One method for determining the amount of corn in early Native American diets is

Question

One method for determining the amount of corn in early Native American diets is the stable isotope ratio analysis (SIRA) technique. As corn photosynthesizes, it concentrates the isotope carbon-13, whereas most other plants concentrate carbon-12. Overreliance on corn consumption can then be correlated with certain diseases, because corn lacks the essential amino acid lysine. Archaeologists use the mass spectrometer to separate the 12C and 13C isotopes in samples of human remains. Suppose you use a velocity selector to obtain singly ionized (missing one electron) atoms of speed 8.95 km/s and want to bend them within a uniform magnetic field in a semicircle of diameter 28.0 cm for the 12C. The measured masses of these isotopes are 1.99×1026 kg (12C) and 2.16×1026 kg (13C).

What strength of magnetic field is required?

What is the diameter of the 13C semicircle?

What is the separation of the 12C and 13C ions at the detector at the end of the semicircle? Express your answer using two significant figures.

Is the distance found in part C large enough to be easily observed?

Explanation / Answer

let,

speed v=8.95 km/sec

mass of the 12C isotope is m1=1.99*10^-26 kg

mass of the 13C isotope is m2=2.16*10^-26 kg

for the 12C isotope diameter of the semi circle d1=28cm

and radius r1=d1/2=28/2 ====> r1=14cm

charge of electron q=1.6*10^-19 C

a)

let

magnetic field is B

q*V*B=m*v^2/r

====>

q*B=m1*v/r1

1.6*10^-19*B=(1.99*10^-26*8.95*10^3)/(14*10^-2)

==> B=7.95*10^-3 T

magnetic field B=7.95mT

b)

from,

q*v*B=m*v^2/r

q*B=m*v/r

====>

r1/r2=m1/m2 ( here, r2=d2/2 is radius of for 13C isotope)

===>

(14*10^-2/r2)=(1.99*10^-26)/(2.16*10^-26)

===> r2=0.152 m

r2=15.2 cm

diameter d2=2*r2=

d2=2*15.2

d2=30.4 cm

c)

separation of the semicircle is dy=d2-d1

dy=30.4-28

dy=2.4 cm

this separation dy=2.4cm can be easily observed

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