Problem 74 - A Complete Resource Book in Chemistry for JEE Main

Question

Electron energy of a photon is given as: \(\Delta \mathrm{E} /\) atom \(=3.03 \times 10^{-19} \mathrm{~J}\) atom \(^{-1}\) then, the wavelength of the photon is (a) \(6.56 \mathrm{~nm}\) (b) \(65.6 \mathrm{~nm}\) (c) \(656 \mathrm{~nm}\) (d) \(0.656 \mathrm{~nm}\) Given, \(\mathrm{h}\) (Planck constant) \(=6.63 \times 10^{-34} \mathrm{Js} \mathrm{c}\) (velocity of light \()=3.00 \times 10^{8} \mathrm{~ms}^{-1}\)

Step-by-Step Solution

Answer

The wavelength of the photon is 656 nm (option c).

Step 1: Understanding the Given Data

We are provided with the energy of a photon per atom, which is \(\Delta E = 3.03 \times 10^{-19} \text{ J}\text{/atom}\). The goal is to find the wavelength of this photon. The given constants include Planck's constant \(h = 6.63 \times 10^{-34} \text{ J s}\), and the speed of light \(c = 3.00 \times 10^8 \text{ m/s}\).

Step 2: Relate Energy to Wavelength

The energy of a photon can be related to its wavelength using the formula: \(E = \frac{h \cdot c}{\lambda}\), where \(\lambda\) is the wavelength. We need to rearrange this equation to solve for \(\lambda\).

Step 3: Solve for Wavelength

Rearrange the formula \(E = \frac{h \cdot c}{\lambda}\) to \(\lambda = \frac{h \cdot c}{E}\). Substitute the given values: \(\lambda = \frac{6.63 \times 10^{-34} \times 3.00 \times 10^8}{3.03 \times 10^{-19}}\).

Step 4: Calculate the Wavelength

Perform the calculation: \[ \lambda = \frac{(6.63 \times 10^{-34}) \cdot (3.00 \times 10^8)}{3.03 \times 10^{-19}} \approx 6.56 \times 10^{-7} \text{ m} = 656 \text{ nm} \]. Therefore, the wavelength of the photon is 656 nm.

Key Concepts

Planck constant

The Planck constant, denoted as \(h\), is a fundamental constant in physics, crucial for understanding the quantum mechanics realm. It represents the proportionality between the energy of a photon and the frequency of its corresponding electromagnetic wave.
In mathematical terms, this is expressed as:

• Energy (E) = h * frequency (f)

Here, \(h\) has a value of approximately 6.63 x 10^{-34} \text{ Js}\. This tiny number indicates how energy at the quantum level is quantized into discrete packets, which are tiny enough to fit the scale of atomic and subatomic particles.
Planck constant plays a pivotal role in the formula \(E = \frac{h \cdot c}{\lambda}\), linking it to wavelength and frequency in light-based phenomena.

speed of light

When you think of the speed of light, picture it as the universe's ultimate speed limit. Denoted as \(c\), its value is about 3.00 × 10^{8} \text{ m/s}.
This speed is constant and remains the same regardless of the observer's relative motion, a core part of Einstein's theory of relativity.

• It's the speed at which light travels in a vacuum.
• At this speed, light can travel around the Earth's equator about 7.5 times in a second.
• This constant is critical in equations involving light, such as \(E = \frac{h \cdot c}{\lambda}\), where it connects the energy of a photon to its wavelength.

wavelength calculation

To calculate a photon's wavelength, you can rearrange the formula \(E = \frac{h \cdot c}{\lambda}\) to solve for \(\lambda\):

• \(\lambda = \frac{h \cdot c}{E} \)

This equation highlights that wavelength is inversely proportional to energy. As energy increases, wavelength decreases, and vice versa.
For example, in the provided exercise, by plugging the given energy \(3.03 \times 10^{-19} \text{ J/atom}\), Planck constant \(6.63 \times 10^{-34} \text{ Js}\), and speed of light \(3.00 \times 10^{8} \text{ m/s}\) into the formula, we find that the wavelength is about 656 nm. This step is crucial for analyzing or designing systems involving electromagnetic radiation, such as light.

photon wavelength

Photon wavelength is the distance between successive peaks of the electromagnetic wave associated with a photon. This measurement helps determine the photon's color in visible light or type within the electromagnetic spectrum.
The wavelength determines various characteristics of the light:

• Short wavelengths correspond to high-energy photons (like X-rays).
• Long wavelengths are associated with lower energy (like radio waves).
• Visible light covers wavelengths from about 400 nm (violet) to 700 nm (red).
• In our example, a calculated wavelength of 656 nm falls in the red spectrum, common in phenomena like hydrogen emissions in space.

Understanding wavelength helps in fields ranging from astronomy to telecommunications.

energy-wavelength relation

The energy-wavelength relation illustrates the inverse relationship between a photon's energy and its wavelength. It's commonly expressed by the equation \(E = \frac{h \cdot c}{\lambda}\).
This equation unveils a few fundamental insights:

• Higher energy translates to shorter wavelengths (i.e., \(\gamma\)-rays), meaning they have more penetrative power.
• Lower energy indicates longer wavelengths (i.e., radio waves), suitable for communication purposes.
• This principle is vital in understanding the behavior of light, synthetic materials manufacturing, and diagnosing emission spectrum of stars.

With this equation, you discern how varying wavelengths manifest in different colors and energies, providing essential insights into both natural phenomena and technological applications.