Question

Mark the correct options.
(a) An atom with a vacancy has smaller energy that a neutral atom.
(b) K X-ray is emitted when a hole makes a jump from the K shell to some other shell.
(c) The wavelength of K X-ray is smaller than the wavelength of L X-ray of the same material.
(d) The wavelength of K_α X-ray is smaller than the wavelength of K_β X-ray of the same material.

Answer 1

(b) K X-ray is emitted when a hole makes a jump from the K shell to some other shell.
(c) The wavelength of K X-ray is smaller than the wavelength of L X-ray of the same material.
 
Energy of a vacant atom is higher than that of a neutral atom.
Hence, option (a) is incorrect.

K X-ray is emitted when an electron makes a jump to the K shell from some other shell. As a result, a positive charge hole is created in the outer shell. As the electron continuously moves to the K shell, the hole moves from the K shell to some other shell. Hence, option (b) is correct.

K X-ray is emitted due to the transition of an electron from the L or M shell to the K shell and L X-ray is emitted due to the transition of an electron from the M or N shell to the L shell. The energy involved in the transition from the L or M shell to the K shell is higher than the energy involved in the transition from the M or N shell to the L shell. Since the energy is inversely proportional to the wavelength, the wavelength of the K X-ray is smaller than the wavelength of the L X-ray of the same material. Hence, option (c) is correct.

If E_K, E_L and E_M are the energies of K, L and M shells, respectively, then the wavelength of K_α X-ray (`lambda_1`) is given by

`lambda_1 = (hc)/(E_K - E_L)`

Here,
h = Planck's constant
c = Speed of light
Wavelength of the K_β X-ray (`lambda_2`) is given by 

`lambda_2 = (hc)/(E_K - E_M)`

As the difference of energies (`E_K - E_M`) is more than `(E_K - E_L)`, `λ_2` is less than `λ_1` Hence, option (d) is not correct.