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) \(2 \times 10^{-25}\) (d) \(5 \times 10^{-18}\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Relationship

The relationship between wavelength \( \lambda \), frequency \( f \), and the speed of light \( c \) is given by the equation \( \lambda = \frac{c}{f} \). We know \( c = 3.0 \times 10^8 \mathrm{~m/s} \) and \( f = 8 \times 10^{15} \mathrm{~Hz} \).

Step 2: Calculate the Wavelength

Using the formula \( \lambda = \frac{c}{f} \), we substitute the values: \[ \lambda = \frac{3.0 \times 10^8}{8 \times 10^{15}} \] Calculating this gives: \[ \lambda = 3.75 \times 10^{-8} \mathrm{~m} \]

Step 3: Convert Wavelength from Meters to Nanometres

To convert meters to nanometres, we use the conversion factor where \(1 \mathrm{~m} = 10^9 \mathrm{~nm}\). Thus, \[ \lambda = 3.75 \times 10^{-8} \times 10^9 \mathrm{~nm}\] Calculating this gives: \[ \lambda = 37.5 \mathrm{~nm} \]

Step 4: Compare to Given Options

The closest value to \( 37.5 \mathrm{~nm} \) from the options given is (a) \(4 \times 10^{1}\), which is equivalent to \( 40 \mathrm{~nm} \).

Key Concepts

Planck constant

The Planck constant is a fundamental principle in the world of quantum physics. It is denoted by the symbol \( h \) and represents the proportionality factor between the energy of a photon and the frequency of its corresponding electromagnetic wave. This constant is key to understanding the quantum nature of light and matter.

Planck's constant has a value of approximately \( 6.63 \times 10^{-34} \text{Js} \). This tiny number might seem insignificant, but it plays a critical role in the calculation of energy levels within atoms and molecules. The equation connecting the energy \( E \) of a photon to its frequency \( f \) is given by:

\[ E = h \cdot f \]

With this equation, you can determine the energy of a photon if you know its frequency, establishing a clear connection between the wave and particle nature of light. Planck's constant effectively bridges the gap between the macroscopic concepts of energy and the microscopic world of quantum particles.

Speed of Light

The speed of light, often represented by the symbol \( c \), is a crucial constant in physics. Light travels incredibly fast, at approximately \( 3.0 \times 10^8 \frac{m}{s} \), which is about 300,000 kilometers per second. This speed is not just impressive; it forms the backbone of many areas of physics including relativity and optics.

Light speed is a universal constant which means it doesn't change regardless of the observer's motion or the light source. It serves as a boundary speed for the transmission of information in our universe. In the context of wavelength calculations, the speed of light helps define the relationship between light's wavelength and frequency.

Knowing the speed of light allows us to predict how light will behave when interacting with different media and when calculating the wavelengths of various spectral lines. Whether in a vacuum or passing through glass, the speed at which light travels profoundly impacts calculations in both classical and quantum physics.

Frequency-Wavelength Relationship

The relationship between frequency and wavelength is foundational to understanding wave phenomena, including light. It is depicted through the equation:

\[ \lambda = \frac{c}{f} \]

Where \( \lambda \) is the wavelength, \( c \) is the speed of light, and \( f \) is the frequency. This equation tells us that wavelength is inversely proportional to frequency. Essentially, as the frequency of a wave increases, its wavelength decreases, and vice versa.

In simpler terms, if you have a high-frequency wave, it will have a shorter wavelength. Conversely, a low-frequency wave will stretch out longer. This fundamental relationship is used across many fields of science and engineering to characterize different types of waves, from radio waves and microwaves to visible light.

By using the speed of light and known frequencies, scientists can accurately calculate wavelengths, helping to reveal processes at both cosmic and microscopic levels. Understanding this relationship allows us to design better communication systems and understand deeper aspects of light and the universe.