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) \(\mathrm{E}_{1}=\mathrm{E}_{2}\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Relationship Between Energy and Wavelength

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

Step 2: Calculate the Energy for Each Wavelength

Let's calculate the energy for \( \lambda_1 = 8000 \, \text{Å} \) and \( \lambda_2 = 16000 \, \text{Å} \). We have:\[ E_1 = \frac{hc}{8000 \, \text{Å}} \] and \[ E_2 = \frac{hc}{16000 \, \text{Å}} \].

Step 3: Compare the Energies

By finding the ratio \( \frac{E_1}{E_2} \), we get:\[ \frac{E_1}{E_2} = \frac{\frac{hc}{8000}}{\frac{hc}{16000}} = \frac{16000}{8000} = 2 \]Thus, \( E_1 = 2E_2 \).

Step 4: Match the Result with Provided Options

From the ratio \( E_1 = 2E_2 \), this matches option (a), \( E_1 = 2E_2 \).

Key Concepts

Planck's Constant

Planck's constant, denoted as \( h \), is a fundamental physical constant that plays a crucial role in quantum mechanics. It helps to describe the quantization of energy levels. In simple terms, this constant is used to determine how much energy is carried by a single particle, such as a photon, of electromagnetic radiation. Planck's constant has a value of approximately \( 6.626 \times 10^{-34} \, \text{Js} \), which is incredibly small. This smallness reflects the tiny energy increments involved in interactions at the atomic and subatomic scale.

Understanding Planck's constant is essential because it relates directly to the frequency and energy of electromagnetic waves. The fundamental equation \( E = hf \) tells us that energy \( E \) is a product of Planck’s constant \( h \) and the frequency \( f \) of the radiation. This means that higher frequency radiation, such as ultraviolet light, carries more energy than lower frequency radiation, like radio waves. Ultimately, Planck's constant allows us to delve into the world of photons and energy changes at the microscopic level.

Speed of Light

The speed of light, denoted by \( c \), is a universal constant that represents the speed at which all massless particles, such as photons, travel in a vacuum. It is approximately \( 3.00 \times 10^8 \, \text{m/s} \).

This constant is vital in the world of physics because it not only governs how fast light can move but also serves as an ultimate speed limit for information and matter across the universe. In our exercise, the speed of light ties directly into the energy equation \( E = \frac{hc}{\lambda} \). Here, \( c \) is multiplied by Planck’s constant and divided by the wavelength \( \lambda \).

By playing its role in the formula, the speed of light helps equate the energy associated with different wavelengths of light. Knowing this speed allows scientists to determine the behavior and relationship of particles traveling through space, reinforcing the importance of \( c \) in connecting energy and wavelength together.

Inverse Proportionality of Wavelength and Energy

In the realm of physics, wavelength \( \lambda \) and energy \( E \) have an inverse relationship, which means that as one increases, the other decreases. This is a central point in understanding electromagnetic radiation. The inverse proportionality is mathematically expressed by the equation \( E = \frac{hc}{\lambda} \).

This relationship indicates that shorter wavelengths are associated with higher energy levels, while longer wavelengths correspond to lower energy levels. For instance, gamma rays have very short wavelengths and thus possess high energies, whereas radio waves have long wavelengths and low energies.

• This concept is critical in fields such as spectroscopy, where it helps in identifying substances by their spectral lines.
• It also underpins technologies like lasers and medical imaging tools (e.g., X-rays), where manipulating wavelength and energy is fundamental.

By comprehending this inverse proportionality, one can predict how different waves behave and interact with matter based on their wavelength and energy, leading to practical applications in science and technology.