At temperature T, the average kinetic energy of any particle is $\frac{3}{2}$ kT. The de Broglie wavelength follows the order :

Question

Observe the following statements
Statement-I: Rutherford model of an atom cannot explain the stability of an atom
Statement-II: The wavelength of X-rays is higher than the wavelength of microwaves
The correct answer is

Answer 1

To solve this problem, we need to analyze the de Broglie wavelength of particles and photons mentioned in the options. The de Broglie wavelength (\lambda) is given by the formula:

\lambda = \frac{h}{p}

where h is Planck's constant and p is the momentum of the particle.

For particles with mass, the momentum p is related to kinetic energy by:

p = \sqrt{2m \cdot KE}

where m is the mass of the particle and KE is its kinetic energy.

According to the problem, the average kinetic energy at temperature T is:

KE = \frac{3}{2}kT

where k is the Boltzmann constant.

First, let's consider photons. A photon of visible light has wavelength \lambda inherent to its nature, which does not depend on temperature or mass. Thus, it retains a constant wavelength that is generally shorter than particles at thermal energy.

Now, let's compare a thermal electron, proton, and neutron. Since they are under thermal condition:

Their kinetic energies are the same at the same temperature, \frac{3}{2}kT.
Protons and neutrons have significantly larger mass than electrons.

According to the de Broglie wavelength formula, a larger mass results in a smaller wavelength given the same momentum. Hence, we can order their de Broglie wavelengths as:

Thermal electron (least mass, largest wavelength)
Thermal neutron/proton (larger mass, smaller wavelength)

Thus, the correct order considering these points is:

Visible photon > Thermal electron > Thermal neutron

This leads to choosing the correct option:

Visible photon > Thermal electron > Thermal neutron

Answer 2