Problem 12 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The work function of sodium metal is \(4.41 \times 10^{-19} \mathrm{~J}\). If photons of wavelength \(300 \mathrm{~nm}\) are incident on the metal, the kinetic energy of the ejected electrons will be \(\left(h=6.63 \times 10^{-34} \mathrm{~J} \mathrm{~s} ; c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)\) \(\times 10^{-21} \mathrm{~J}\)

Step-by-Step Solution

Answer

The kinetic energy of the ejected electrons is \( 222 \times 10^{-21} \mathrm{~J} \).

Step 1: Define the Problem

We need to find the kinetic energy of electrons ejected from sodium metal when photons of a given wavelength are incident on it. We will use the equation of the photoelectric effect, which is \( KE = E_{photon} - \phi \), where \( KE \) is the kinetic energy of the ejected electrons, \( E_{photon} \) is the energy of the incoming photon, and \( \phi \) is the work function of the metal.

Step 2: Calculate Photon Energy

First, calculate the energy of the incoming photon using the formula \( E_{photon} = \frac{hc}{\lambda} \), where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength of the photon. Substitute the given values: \( h = 6.63 \times 10^{-34} \mathrm{~J \cdot s} \), \( c = 3 \times 10^{8} \mathrm{~m/s} \), and \( \lambda = 300 \mathrm{~nm} = 300 \times 10^{-9} \mathrm{~m} \).

Step 3: Substitute and Solve for Photon Energy

Substitute the values: \[ E_{photon} = \frac{6.63 \times 10^{-34} \times 3 \times 10^{8}}{300 \times 10^{-9}} \] Simplify this expression: \( E_{photon} = \frac{19.89 \times 10^{-26}}{300 \times 10^{-9}} = 6.63 \times 10^{-19} \mathrm{~J} \).

Step 4: Calculate Kinetic Energy

Subtract the work function \( \phi \) from the photon energy \( E_{photon} \) to find the kinetic energy \( KE \): \( KE = E_{photon} - \phi = 6.63 \times 10^{-19} \mathrm{~J} - 4.41 \times 10^{-19} \mathrm{~J} \).

Step 5: Final Calculation

Perform the subtraction: \( KE = 6.63 \times 10^{-19} \mathrm{~J} - 4.41 \times 10^{-19} \mathrm{~J} = 2.22 \times 10^{-19} \mathrm{~J} \). Express this in terms of \( \times 10^{-21} \mathrm{~J} \): \( KE = 222 \times 10^{-21} \mathrm{~J} \).

Key Concepts

Work Function

The work function is the minimum energy needed to eject an electron from the surface of a material. It is a characteristic property of the material and is denoted by the Greek letter \( \phi \). For sodium metal in this exercise, it is given as \( 4.41 \times 10^{-19} \text{ J} \).

• A higher work function means that more energy is required to remove an electron.
• The work function is a barrier that must be overcome for photoelectric emission to occur.

Understanding the work function is crucial because it determines how much "extra" energy will be available to the electrons as kinetic energy after they are ejected.

Photon Energy

Photon energy is the energy carried by a single photon, which is determined by its wavelength. The equation for photon energy is \( E_{photon} = \frac{hc}{\lambda} \), where \( h \) is Planck’s constant \( 6.63 \times 10^{-34} \text{ J} \cdot \text{s} \), \( c \) is the speed of light \( 3 \times 10^{8} \text{ m/s} \), and \( \lambda \) is the wavelength of the photon.

• For a photon with a wavelength of \( 300 \text{ nm} \), \( E_{photon} \) is calculated to be \( 6.63 \times 10^{-19} \text{ J} \).
• This energy is reliant on the frequency and inversely proportional to the wavelength—shorter wavelengths have higher energies.

Calculating photon energy is the first step in determining whether the photon's energy is enough to overcome the metal's work function and what the kinetic energy of the ejected electron will be.

Kinetic Energy

Kinetic energy of an ejected electron in the photoelectric effect is the energy that remains once the work function has been overcome by the incoming photon's energy. It can be calculated using the formula \( KE = E_{photon} - \phi \).
In this exercise, you've already calculated photon energy as \( 6.63 \times 10^{-19} \text{ J} \) and the work function of sodium as \( 4.41 \times 10^{-19} \text{ J} \).

• By subtracting the work function from the photon energy, we find that \( KE = 2.22 \times 10^{-19} \text{ J} \).
• This energy is what enables ejected electrons to move, and the magnitude of kinetic energy determines their speed.

Understanding kinetic energy helps us make predictions about the behavior of electrons once they have been freed from the material, including how fast they will be moving.