Moseley's Law

The fundamental principle of atomic physics, discovered by Henry Moseley in 1913, is referred to as Moseley’s Law where characteristic X-ray spectra emitted by elements are concerned. The law states that if an element is producing a certain kind of X-rays then the frequency with which these X-rays appear will be proportional to that rough number square. This meant that they had to re-evaluate their periodic table organization around how heavy these atoms were but could now depend on their occurrences in nature as well for factual data regarding their makeup. In this article, we will discuss the concept of Moseley's Law. It sets up the base for understanding the atomic structure, X-ray spectroscopy and the general behaviour of elements.

Moseley's Law

During the time when the periodic table is arranged with atomic weight, Moseley measured the frequency of characteristic X-rays from a large number of elements and plotted the square root of the frequency against its position number in the periodic table. He discovered that the plot is very close to a straight line. A portion of Moseley's plot is shown in the figure where $\sqrt{\nu}$ of $\mathrm{K}_0$ X-rays is plotted against the position number. From this linear relation, Moseley concluded that there must be a fundamental property of the atom which increases by regular steps as one moves from one element to the other. This quantity was later identified to be the number of protons in the nucleus and was referred to as the atomic number.

The frequency $\nu$ of a characteristic X-ray of an element is related to its atomic number Z by

$\sqrt{\nu}=a(Z-b)$

where a and b are constants called proportionality and screening (or shielding) constants. For the K series, $a=\sqrt{\frac{3 R c}{4}}$ and that of b is 1. Here R is Rydberg's constant and c is the speed of light (as in Bohr's model).

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Solved Examples Based on Moseley's Law

Example 1: If the $K_\alpha$ radiation of M₀ (z=42) has a wavelength of 0.71A⁰. Find the wavelength of the corresponding radiation of Cu (z=29)

1) $1 A^{\circ}$
2) $2 A^{\circ}$
3) $1.52 A^{\circ}$
4) $1.25 \mathrm{~A}^{\circ}$

Solution:

Moseley's law

$\sqrt{\nu}=a(z-b)$

wherein

$a=\sqrt{\frac{3 R C}{4}}$

b=1 for

$\begin{aligned} & K_\alpha \text { lines } \\ & \sqrt{\nu}=a(z-b) \\ & \text { or } \quad \sqrt{\frac{c}{\lambda}}=a(z-b) \\ & b \simeq 1 \\ & \Rightarrow \sqrt{\frac{c}{\lambda_0}}=a(42-1) \\ & \Rightarrow \sqrt{\frac{c}{\lambda_{c u}}}=a(29-1) \\ & \frac{\lambda_{C u}}{\lambda_{M o}}=\left(\frac{41}{28}\right)^2 \Rightarrow \lambda_{C u}=\left(\frac{41}{28}\right)^2 \times 0.71 A^{\circ} \\ & \lambda_{C u}=1.52 A^{\circ}\end{aligned}$

Hence, the answer is the option (3)

Example 2: According to Moseley's law, the ratio of slopes of the graph between the square root of frequency and mass number for $K_\beta$ and $K_\alpha$ is

1) 1.089

2) 2.234

3) 0.132

4) 2.549

Solution:

Moseley's law for frequency of $X$-ray is given by

$
\sqrt{V}=\sqrt{C R\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)}(Z-b)
$

for $K$-series $b=1$. for $K$ a line $\mathrm{n}_1=1$ and $\mathrm{n}_2=2$; for $\mathrm{K}_\beta$ line $\mathrm{n}_1=1$ and $\mathrm{n}_2=3$ For Ka line the slope is

$
\sqrt{C R \frac{3}{4}}
$

and for $\mathrm{K} \beta$ line the slope is

$
\begin{aligned}
& \sqrt{C R \frac{8}{9}} \\
& \sqrt{\frac{8}{9} \times \frac{4}{3}}=\sqrt{\frac{32}{27}}=1.089
\end{aligned}
$

Hence, the answer is (1.089).

Example 3: If $\lambda_{\mathrm{Cu}}$ is the wavelength of $\mathrm{K}_\alpha$ the X-ray line of copper (atomic number 29) and $\lambda_{\mathrm{Mo}}$ is the wavelength of the $\mathrm{K}_\alpha$ X-ray line of molybdenum (atomic number 42 ), then the ratio $\lambda_{\mathrm{Cu}} / \lambda_{\mathrm{Mo}}$ is close to:

1) 1.99

2) 2.14

3) 0.50

4) 0.48

Solution:

$\frac{\lambda_{\mathrm{Cu}}}{\lambda_{\mathrm{Mo}}}=\left(\frac{\mathrm{Z}_{\mathrm{Mo}}-1}{\mathrm{Z}_{\mathrm{Cu}}-1}\right)^2=\frac{41 \times 41}{28 \times 28}=\frac{1681}{784}=2.144$

Hence, the answer is the option (2).

Summary

Moseley's Law, discovered in 1913, establishes a relationship between the frequency of characteristic X-rays emitted by elements and their atomic numbers. This law revealed that the atomic number, rather than atomic weight, determines an element's position in the periodic table. It provides a foundational understanding of atomic structure and X-ray spectroscopy, with practical applications in determining element identities through X-ray emissions.