Question

Your forensic chemistry group, working closely with local law enforcement agencies, has acquired a mass spectrometer similar to that discussed in the text. It employs a uniform magnetic field that has a magnitude of 0.68 T. To calibrate the mass spectrometer, you decide to measure the masses of various carbon isotopes by measuring the position of impact of the various singly ionized carbon ions that have entered the spectrometer with a kinetic energy of 23 keV. A wire chamber with position sensitivity of 0.46 mm is part of the apparatus. What will be the limit on its mass resolution (in kg) for ions in this mass range, that is, those whose mass is on the order of that of a carbon atom?
Answer in kg

Answer 1

given uniform magnetic field B = 0.68 T
kE = 23*10^3 eV
d = 0.46 mm

mass resolution limit = dm
hence

from kE we get
kE = 0.5mv^2
23*10^3*1.6*10^-19 = 0.5*m*v^2
mv^2 = 7360*10^-18

now, for length l of spectrometer
the deflection d is given by
d = R(1 - cos(wt))
now, Rsin(wt) = l
and
for speed v of charged particel of charge q and mass m
qvB = mv^2/R
qB = mv/R

amd v = w*R
hence

d = (mv/qB)(1 - cos(t*qB/m)) = (mv/qB)(1 - sqroot(1 - l^2*q^2B^2/m^2v^2))

hecne mass resolutionis given by
dm = d*qB = 0.46*10^-3*0.68*1.6*10^-19 = 50048*10^-27 kg
mass of one carbon atom = 12*1.6*10^-27 kg
hence
dm = 2606.66 atoms of carbon atom