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

`0.0084575\ \text{T}`

`0.272\ \text{m}`

2.2 cm easily observable

Step-by-step explanation:

`m_1` = Mass of 12 C =
`1.99* 10^(-26)\ \text{kg}`

`m_2` = Mass of 13 C =
`2.16* 10^(-26)\ \text{kg}`

`r_1` = Radius of 12 C =
`(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=(m_1v^2)/(r)\\\Rightarrow B=(m_1v)/(rq)\\\Rightarrow B=(1.99* 10^(-26)* 8500)/(12.5* 10^(-2)* 1.6* 10^(-19))\\\Rightarrow B=0.0084575\ \text{T}`

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

Radius is given by

`r=(mv)/(qB)`

`r\propto m`

So

`(r_2)/(r_1)=(m_2)/(m_1)\\\Rightarrow r_2=(m_2)/(m_1)r_1\\\Rightarrow r_2=(2.16* 10^(-26))/(1.99* 10^(-26))* 12.5* 10^(-2)\\\Rightarrow r_2=0.136\ \text{m}`

The required diameter is
`2* 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.