

==>2*1.6*10^(-19)*V=4.4961*10^(-14)

==>V=1.4050*10^5 volts