Elements that appear in the same column of the periodic table often share simila

Question

Elements that appear in the same column of the periodic table often share similar chemical properties. In the case of the alkaline earth metals, this is troublesome since the body treats calcium (necessary for proper bone growth) and radium (a radioatictive element) as chemically similar, storing both in bone marrow. The radium then bombards nearby bone cells with alpha particles, causing them to "crumble." Radium poisoning investigations often center on the identification of radium and its isotopes in bone samples using a mass spectrometer. Pictured is a schematic of a simplified mass spectrometer, showing the paths of calcium, barium (another alkaline earth metal) and radium isotopes entering the chamber. The region shown is immersed in a constant magnetic field of 0.352 T pointing out of the plane of the schematic. Motion of the positively-charged isotopes toward the right was initiated by a potential difference of 3082 V on the two plates shown. Using the data shown in the table below, calculate the path radius of the Ca ion.

Ca+ Mass = 0.666x10^-25 kg

Ca+ Charge = +1.602x10^-19 C

Explanation / Answer

The calcium ion is accelerated through a potential of 3082 V. The energy acquired by the ion is expressed as

W = charge(Q) * voltage(V)
= 1.60 * 10^-19 C * 3O82 V
= 4.937* 10^-16J

This energy is equivalent to kinetic energy, so

W = 0.5 m v^2,
v = sqrt( 2 W / m)
v = sqrt ( 2 * 4.937 * 10^-16 / (0.666 * 10^-25))
= 121000 ms^-1

In a magnetic field a charged ion will experience a force, as the ion moves in a direction perpendicular to the field the force will always be perpendicular to the ions direction of motion. The magnitude of the force will not change, and so the ion will be made to move in a circular path.

The equation for circular motion is:
F = mv^2 / r

The force experienced by the ion in the magnetic field is:
F = BQv

Equating the two equations
BQv = mv^2 / r

Rearranging
r = mv / (BQ)

r = 0.666 * 10^-25 kg * 121000 ms^-1 / (0.352 T * 1.60 * 10^-19 C)
= 14.2 cm

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