## Related questions with answers

Conditions in the ion source of a mass spectrometer can be adjusted to allow formation of multiply ionized atoms. (a) Rewrite the above equation for the case of an $n$ times ionized atom, and solve for $R$. (b) Taking the masses of ${ }^4 \mathrm{He},{ }^{12} \mathrm{C}$, and ${ }^{16} \mathrm{O}$ to be $4 \mathrm{u}, 12 \mathrm{u}$, and $16 \mathrm{u}$, respectively, write down expressions for the radii, $R$, for $\mathrm{He}^{+}, \mathrm{C}^{3+}$, and $\mathrm{O}^{4+}$. (c) Using the actual masses of the atoms, find the fractional differences in these three radii (He versus $\mathrm{C}$ and $\mathrm{C}$ versus $\mathrm{O}$ ). The tiny differences in these radii can be measured very accurately and allow precise comparisons of the three masses involved (one of which, ${ }^{12} \mathrm{C}$, defines the atomic mass scale).

Solution

VerifiedWhen a potential difference $V_0$ accelerates an ion with mass $m$, the ion follows a circular path with radius $R$ when moving against a constant magnetic field $B$.

$m=\dfrac{eB^2R^2}{2V_0}\tag{16.44}$

In the case of a multiply ionized atom, this formula will be different.

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