Moseley’s Law: Definition, Statement, Derivation and Applications

Moseley's Law applies to distinctive x-rays released by atoms. Essential attribute of elements, according to Moseley's Law, is their atomic number (rather than their atomic weight). Frequency of an emitted X-ray and atomic number of an element are linked by this law. Henry Moseley evaluated the energy radiated by an electron as it migrates from low-level orbitals using the structure of Bohr's atomic model. 

Key Terms: Moseley’s Law, Atoms, Elements, Henry Moseley, Atomic Number.

Read More: Number of Moles Formula

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What is Moseley’s Law?

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Moseley's law is an empirical law that is applicable to distinctive x-rays released by atoms. Henry Moseley discovered and published the law in 1913-1914. Until Moseley's study, "atomic number" was simply an element's position in the periodic table and had not been linked to any quantifiable physical quantity. 

Atomic number is approximately proportional to the square root of frequency of the emitted x-ray. In the typical X-ray spectrum, frequency of a spectral line varies directly as the square of the atomic number of the element generating it. 

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Moseley Law Statement

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Moseley’s Law Statement asserts that: 

“The square root of the frequency of the x-ray emitted by an atom is proportional to its atomic number.”

According to the law, 

E (kev) = K (Z - 1)²

Where Z= atomic number

K = 1.042 x 10^-2 for K-shell

K = 1.494 x 10^-3 for L-shell

and K = 3.446 x 10^-4 for M-shell

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A simple idea is that the effective charge of the nucleus decreases by 1 when it is being screened by an unpaired electron that persists behind in the K-shell. In any case, Bohr’s formula for Moseley’s K-alpha X-ray transitions is as follows:

E = hv

= E - E = mq4 (Z−1)28h2ε2 112 - 122..... (2)

Now, dividing both the sides by ‘h’ and converting ‘E’ to ‘f’ in equation (2), we get:

f = v = (34) mq4 (Z−1)28h2ε2 = (2.48 ∗ 10Hz (Z - 1).... (3)

Here, one must know that equation (3) is the Moseley equation.

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Moseley’s Law Experiment

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• X-ray spectrometers are the primary building blocks of the X-ray crystallography process.
• Moseley's method of working with X-ray spectrometers is as follows:
• A glass-bulb electron tube was utilized, and electrons were fired at a metallic object, which was a sample of the pure element in his study, within this evacuated tube.
• The ionization of electrons from the element's inner electron shells was generated by the firing of electrons on a metallic object. The rebound of electrons into the inner shell holes resulted in the emission of X-ray photons leaving the tube in a semi-beam through an aperture in the exterior X-ray shielding.
• These radiated X-rays were subsequently diffracted by a standardized salt crystal, with angular results emanating in the form of photographic lines by the exposure of an X-ray film mounted at a given distance outside the vacuum tube.
• Moseley then used Bragg's formula to compute the wavelength of the emitted X-rays after first guessing the mean distances between atoms in the metallic crystal based on its density.

Moseley’s Experiment

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Moseley’s Law Experiment Analysis

To determine the following:

• First, we must confirm Moseley's law with six known element samples. Since the energy is the typical X-ray, which is proportional to (Z - n) 2, and channel number N is directly proportional to E, so N is proportional to (Z - n). As a result, N kZ = bg n.
• Create a graph that plots N vs. Z for the six known examples. This graph shows how to find the best values of k and n. Examine your spectra carefully now, and consider the uncertainties in your data. Create a workable approach for calculating the uncertainty in n and k.
• Determine Z for the unknowns by comparing their peak positions to your results from the six known samples, as well as the uncertainty associated with your findings.

So, by studying an element's X-ray characteristics, we may determine the atomic number of material.

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Moseley’s Law Derivation

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So according to Henry Moseley,

We have ν = a (Z–b)…(1)

Let a transitive state occur from n₁ to n₂ state as per Bohr’s theory. The energy of an emitted photon is

hν = RChz₂ (1n₂₁–1n₂₂)

where,

n₁ = quantum number of final energy level

n₂ = quantum number of initial energy level

ν = frequency for the kα lines

z = Atomic number

h=6.63∗10⁻³⁴Js

R = Rydberg constant

C = Constant

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Applications of Moseley’s Law

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• The law led to the discovery of new elements like Hafnium (72), Technetium (43), Rhenium (75), etc.
• Using the concept of Atomic Number instead of Atomic Mass for arranging elements in the periodic table helped a lot in solving many discrepancies.

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Moseley’s Law Solved Examples

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Example 1. If λCu is the wavelength of kα X-ray line of copper (atomic number 29) and λMo is the wavelength of the kα X-ray line of molybdenum (atomic number 42), the ratio λCu/λMo is close to.

Solution. The wavelength of kα X-ray line is related to atomic number Z by Moseley’s Formula

λ = (Z−1)⁻²

∴ λ_cu/λ_mo = (Z_mo −1/Zcu −1)²

 λ_cu/λ_mo = (42 −1/29 −1)² = 2.144

The elements with higher atomic number (molybdenum in this example) give high energy X-rays (short wavelengths).

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Example 2. Which of the following statements is wrong in the context of X-rays generated from an X-ray tube?

1. The wavelength of characteristic X-rays decreases when the atomic number of the target increases.
2. The cut-off wavelength of the continuous X-rays depends on the atomic number of the target.
3. Intensity of the characteristic X-rays depends on the electrical power given to the X-ray tube.
4. The cut-off wavelength of the continuous X-rays depends on the energy of the electrons in the X-ray tube.

Solution. The frequency of characteristic X-rays is related to atomic number Z by Moseley’s law. The wavelength of emitted X-rays decreases with an increase in Z.

The cut-off wavelength of continuous X-rays corresponds to the maximum energy of electrons in an X-ray tube. It is given by

hc / λ = eV

Where V is the accelerating potential. The intensity of X-rays depends on the number of electrons striking the target per second, which, in turn, depends on the electrical power given to the X-ray tube as the energy of each electron is eV.

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Things to Remember

• Moseley's law is a rule of thumb that applies to the unique x-rays emitted by atoms.
• According to Mosley's law, the atomic number is roughly proportional to the square root of the emitted x-ray frequency.
• According to Moseley's Law, the atomic number of an element is its most important property.
• This law connects the frequency of an emitted X-ray and the atomic number of an element.
• The frequency of a spectral line in a typical X-ray spectrum varies directly as the square of the atomic number of the element that generates it.
• The atomic number is proportional to the square root of the frequency of the x-ray emitted by an atom.
• New elements such as Hafnium (72), Technetium (43), Rhenium (75), and others were discovered as a result of Mosley's law.

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