Two Questions (MCQ) on X-rays

(1) The atomic numbers of the target materials in two X-ray tubes are in the ratio 1:3. The X-ray photon energies of the K_α lines of these materials are in the ratio

(a) 1:3 (b) 3:1 (c) 1:√3 (d) 1:9 (e) 1:1

According to Mosley’s law, the frequency ‘ν’ of a particular characteristic X-ray (such as K_α) is directly proportional to the square of the atomic number of the target. Since the photon energy is hν, it follows that the energy is directly proportional to the square of the atomic number. Therefore, the energies are in the ratio 1:9.

(2) The minimum wave length of X-rays produced by an X-ray tube operating at an anode voltage of 24.8 kV is very nearly

(a) 1.8 Ǻ (b) 1.5 Ǻ (c) 1 Ǻ (d) 0.8 Ǻ (e) 0.5 Ǻ

The minimum wave length X-ray photon will have the entire energy of the impinging electron, which is 24.8 keV.

Since the product of the photon energy in eV and the wave length in Angstrom is 12400, the minimum wave length is 12400 ÷ 24800 = 05 Ǻ.

[You may use the equation hc/λ = 24800 eV= 24800×1.6×10^–19 joule to calculate λ in metre after substituting for Planck’s constant ‘h’ and the speed of light ‘c’, but it will be time consuming].

Two IIT-JEE 2007 Questions on X-rays

The following multiple choice question appeared in IIT-JEE 2007 question paper:

Electrons with de Broglie wave length λ fall on the target in an X-ray tube. The cut off wave length of the emitted X-rays is

(a) λ₀ = 2mcλ²/h (b) λ₀ = 2h/mc (c) λ₀ = 2m²c² λ³/h² (d) λ₀ = λ

X-rays of cut off (minimum) wave length for a given target voltage are obtained when the entire kinetic energy of the incident electron is converted into X-ray photon energy. The kinetic energy of the electron is p²/2m where ‘p’ is its momentum and ‘m’ is its mass.

But p = h/λ where ‘h’ is Planck’s constant [de Broglie relation] so that we have

h²/2mλ² = hc/λ₀

[The term on the right hand side is the energy of the X-ray photon of cut off wave length λ₀].

From the above, λ₀ = 2mcλ²/h

Here is an assertion-reason type multiple choice question which appeared in IIT-JEE 2007 question paper:

STATEMENT-1

If the accelerating potential in an X-ray tube is increased, the wave lengths of the characteristic X-rays do not change

because

STATEMENT-2

When an electron beam strikes the target in an x-ray tube, part of the kinetic energy is converted in to X-ray energy.

(a) Statement-1 is True, Statement-2 is True; Statement-2 is a correct

explanation for Statement-1

(b) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct

explanation for Statement-1

(c) Statement-1 is True, Statement-2 is False

(d) Statement-1 is False, Statement-2 is True

Wave lengths of characteristic X-rays are characteristic of the target material (since they depend only on the energy levels of the atoms in the target material). Therefore, their wave lengths do not change when the accelerating voltage is increased. Statement-1 is therefore true.

Statement-2 is true. But, it is not a correct explanation for statement-1. Characteristic X-rays originate because of electron transitions between discrete energy levels of the atoms in the target. Statement-2 is only a general statement regarding the process of X-ray production.

Questions on Bohr Model of Hydrogen Atom

Questions based on the Bohr model of hydrogen atom are inevitable in any Medical and Engineering test paper. Consider the following two questions which appeared in A.I.I.M.S.- 2005 test paper:
(1) Solid targets of different elements are bombarded by highly energetic electron beams. The frequency (f) of the characteristic X-rays emitted from different targets varies with atomic number Z as
(a) f α √Z (b) f α Z² (c) f α Z (d) f α Z^3/2
As you might have noted, X-rays are produced by electron transitions from outer orbits to inner orbits. But atoms of high atomic number are required for the production of X-rays since the X-ray photon has much greater energy compared to light photon. The energy of the electron in an orbit is directly proportional to Z². [Note that in a hydrogen like atom, the energy is -13.6 Z²/n² electron volt]. The energy difference between levels also is directly proportional to Z². Since the energy difference is equal to hν where ‘ν’ is the frequency of the radiation emitted, the correct option is (b).
[More rigorous treatment shows that ν α (Z-b)² where b is a constant for a given spectral series].
(2) The ground state energy of hydrogen atom is -13.6 eV. What is the potential energy of the atom in this state?
(a) 0 eV (b) -27.2 eV (c) 1eV (d) 2 eV
The correct option is (b) since the potential energy is twice the kinetic energy. You should note that in all cases of central field motion under inverse square law attractive force, the total energy and potential energy are negative and the potential energy is twice the total energy.
Suppose we modify this question as follows:
The ground state energy of hydrogen atom is -13.6 eV. What is the kinetic energy of the electron in this state?
(a) -13.6 eV (b) -27.2 eV (c) 0 eV (d) 13.6 eV

The correct option is (d) since the kinetic energy is always positive and its magnitude is the same as that of the total energy. This is true in all cases of central field motion under an attractive inverse square law force, as in the case of the motion of a satellite around a planet.

You will find more multiple choice questions (with solution) at physicsplus: Questions on Bohr Atom Model