Problem 70 - 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 \(2: 1\) (Answer b).

Step 1: Understanding Wavelength and Energy Relationship

The energy of a photon is inversely proportional to its wavelength. The formula relating energy and wavelength is given by \[ E = \frac{hc}{\lambda} \]where \(E\) is the energy, \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength. This suggests that a shorter wavelength means higher energy.

Step 2: Calculating Energy Ratio Between Two Wavelengths

We need to find the ratio of energies \[ \frac{E_1}{E_2} = \frac{\frac{hc}{\lambda_1}}{\frac{hc}{\lambda_2}} = \frac{\lambda_2}{\lambda_1} \]Substituting the given wavelengths, \[ \frac{E_1}{E_2} = \frac{6000 \text{ Å}}{3000 \text{ Å}} = \frac{6000}{3000} = 2 \]This indicates that the energy ratio is \(2:1\).

Key Concepts

Wavelength and Energy Relationship

The relationship between a photon's wavelength and its energy is one of inversely proportional characteristics. This means that as the wavelength of a photon decreases, its energy increases, and vice versa.

Understanding this relationship is crucial because it underpins how different types of electromagnetic waves behave. For instance, X-rays have very short wavelengths compared to visible light, and therefore they carry much higher energy. Conversely, radio waves have long wavelengths and consequently lower energy.

The formula that links energy and wavelength is expressed through\[E = \frac{hc}{\lambda}\]Here:

• \(E\) represents the energy of the photon.
• \(h\) is Planck's constant, which is a fundamental constant in physics.
• \(c\) stands for the speed of light in a vacuum.
• \(\lambda\) (lambda) is the symbol for wavelength.

By using this formula, one can determine how shifts in wavelength affect the energy level of a photon.

Planck's Constant

Planck's constant is a vital component of quantum physics and plays a key role in our understanding of photon energy. It represents the proportionality constant between the energy of a photon and the frequency of its corresponding electromagnetic wave.

The value of Planck's constant is approximately \(6.626 \times 10^{-34} \text{ Js}\). This seemingly small number underscores the tiny energy carried by each individual photon at ordinary frequencies, but it becomes significant when we deal with processes at the atomic scale.

To better understand its contributions to physics, consider this:

• Before Planck’s constant was introduced, classical physics could not accurately describe phenomena at the atomic level, such as black body radiation.
• Planck introduced this constant to calculate the energy of these minuscule quantum particles, accurately predicting experimental results.
• It is essential in formulas that link light energy to wavelength and frequency, as seen in the equation for photon energy \(E = hf\), where \(f\) is the frequency.

Planck's constant, therefore, is foundational to our comprehension of not just photons but all quantum entities.

Energy Ratio Calculation

The process of calculating the energy ratio between photons of different wavelengths involves understanding the inverse relationship between their wavelength and energy. Given two wavelengths, you can determine their energy ratio by using the formula provided in the wavelength-energy relation.

To find the energy ratio \(\frac{E_1}{E_2}\) between two different wavelengths \(\lambda_1\) and \(\lambda_2\), you manipulate the energy formula like this:\[\frac{E_1}{E_2} = \frac{\frac{hc}{\lambda_1}}{\frac{hc}{\lambda_2}} = \frac{\lambda_2}{\lambda_1}\]This shows that the ratio of energies is simply the inverse of the ratio of their wavelengths.

Consider the exercise, where we have two wavelengths: \(3000 \text{ Å}\) and \(6000 \text{ Å}\). Substituting these values into the formula, we get:

• \(\frac{E_1}{E_2} = \frac{6000 \text{ Å}}{3000 \text{ Å}} = 2\)

This result implies that the photon with the \(3000 \text{ Å}\) wavelength has twice the energy of the photon with the \(6000 \text{ Å}\) wavelength, corresponding to an energy ratio of \(2:1\).

Understanding these calculations can enhance your grasp of how photons of different wavelengths interact in various applications, from simple light observations to complex quantum experiments.