Problem 62 - A Complete Resource Book in Chemistry for JEE Main

Question

The relationship between energy \(\mathrm{E}\), of the radiation with a wavelength \(8000 \AA\) and the energy of the ra diation with a wavelength \(16000 \AA\) is (a) \(\mathrm{E}_{1}=2 \mathrm{E}_{2}\) (b) \(\mathrm{E}_{1}=4 \mathrm{E}_{2}\) (c) \(\mathrm{E}_{1}=6 \mathrm{E}_{2}\) (d) \(E_{1}=E_{2}\)

Step-by-Step Solution

Answer

The correct answer is (a) \( E_1 = 2E_2 \).

Step 1: Understanding the Energy-Wavelength Relationship

The energy of a photon is related to its wavelength by the equation \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength of the photon.

Step 2: Calculate Energy for Wavelength 8000 Ångströms

For radiation with a wavelength of 8000 \( \text{Å} \), the energy \( E_1 \) is given by: \[ E_1 = \frac{hc}{8000 \ Å} \] We can rewrite it using the proportionality relationship:\[ E_1 \propto \frac{1}{8000} \]

Step 3: Calculate Energy for Wavelength 16000 Ångströms

For radiation with a wavelength of 16000 \( \text{Å} \), the energy \( E_2 \) is given by: \[ E_2 = \frac{hc}{16000 \ Å} \] We can rewrite it using the proportionality relationship:\[ E_2 \propto \frac{1}{16000} \]

Step 4: Find the Relationship between \( E_1 \) and \( E_2 \)

Using the proportional relationship of energy to the inverse of wavelength, compare \( E_1 \) and \( E_2 \):\[ \frac{E_1}{E_2} = \frac{\frac{1}{8000}}{\frac{1}{16000}} \]Simplify this:\[ \frac{E_1}{E_2} = \frac{16000}{8000} = 2 \]Thus, \( E_1 = 2E_2 \).

Key Concepts

Photon Energy Calculation

When it comes to understanding the energy of a photon, it's all about the connection between three important values: energy, wavelength, and certain constants. A photon's energy can be calculated using the formula:

• Energy, E = \( \frac{hc}{\lambda} \)

Here, \( h \) refers to Planck's constant and \( c \) is the speed of light, while \( \lambda \) is the wavelength of the photon. This formula shows us how energy and wavelength are interconnected.
The nature of this relationship is inversely proportional, meaning as the wavelength increases, the energy decreases and vice versa. This fundamental concept is crucial for comprehending how different wavelengths affect the energy levels of photons encountered in various types of electromagnetic radiation.

Planck's Constant

Planck's constant is an essential value in the world of physics, and it plays a significant role in our calculation of photon energy. Represented as \( h \), its value is approximately \( 6.626 \times 10^{-34} \) J·s.
Planck's constant connects the energy of photons to their frequency, reinforcing the quantum nature of light.

• The constant signifies that energy is quantized, meaning it's divided into discrete 'packets' or photons.
• This supports the particle-like behavior of light, similar to microscopic particles like electrons.

This concept was trailblazing when it was first introduced by Max Planck in 1900 and paved the way for the development of quantum mechanics, reshaping our understanding of atomic and subatomic processes.

Speed of Light

The speed of light is another constant crucial to the equation governing photon energy. Denoted by \( c \), its precisely measured value is approximately \( 3.00 \times 10^8 \) meters per second.
This speed is the limit at which all light waves travel in a vacuum and is an integral factor in associating energy with wavelength in the formula \( E = \frac{hc}{\lambda} \).

• This constant ensures that the relationship between wavelength and frequency is steady and predictable.
• The speed of light's constancy across the vacuum makes it a reliable building block for equations that describe electromagnetic phenomena.

By integrating this constant, we can accurately predict the behavior of light moving through different mediums and transitions in energy levels.

Wavelength Inversion Proportionality

The concept of wavelength inversion proportionality is pivotal to understanding how energy and wavelength are related. The equation \( E = \frac{hc}{\lambda} \) underscores that energy varies inversely with wavelength.
The shorter the wavelength, the higher the energy of the photon.

• For an 8000 Å photon, the energy is calculated as \( E = \frac{hc}{8000 \text{ Å}} \), whereas for a 16000 Å photon, \( E = \frac{hc}{16000 \text{ Å}} \).
• This means that if the wavelength doubles, as in going from 8000 Å to 16000 Å, the energy halves.

Practically, this means that shorter wavelengths like ultraviolet carry more energy than longer wavelengths like infrared.
This inversion relationship is a key reason why higher-energy phenomena (like X-rays) can be more penetrating compared to longer-wavelength forms of light.