Question

in a mass spectrometer, germanium atoms have radii of curvature equal to 21.0, 21.6, 21.9, 22.2, and 22.8 cm. the largest radius corresponds to an atomic mass of 76 u. part a what are the atomic masses of the other isotopes? express your answers using two significant figures separated by commas. m21.0, m21.6, m21.9, m22.2 =

Explanation:

Step1: Establish the proportional - relationship

In a mass spectrometer, the radius of curvature $r$ of a charged particle is proportional to the square - root of its mass $m$, i.e., $r\propto\sqrt{m}$. So, $\frac{r_1}{r_2}=\sqrt{\frac{m_1}{m_2}}$, or $m_1 = m_2(\frac{r_1}{r_2})^2$.
Let $r_1$ be the radius of the isotope with known mass $m_2 = 76\ u$ and $r_2$ be the radius of the other isotopes.

Step2: Calculate $m_{21.0}$

$m_{21.0}=76(\frac{21.0}{22.8})^2$
$m_{21.0}=76\times(\frac{21.0}{22.8})^2=76\times\frac{441}{519.84}\approx64\ u$

Step3: Calculate $m_{21.6}$

$m_{21.6}=76(\frac{21.6}{22.8})^2$
$m_{21.6}=76\times\frac{466.56}{519.84}\approx68\ u$

Step4: Calculate $m_{21.9}$

$m_{21.9}=76(\frac{21.9}{22.8})^2$
$m_{21.9}=76\times\frac{479.61}{519.84}\approx70\ u$

Step5: Calculate $m_{22.2}$

$m_{22.2}=76(\frac{22.2}{22.8})^2$
$m_{22.2}=76\times\frac{492.84}{519.84}\approx72\ u$

Answer:

64, 68, 70, 72