A mass spectrometer is used to separate uranium ions of mass 3.92x10^-25 kg and

Question

A mass spectrometer is used to separate uranium ions of mass 3.92x10^-25 kg and charge of 3.2x10^-19 C from related species. The ions are accelerated through a potential difference of 100 kV and then pass into a uniform magnetic fields where they bent in a path of radius 1.00 m. After traveling through 180 degrees and passing through a slit of width 1.0 mm and height 1.0 cm, they are collected in a cup.

A) What is the magnitude of (the perpendicular) magnetic field in the separator?

B) If the machine is used to separate out 100 mg of material per hour, calculate the current of the desired ions in the machine

C) If the machine is used to separate out 100 mg of material per hour, calculate the thermal energy in the cup 1.00 h.

Explanation / Answer

A particle with charge z, mass m, and speed v moving perpendicular to a magnetic field B will orbit in a circular path of radius r given by:

r = m*v/(B*z)

B = m*v/(r*z)

A charged particle accelerated through a voltage V has a kinetic energy equal to:

(1/2)*m*v^2 = z*V

so the speed of the accelerated particle is:

v = sqrt(2*z*V/m)

Plug this into the expression above for the magnetic field to get:

B = sqrt(2*m*V/(z*r^2))

In this case,

V = 10^5 volts

m = (235 gm/mol)/(6.022*10^23 atoms/mol) = 3.902*10^25 kg

z = 2*1.602*10^-19 C = 3.204*10^-19 C

r = 1 m

Plugging all this into the equation for B, we find that:

B = 0.494 T. This is the answer to part (a).

For part (b),

we need to calculate the number of ions reaching the detector per second:

100mg/hr = 0.1 gm/hr

(0.1gm/hr)/(235 gm/mol) = 4.255*10 ^-4 mol/hr

(4.255*10^-4 mol/hr)*6.022*10^23 ions/mol = 2.563*10^20 ions/hr

(2.563*10^20 ions/hr) * (1/60^2 hr/sec) * (2 elementary charges/ion) * (1.602*10^-19 C/elem charge) = 2.281*10^-2 A

For part (c),

we'll ignore any resistive heating of the Faraday cup, and assume that all the heating is caused by the deposition of the kinetic energy of the incoming ions.

The kinetic energy of each ion is given by:

z*V = (2*1.602*10^-19 C/ion)*10^5 V = 3.204*10^-14 J/ion

We already calculated that there are

2.563*10^20 ions/hr = 7.118*10^16 ions/sec reaching the collector, so the total energy being dissipated (as heat) in the collector per unit time is:

7.118*10^16 ions/sec * 3.204*10^-14 J/ion = 2.281*10^3 W

So,

Energy = Pt

= 2.281*10^3 x (1*60*60s)   (converted 1h to sec)

= 8.2*10^6 J

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