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) \(E_{1}=6 E_{2}\) (d) \(\mathrm{E}_{1}=\mathrm{E}_{2}\)

Step-by-Step Solution

Answer

(a) \( E_1 = 2E_2 \)

Step 1: Understand the Relationship Between Energy and Wavelength

The energy of radiation is inversely proportional to its wavelength. This relationship is given by the equation \( E = \frac{hc}{\lambda} \), where \(E\) is the energy, \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength of the radiation.

Step 2: Express Energy for Each Wavelength

For the radiation with wavelength \(\lambda_1 = 8000 \ \text{Å} \), the energy is \( E_1 = \frac{hc}{8000} \). Similarly, for wavelength \(\lambda_2 = 16000 \ \text{Å} \), the energy is \( E_2 = \frac{hc}{16000} \).

Step 3: Set Up the Ratio of Energies

We need to find \( \frac{E_1}{E_2} \). Substitute the expressions for \(E_1\) and \(E_2\) from Step 2: \[ \frac{E_1}{E_2} = \frac{\frac{hc}{8000}}{\frac{hc}{16000}} \].

Step 4: Simplify the Ratio

\(h\) and \(c\) appear in both the numerator and the denominator, so they cancel each other out. This simplifies to: \[ \frac{E_1}{E_2} = \frac{16000}{8000} \].

Step 5: Calculate the Ratio

Thus, \( \frac{E_1}{E_2} = \frac{16000}{8000} = 2 \). Therefore, \( E_1 = 2E_2 \).

Key Concepts

Planck's Constant

Planck's constant is one of the most fundamental constants in physics, symbolized by the letter \( h \). Its value is \( 6.62607015 \times 10^{-34} \, \text{Js} \). This constant is crucial in the field of quantum mechanics because it relates the energy of photons to their frequency.

Here's how it works: when you have a light wave or any kind of electromagnetic radiation, the energy of this radiation is determined by the frequency of the wave. Using the formula \( E = h \times f \), where \( E \) is the energy, \( h \) is Planck's constant, and \( f \) is the frequency, we can calculate the exact energy of a photon.

• Why Use Planck's Constant?: Planck's constant helps us understand the particle-like behavior of light and its energy quantization.
• Practical Application: It is crucial in technologies like quantum computing, which rely on the principles of quantum mechanics.

Though it's a tiny value, Planck's constant has huge implications in our understanding of atomic and subatomic particles.

Speed of Light

The speed of light in a vacuum is an essential part of modern physics. Represented by the letter \( c \), the speed of light is approximately \( 3 \times 10^8 \, \text{m/s} \). This speed is a universal constant and plays a vital role in the equations that govern electromagnetic waves.

Light travels incredibly fast, which allows it to be a constant in equations like \( E = \frac{hc}{\lambda} \). This equation illustrates how light speed aids in linking energy and wavelength. The faster light travels, the more energy it can carry, provided the wavelength remains constant.

• Why Does Speed Matter?: The speed of light influences how we measure distances in space and how we perceive time and space through theories like relativity.
• Technological Importance: Understanding and using the speed of light is vital for technologies such as GPS and fiber optic communications.

The constant nature of light speed makes it a perfect reference point for many scientific inquiries and formulas.

Inverse Proportionality of Energy and Wavelength

In physics, inverse proportionality occurs when one value increases as another decreases. In the context of electromagnetic waves, the energy of the radiation is inversely proportional to its wavelength. This means as the wavelength of a wave increases, its energy decreases, and vice versa.

The relationship is mathematically expressed by the equation \( E = \frac{hc}{\lambda} \). Here, \( E \) is the energy, \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength. This equation shows that if the wavelength \( \lambda \) is larger, then the fraction becomes smaller, resulting in lower energy.

• Understanding the Equation: Makes it clear why energy in long wavelengths like radio waves is much less compared to short wavelengths like X-rays.
• Everyday Examples: Think about microwaves in a kitchen: they have longer wavelengths and hence lower energy, making them safe for cooking.

Grasping this inverse relationship is crucial in fields such as optics, astronomy, and various radiative technologies.