Question

The amount of meat in prehistoric diets can be determined by measuring the ratio of the isotopes nitrogen-15 to nitrogen-14 in bone from human remains. Carnivores concentrate 15 N + 15N+ , so this ratio tells archaeologists how much meat was consumed by ancient people. Suppose you use a velocity selector to obtain singly-ionized (missing one electron) isotopes of speed 8.50 km/s km/s . The spectrometer is calibrated by where 12 C + 12C+ arrive after traveling around a semicircle. Suppose the 12 C + 12C+ ions travel along a semicircle with diameter 30.0 cm cm . Use the actual masses of these ions in your calculations: 1.99×10−26 kg kg for 12 C + 12C+ 2.32×10−26 kg kg for 14 N + 14N+ 2.49×10−26 kg kg for 15 N +

Find the separation of the 14 N + 14N+ and 15 N + 15N+ isotopes at the detector in centimeters.

Answer 1

Step 1: Find value of uniform magnetic field, using force balance on the 12C atoms:

Force on Carbon-12 ions will be:

Force due to circular motion = Force due to magnetic field

Fc = Fm

m1*V^2/R1 = q*V*B

B = m1*V/(q*R1)

m1 = mass of Carbon-12 = 1.99*10^-26 kg

V = Speed of ions = 8.50 km/sec = 8500 m/s

q = charge on simgly-ionized atoms = +e = 1.6*10^-19 C

R1 = radius of circular path of Carbon-12 = diameter/2 = 30/2 = 15 cm = 0.15 m

So,

B = 1.99*10^-26*8500/(1.6*10^-19*0.15)

B = 7.048*10^-3 T = Uniform magnetic field

Step 2: Find out the radius of path followed by Nitrogen-14:

Since we know that:

m2*V^2/R2 = q*V*B

R2 = m2*V/(q*B)

m2 = mass of Nitrogen-15 = 2.32*10^-26 kg

R2 = 2.32*10^-26*8500/(1.6*10^-19*7.048*10^-3)

R2 = 0.1749 m = 17.49 cm

Step 3: Find out the radius of path followed by Nitrogen-15:

Since we know that:

m2*V^2/R2 = q*V*B

R2 = m2*V/(q*B)

m2 = mass of Nitrogen-15 = 2.49*10^-26 kg

R2 = 2.49*10^-26*8500/(1.6*10^-19*7.048*10^-3)

R2 = 0.1877 m = 18.77 cm

Step 4: diameter of semicircle path of Nitrogen-14 will be:

d1 = 17.49*2 = 34.98 cm

diameter of semicircle path of Nitrogen-15 is:

d2 = 18.77*2 = 37.54 cm

So separation between 14N and 15N isotopes will be:

d = d2 - d1 = 37.54 - 34.98

d = 2.56 cm

Let me know if you've any query.