A mass spectrometer is being used to separate common oxygen-16 from the much rarer oxygen-18, taken from a sample of old glacial ice. (The relative abundance of these oxygen isotopes is related to climatic temperature at the time the ice was deposited.) The ratio of the masses of these two ions is 16 to 18, the mass of oxygen-16 is 2.66 ✕ 10-26 kg, and they are singly charged and travel at 3.70 ✕ 106 m/s in a 2.00 T magnetic field. What is the separation (in m) between their paths when they hit a target after traversing a semicircle?

Given:

m16=2.66 x 10^-26 kg

v=3.7 x 10^6 m/s

B = 2.0

These are singly charged hence

q=1.6 x 10^-19 C

The ratio of these two masses is 16 to 18.

Let the mass of m18 be x

m18/m16 = x / (2.66 x 10^-26)

18/16 = x / (2.66 x 10^-26)

x = [18 x (2.66 x 10^-26) ]/16

=2.99 X 10^-26 kg

When they hit a target after traversing a semicircle the distance between them is the difference of their diameter. (∆d)

∆d=2r18-2r16

Where r18 is the radius of the m18 mass

r16 is the radius of the m16 mass.

∆d=2 (r18-r16).

The radius of an object transversing in a magnetic field is given by the below formula.

r = mv/qB

m is the mass of the object.

v is the velocity of the object

q is the charge carried by the object

B magnetic field

∆d=2 (r18-r16). Substituting the value for r from the above formula.

∆d=2[ (m18-m16) v]/qB

m18-m16=

(2.99-2.66) x10^-26=0.33x 10^-26

qB=1.6x10^-19 x2=3.2 x10^-19

Substituting these values in the ∆d formula we get

∆d=

2x3.7x10^6x0.33x10^-26/3.2x10^-19

=2.442 x 10^-20/3.2x 10^-19

=0.76 x10-1m