Problem 58 - A Complete Resource Book in Chemistry for JEE Main

Question

The value of Planck constant is \(6.63 \times 10^{-34} \mathrm{Js}\). he velocity of light is \(3.0 \times 10^{8} \mathrm{~m} \mathrm{~s}^{-1} .\) which value is closest to the wavelength in nanometres of a quantum of light with frequency of \(8 \times 10^{15} \mathrm{~s}^{-1}\) ? (a) \(4 \times 10^{1}\) (b) \(3 \times 10^{7}\) (c) \(2 \times 10^{-2 s}\) (d) \(5 \times 10^{-18}\)

Step-by-Step Solution

Answer

The wavelength is closest to \(4 \times 10^{1}\) nanometers, option (a).

Step 1: Identify Given Values

We are given the following: Planck's constant, \( h = 6.63 \times 10^{-34} \text{ Js} \); Speed of light, \( c = 3.0 \times 10^{8} \text{ m/s} \); Frequency of the quantum of light, \( f = 8 \times 10^{15} \text{ s}^{-1} \). We need to find the wavelength \( \lambda \) in nanometers.

Step 2: Use the Formula for Wavelength

The formula relating speed of light \( c \), frequency \( f \), and wavelength \( \lambda \) is \( c = \lambda \times f \). Thus, the wavelength can be calculated using \( \lambda = \frac{c}{f} \).

Step 3: Calculate Wavelength in Meters

Substitute the given values into the formula: \( \lambda = \frac{3.0 \times 10^{8} \text{ m/s}}{8 \times 10^{15} \text{ s}^{-1}} = 3.75 \times 10^{-8} \text{ m} \).

Step 4: Convert Meters to Nanometers

To convert the wavelength from meters to nanometers, recall that \(1 \text{ m} = 10^{9} \text{ nm}\). Thus, \( \lambda = 3.75 \times 10^{-8} \text{ m} \times 10^{9} \text{ nm/m} = 37.5 \text{ nm} \).

Key Concepts

Planck's constant

One of the most fundamental constants in quantum mechanics is Planck's constant, denoted by the symbol \( h \). It plays a critical role in the quantum world. You can think of it as the bridge between the wave and particle nature of light. Planck's constant helps us calculate the energy of photons when we know their frequency.
The value of Planck's constant is \( 6.63 \times 10^{-34} \text{ Js} \). This incredibly small number shows that the energy packets or 'quanta' of light are tiny.

• The energy of a photon can be determined using the formula: \( E = h \times f \), where \( E \) is energy and \( f \) is frequency.
• Since the value is so small, it underscores why quantum effects aren't noticeable in our macroscopic world.

Understanding Planck's constant gives insight into why energy at a quantum level is quantized, meaning it comes in small but discrete amounts.

speed of light

The speed of light, denoted as \( c \), is one of the most important constants in physics. Light travels incredibly fast in a vacuum at approximately \( 3.0 \times 10^{8} \text{ m/s} \). This speed isn't just for light visible to our eyes but applies to all electromagnetic waves, like radio waves and X-rays.
Knowing the speed of light is crucial because it helps physicists understand the behavior of light and how it interacts with matter.

• It allows us to calculate how long it takes for light to travel from one point to another, such as from the sun to Earth.
• The speed of light constant forms the basis of many equations in physics, like Einstein's famous equation \( E=mc^2 \).

With its significant role, the speed of light is a vital constant that allows us to connect our understanding of light's wave nature with its particle-like properties.

frequency-wavelength relation

The frequency-wavelength relation is a fundamental concept that connects two critical properties of waves: frequency and wavelength. The formula \( c = \lambda \times f \) links these properties together, where \( c \) is the speed of light, \( \lambda \) is the wavelength, and \( f \) is the frequency.
This relationship is integral to understanding how waves behave and interact. If you know either the wavelength or the frequency of a light wave, you can easily calculate the other using this key equation.

• Wavelength (\( \lambda \)) is the distance between consecutive crests of a wave, typically measured in meters.
• Frequency (\( f \)) is the number of wave cycles that pass a given point per second, measured in Hertz (Hz).
• As frequency increases, wavelength decreases, and vice versa, maintaining the speed of light as a constant in a vacuum.

By using this relation, scientists can interpret many phenomena in optics and telecommunications, understanding how waves transmit information and energy.