Animated Solution for Physics - Dual Nature of Matter and Radiation: Four physical quantities are listed in Column I. Their values are listed in Column II in a random order

In this fascinating problem, we are tasked with matching four distinct physical phenomena with their typical energy scales. This exercise is a beautiful journey through the vastness of physics, taking us from the macroscopic thermal motion of the air we breathe to the intense, microscopic forces binding the atomic nucleus.

Let's break down each phenomenon and estimate its energy scale.

Thermal Energy

Imagine the air molecules in your room right now. They are constantly in motion, bouncing off walls and each other. This chaotic motion is what we perceive as temperature. The average thermal energy of a particle at a temperature $T$ is on the order of $kT$, where $k$ is the Boltzmann constant ($8.62\times 10^{-5}\text{ eV/K}$).

At a typical room temperature of about $300\text{ K}$, we can estimate this energy:

Looking at our options, this perfectly aligns with $0.02\text{ eV}$.

Binding Energy

Now, let's zoom into the nucleus of an atom. The protons and neutrons are held together by the strong nuclear force, the most powerful force in the universe. The binding energy per nucleon is the average energy required to remove a single nucleon from the nucleus.

For heavy nuclei like Uranium (${}^{238}\text{U}$), this value is incredibly high. While the maximum binding energy per nucleon is about $8.8\text{ MeV}$ for Iron (${}^{56}\text{Fe}$), it gradually drops to about $7.6\text{ MeV}$ for heavier elements.

This is millions of times stronger than the chemical bonds holding molecules together! Thus, the binding energy of heavy nuclei per nucleon matches with $7\text{ MeV}$.

X-Rays

X-rays are highly energetic electromagnetic waves, capable of passing through human tissue to reveal the bones beneath. Their wavelengths typically range from $0.1\text{ A˚}$ to $100\text{ A˚}$.

To quickly convert wavelength to energy in electron-volts, we use the incredibly handy formula:

If we take a typical X-ray wavelength of $1\text{ A˚}$, the energy is:

This falls perfectly into the $10\text{ keV}$ range provided in the options.

Visible Light

Finally, let's consider the light our eyes can detect. Visible light photons have much longer wavelengths than X-rays, ranging from about $4000\text{ A˚}$ (violet) to $7000\text{ A˚}$ (red).

Using our trusty formula again, let's plug in a mid-range wavelength, say $6200\text{ A˚}$ (orange-red light):

This perfectly matches our final remaining option, $2\text{ eV}$.

By understanding the fundamental formulas and typical scales of these phenomena, we can confidently match them: - Thermal energy $\to$ $0.02\text{ eV}$ - Binding energy $\to$ $7\text{ MeV}$ - X-ray photon $\to$ $10\text{ keV}$ - Visible light $\to$ $2\text{ eV}$