Problem 15 - A Complete Resource Book in Chemistry for JEE Main

Question

If a source of power \(4 \mathrm{Kw}\) produces \(10^{20}\) photon Per second, the radiation belongs to a part of the spectrum called (a) \(\gamma\)-Rays (b) U.V Rays (c) Microwaves (d) X-Rays

Step-by-Step Solution

Answer

The radiation belongs to part of the spectrum called X-Rays.

Step 1: Calculate Energy per Photon

First, we need to find the energy per photon using the power and the number of photons. The power is given as 4 KW (or 4000 W), and it produces \(10^{20}\) photons per second. Using the equation for power \( P = n \times E\), where \( P \) is power, \( n \) is the number of photons, and \( E \) is the energy per photon, we have: \[ E = \frac{P}{n} = \frac{4000 \text{ W}}{10^{20} \text{ photons}} = 4 \times 10^{-17} \text{ J/photon} \]

Step 2: Convert Energy per Photon to Wavelength

Next, convert the energy of a photon to its corresponding wavelength. We use the energy-wavelength relation, which is: \[ E = \frac{hc}{\lambda} \] Solving for \( \lambda \), the wavelength, gives \[ \lambda = \frac{hc}{E} = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{4 \times 10^{-17}} \] Calculating the above expression, we find \[ \lambda \approx 4.97 \times 10^{-11} \text{ meters} \]

Step 3: Identify the Radiation Type

Finally, identify the part of the electromagnetic spectrum corresponding to the calculated wavelength. The wavelength found (\(4.97 \times 10^{-11} \text{ m}\)) is in the range for X-Rays, which is typically from \(0.01 \) to \(10 \) nanometers (\(10^{-11} \) to \(10^{-9} \) meters). Therefore, the radiation is X-Rays.

Key Concepts

Photon Energy

Understanding photon energy is crucial when discussing electromagnetic radiation. Photons are tiny packets of energy, and the energy of a photon depends on the electromagnetic wave frequency. Photon energy can be calculated using the equation for power expressed as \( P = n \times E \), where:

• \( P \) is the power in watts (W),
• \( n \) is the number of photons,
• \( E \) is the energy per photon in joules (J).

Substituting the given values from the problem, \( P = 4000 \text{ W} \) and \( n = 10^{20} \). By rearranging the formula as \( E = \frac{P}{n} \), we calculate the energy per photon to be \( 4 \times 10^{-17} \text{ J/photon} \). This small amount of energy signifies what just a single photon can carry in a wave of electromagnetic radiation.

Wavelength Calculation

Converting photon energy to wavelength is another fundamental task in understanding the electromagnetic spectrum. This conversion uses the energy-wavelength relationship given by \[ E = \frac{hc}{\lambda} \]where:

• \( E \) is energy per photon,
• \( h \) is Planck's constant \(6.626 \times 10^{-34} \text{ J}\cdot \text{s}\),
• \( c \) is the speed of light in a vacuum \(3 \times 10^8 \text{ m/s} \),
• \( \lambda \) is the wavelength in meters (m).

Solving for wavelength \( \lambda \) involves rearranging the formula to\[ \lambda = \frac{hc}{E} \]Plugging in the values for \( h \), \( c \), and \( E \), we calculate the wavelength \( \lambda \) to be approximately \( 4.97 \times 10^{-11} \text{ meters} \). This wavelength places the radiation within the specific range in the electromagnetic spectrum.

Radiation Types

Different radiation types correspond to different wavelengths within the electromagnetic spectrum, each having unique applications and characteristics. The electromagnetic spectrum is divided into categories:

• **Gamma Rays**: These have the shortest wavelengths (less than \( 10^{-11} \text{ m}\)) and the highest energy, used in medical treatments and astronomy.
• **X-Rays**: Typically range from \( 10^{-11} \) to \( 10^{-9} \text{ m} \). They are used in medical imaging and materials analysis.
• **Ultraviolet (UV) Rays**: Wavelengths between \( 10^{-9} \) and \( 10^{-7} \text{ m} \) are common in sterilization and detecting forged documents.
• **Microwaves**: Range from \( 10^{-3} \) to several centimeters. Used in cooking and some wireless communications.

In our problem, the calculated wavelength \( 4.97 \times 10^{-11} \text{ meters} \) falls within the X-Ray segment of the spectrum. X-Rays are crucial in medical fields and security due to their ability to penetrate various materials.