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 a mass spectrometer to separate the 12 C and 13 C isotopes in samples of human remains. Suppose you use a velocity selector to obtain singly ionized (missing one electron) atoms of speed 8.50 km/s, and you want to bend them within a uniform magnetic field in a semicircle of diameter 25.0 cm for the 12 C. The measured masses of these isotopes are 1.99×10−26kg(12C) and 2.16×10−26kg(13C).
(a) What strength of magnetic field is required?
(b) What is the diameter of the 13 C semicircle?
(c) What is the separation of the 12 C and 13 C ions at the detector at the end of the semicircle? Is this distance large enough to be easily observed?

Answer 1

Answer:

$0.0084575\ \text{T}$

$0.272\ \text{m}$

2.2 cm easily observable

Explanation:

$m_1$ = Mass of 12 C = $1.99\times 10^{-26}\ \text{kg}$

$m_2$ = Mass of 13 C = $2.16\times 10^{-26}\ \text{kg}$

$r_1$ = Radius of 12 C = $\dfrac{25}{2}=12.5\ \text{cm}$

B = Magnetic field

v = Velocity of atom = 8.5 km/s

$r_2$ = Radius of 13 C

The force balance of the system is

$qvB=\dfrac{m_1v^2}{r}\\\Rightarrow B=\dfrac{m_1v}{rq}\\\Rightarrow B=\dfrac{1.99\times 10^{-26}\times 8500}{12.5\times 10^{-2}\times 1.6\times 10^{-19}}\\\Rightarrow B=0.0084575\ \text{T}$

The required magnetic field is $0.0084575\ \text{T}$

Radius is given by

$r=\dfrac{mv}{qB}$

$r\propto m$

So

$\dfrac{r_2}{r_1}=\dfrac{m_2}{m_1}\\\Rightarrow r_2=\dfrac{m_2}{m_1}r_1\\\Rightarrow r_2=\dfrac{2.16\times 10^{-26}}{1.99\times 10^{-26}}\times 12.5\times 10^{-2}\\\Rightarrow r_2=0.136\ \text{m}$

The required diameter is $2\times 0.136=0.272\ \text{m}$

Separation is given by

$2(r_2-r_1)=2(0.136-0.125)=0.022\ \text{m}$

The distance of separation is 2.2 cm which is easily observable.