Question

The Kα X-rays of aluminium (Z = 13) and zinc (Z = 30) have wavelengths 887 pm and 146 pm respectively. Use Moseley's law √v = a(Z − b) to find the wavelengths of the Kα X-ray of iron (Z = 26).

Answer 1

The wavelengths of the Kα X-ray of iron is 198 pm

Given:

• Kα X-rays of aluminium’s wavelength, λ1 = 887 pm

• X-rays of aluminium frequency is shown as

• νa = cλνa = $3 \times 108887 \times 10-12 \mathrm{va}=33.82 \times 1016 \mathrm{Hz}$

• Kα X-rays of zinc’s wavelength, λ2 = 146 pm

• X-rays of zinc’s frequency is shown as

• $\mathrm{v} z=3 \times 108146 \times 10-12 \mathrm{vz}=2.055 \times 1018 \mathrm{Hz}$

• It is known that ν= a(Z − b)

Aluminium,

• $5.815 \times 108=a(13-b)$... (1)

Zinc,

• $1.4331 \times 109=a(30-b)$   …(2)

• When (2) is divided by (1),

• $13-b-b 30=5.815 \times 10-11.4331=0.4057$

• $\Rightarrow 12.171-0.4057 \mathrm{b}+\mathrm{b}=13 \mathrm{b}=0.51323 \times 108=5 \times 107$

For Fe, Frequency ν’ is shown as  

• $v^{\prime}=5 \times 107(26-1.39)$

• $=123.05 \times 107$

• ν’ = cλ

where,  

• c = light’s speed

• λ = light’s wavelength  

• Therefore, $c \lambda=5141.3 \times 1014$

• $\Rightarrow \lambda=198 \times 10-12=198 \mathrm{pm}$