Atomic Mass of Rare Carbon Isotope: Solving the Puzzle

[phy112]

Homework Statement

Natural carbon consists of two different isotopes (excluding 14C, which is present in only trace amounts). The isotopes have different masses, which is due to different numbers of neutrons in the nucleus; however, the number of protons in the same, and subsequently the chemical properties are the same. The most abundant isotope has an atomic mass of 12.0000 u. When natural carbon is placed in a mass spectrometer, two lines are formed on the photographic plate. The lines show that the more abundant isotope moved in a circle of radius 6.00 cm, while the rarer isotope moved in a circle of radius 6.24 cm. What is the atomic mass of the rarer isotope? (The ions are accelerated through the same potential difference before entering the magnetic field.)

Homework Equations

The Attempt at a Solution

would you just do a ratio of some sort i.e 6/6.24 =12/x??

 

[Gib Z]

Well, yes that ratio ends up giving the correct answer but the key to why is finding the expression that gives the radius. We equate $$F= qvB$$ to the centripetal force $$F = \frac{mv^2}{r}$$, and solve for r, giving $$r = \frac{mv}{qB}$$.

We can deduce their velocities and charges are the same from the given information, and may assume B is constant as well. Hence, r is directly proportional to m, enabling you to use that ratio formula.

 

[vrobins1]

Well, yes that ratio ends up giving the correct answer but the key to why is finding the expression that gives the radius. We equate $$F= qvB$$ to the centripetal force $$F = \frac{mv^2}{r}$$, and solve for r, giving $$r = \frac{mv}{qB}$$.

We can deduce their velocities and charges are the same from the given information, and may assume B is constant as well. Hence, r is directly proportional to m, enabling you to use that ratio formula.

This logically all makes sense to me, but I am still confused as to how to get the final answer. Does that mean that r = m?

 

[Gib Z]

It doesn't mean r = m, we have a formula relating those two and there's more things ! It means r is *proportional* to m, meaning if there is an increase or decrease in r by a certain factor, the same will happen for m.