Problem 69 - A Complete Resource Book in Chemistry for JEE Main

Question

The energy ratio of a photon of wavelength \(3000 \AA\) and \(6000 \AA\) is (a) \(1: 1\) (b) \(2: 1\) (c) \(1: 2\) (d) \(1: 4\)

Step-by-Step Solution

Answer

The energy ratio is (b) \(2: 1\).

Step 1: Understand the Formula

The energy of a photon is given by 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 of the photon.

Step 2: Write the Energy Expressions

For the given problem, we have two wavelengths: \(3000 \ \text{Å}\) and \(6000 \ \text{Å}\). Convert these to meters: \( \lambda_1 = 3000 \times 10^{-10} \ \text{m} \) and \( \lambda_2 = 6000 \times 10^{-10} \ \text{m} \). Therefore, the energies are \( E_1 = \frac{hc}{\lambda_1} \) and \( E_2 = \frac{hc}{\lambda_2} \).

Step 3: Calculate the Energy Ratios

We can find the ratio of the energies by dividing the energy expressions: \( \frac{E_1}{E_2} = \frac{hc/\lambda_1}{hc/\lambda_2} = \frac{\lambda_2}{\lambda_1} \). Therefore, \( \frac{E_1}{E_2} = \frac{6000 \times 10^{-10}}{3000 \times 10^{-10}} = \frac{6000}{3000} = 2 \).

Step 4: Interpret the Result

From the calculation, the energy ratio \( \frac{E_1}{E_2} = 2 \) means that the energy of the photon with wavelength \(3000 \ \text{Å}\) is twice that of the photon with wavelength \(6000 \ \text{Å}\). Hence, the energy ratio is \(2:1\).

Key Concepts

Wavelength

Wavelength is a fundamental concept in understanding photon energy. It is essentially the distance between two identical points in a wave—like the crest of one wave to the crest of the next. Wavelength is denoted by the Greek letter \( \lambda \). The units commonly used for wavelength are meters, but in many scientific contexts, especially involving photons, Angstroms (\( \text{Å} \)) are used. One might ask, why does wavelength matter? In the equation for photon energy \( E = \frac{hc}{\lambda} \), we see wavelength inversely related to energy. This means:

• Shorter wavelengths correspond to higher energy photons.
• Longer wavelengths correspond to lower energy photons.

To convert a wavelength given in Angstroms to meters: \[ 1 \ \text{Å} = 10^{-10} \ \text{m} \]This conversion is crucial when working with formulas in physics, as the standard unit systems usually operate in meters.

Planck's Constant

Planck’s constant, denoted by \( h \), is a key number in quantum mechanics. It is a tiny constant whose value is approximately \( 6.626 \times 10^{-34} \ \text{Joule-seconds} \). This constant fundamentally sets the scale of quantum phenomena, like the energy levels of photons.Why is Planck’s constant important in photon energy? In the formula \( E = \frac{hc}{\lambda} \), \( h \) is integral in calculating the energy possessed by a photon:

• It ensures that energy is quantized, meaning energy can only exist in discrete amounts.
• This quantization is what allows us to calculate photon energy accurately.

Planck's constant illustrates how the quantum world deviates from classical predictions, thus helping explain phenomena that cannot be described by classical physics alone.

Speed of Light

The speed of light, represented by \( c \), is an essential constant in physics. Its accepted value is approximately \( 3.00 \times 10^8 \ \text{m/s} \). It is not only pivotal in the formula for photon energy, but it also represents the maximum speed at which information or matter can travel in the universe.In the context of photon energy, the speed of light ties together with Planck’s constant and wavelength in the equation \( E = \frac{hc}{\lambda} \):

• This formula states that energy is directly proportional to the speed of light when considering a constant wavelength.
• Hence, light's high speed contributes to the high energy of photons.

The speed of light also reinforces Einstein's theory of relativity, demonstrating how closely related energy and mass are linked with the speed of light as a conversion factor.