Problem 85 - A Complete Resource Book in Chemistry for JEE Main

Question

A \(600 \mathrm{~W}\) mercury lamp emits monochromatic radiation of wavelength \(331.3 \mathrm{~nm}\). How many photons are emitted from the lamp per second? \(\left(\mathrm{h}=6.626 \times 10^{-34} \mathrm{~J} \mathrm{~s} ;\right.\) velocity of light \(=3 \times 10^{8} \mathrm{~ms}^{-1}\) (a) \(1 \times 10^{19}\) (b) \(1 \times 10^{20}\) (c) \(1 \times 10^{21}\) (d) \(1 \times 10^{23}\)

Step-by-Step Solution

Answer

Approximately \(1 \times 10^{21}\) photons are emitted per second.

Step 1: Calculate Energy of a Single Photon

To find the energy of a single photon, use the formula \(E = \frac{hc}{\lambda}\), where \(h\) is Planck's constant \(6.626 \times 10^{-34} \mathrm{~J} \mathrm{~s}\), \(c\) is the speed of light \(3 \times 10^{8} \mathrm{~m/s}\), and \(\lambda\) is the wavelength \(331.3 \mathrm{~nm} = 331.3 \times 10^{-9} \mathrm{~m}\). Substitute the values to get \(E = \frac{(6.626 \times 10^{-34} \mathrm{~J} \mathrm{~s})(3 \times 10^{8} \mathrm{~m/s})}{331.3 \times 10^{-9} \mathrm{~m}}\).

Step 2: Solve for Energy of a Single Photon

\[E = \frac{(6.626 \times 10^{-34})(3 \times 10^{8})}{331.3 \times 10^{-9}}\]Calculate the value:\[E \approx 6.003 \times 10^{-19} \mathrm{~J}\] Therefore, the energy of a single photon is approximately \(6.003 \times 10^{-19} \mathrm{~J}\).

Step 3: Calculate Number of Photons Emitted per Second

We know the power of the lamp is \(600 \mathrm{~W}\), which means it emits \(600 \mathrm{~J}\) of energy per second. To find the number of photons emitted per second, divide the total energy by the energy of a single photon:\[\text{Number of photons} = \frac{600 \mathrm{~J}}{6.003 \times 10^{-19} \mathrm{~J/photon}}\].

Step 4: Solve for Number of Photons

Calculate the number:\[\text{Number of photons} \approx \frac{600}{6.003 \times 10^{-19}} \approx 1 \times 10^{21}\]Thus, approximately \(1 \times 10^{21}\) photons are emitted per second.

Key Concepts

Monochromatic Radiation

Monochromatic radiation refers to light or electromagnetic radiation that has a single wavelength or frequency. The term "monochromatic" originates from Greek, meaning "one color." This is because each wavelength or frequency of electromagnetic radiation corresponds to a specific color in the visible spectrum. However, monochromatic radiation can exist beyond visible light, such as in ultraviolet or infrared spectrum.

In practical terms, a monochromatic light source emits light of a consistent wavelength. This is instrumental in various scientific applications, like spectroscopy, where precise measurements of spectral lines are essential. By focusing on a single frequency or wavelength, monochromatic radiation simplifies the analysis of light-matter interactions, allowing scientists and engineers to conduct more accurate experiments and develop technologies like lasers.

Planck's Constant

Planck's constant is a fundamental quantity in physics, symbolized by the letter \( h \), with a value of approximately \( 6.626 \times 10^{-34} \text{ J s} \). It plays a crucial role in quantum mechanics by relating the energy of a photon to its frequency.

The equation that involves Planck's constant is \( E = h u \), where \( E \) is the energy, \( u \) (nu) is the frequency, and \( h \) is Planck's constant. This relationship shows that energy and frequency are directly proportional, meaning higher frequency radiation has greater energy. Understanding this constant is essential in modern physics, as it marks the transition from classical to quantum mechanics, providing insights into energy levels of atoms and the behavior of subatomic particles.

Wavelength and Frequency

Wavelength and frequency are key components in describing any wave, including electromagnetic waves. Wavelength \( \lambda \) is the distance between two consecutive peaks or troughs in a wave. It's commonly measured in meters, but for tiny scales like light, nanometers (nm) are often used. In the original exercise, the given wavelength is \( 331.3 \text{ nm} \).

Frequency \( u \), on the other hand, measures how many oscillations or cycles happen per second, expressed in hertz (Hz). Wavelength and frequency are inversely related, as seen in the equation \( c = u \lambda \), where \( c \) is the speed of light, approximately \( 3 \times 10^8 \text{ m/s} \). When you know the wavelength, you can find the frequency, and vice versa. This relationship underlies many principles of physics, including the calculation of the energy of photons.

Energy of a Photon

The energy of a photon is an essential concept in understanding how light and electromagnetic waves interact with matter. A photon's energy is defined by the equation \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength.

This formula shows how energy is inversely proportional to wavelength. Thus, shorter wavelengths result in higher energy photons, which is why ultraviolet light can cause more damage than visible light due to its shorter wavelength and higher energy.

In practical applications like the exercise, knowing how to calculate a photon's energy enables determining the number of photons emitted by a light source given a certain power. Understanding photon's energy, therefore, is key for fields ranging from medicine, where photon's energy affects treatments like laser surgery, to telecommunications, where it can influence data transmission.