Problem 13 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Ejection of the photoelectron from metal in the photoelectric effect experiment can be stopped by applying \(0.5 \mathrm{~V}\) when the radiation of \(250 \mathrm{~nm}\) is used. The work function of the metal is : (a) \(4 \mathrm{eV}\) (b) \(5.5 \mathrm{eV}\) (c) \(4.5 \mathrm{eV}\) (d) \(5 \mathrm{eV}\)

Step-by-Step Solution

Answer

The work function is 4.5 eV.

Step 1: Understand the Photoelectric Equation

The photoelectric effect can be expressed with the equation \(hu = W + KE_{max}\), where \(h\) is Planck's constant, \(u\) is the frequency of the incident light, \(W\) is the work function of the metal, and \(KE_{max}\) is the maximum kinetic energy of the ejected photoelectron. The stopping potential \(V\) gives \(KE_{max} = eV\).

Step 2: Calculate the Frequency of the Incident Light

Convert the given wavelength \(\lambda = 250 \text{ nm}\) into meters to use in calculations: \(250 \text{ nm} = 250 \times 10^{-9} \text{ m}\). Use the relation \(c = \lambda u\) to find frequency \(u\): \(u = \frac{c}{\lambda}\), where \(c = 3 \times 10^8 \text{ m/s}\).

Step 3: Compute the Photon Energy

Substitute the frequency into the photon energy equation: \(E = hu\), where Planck's constant \(h = 6.626 \times 10^{-34} \text{ J s}\). Calculate \(E\) and convert it to electron volts (1 eV = \(1.602 \times 10^{-19} \text{ J}\)).

Step 4: Consider the Stopping Potential

The stopping potential is given as \(V = 0.5 \text{ V}\). This relates to the kinetic energy of the ejected electron as \(KE_{max} = eV = 0.5 \text{ eV}\).

Step 5: Determine the Work Function

Rearrange the photoelectric equation to solve for the work function \(W = E - KE_{max}\). Subtract the stopping potential energy from the photon energy.

Step 6: Final Calculation and Comparison

Subtract the electron's kinetic energy due to the stopping potential from the photon energy to get \(W\). Compare \(W\) to the given options to determine the correct answer.

Key Concepts

Work Function

The work function is a critical concept in understanding the photoelectric effect. It is the minimum energy needed to remove an electron from the surface of a metal. This energy threshold relies on the type of metal and its surface characteristics. In simpler terms, if the energy from photons (light particles) is enough to overcome this barrier, the electron will be ejected from the metal surface.In the given exercise, to find the work function, we rely on the equation from the photoelectric effect: - The equation is: \( hu = W + KE_{max} \) - Here, \( h \) is Planck's constant, \( u \) is the frequency, \( W \) is the work function, and \( KE_{max} \) is the maximum kinetic energy of the ejected electron.By understanding the work function, you can determine how much energy is needed to release an electron, while also comparing it against the photon energy to see if ejection is possible. This explains why light with sufficient energy can dislodge electrons from certain metals.

Stopping Potential

Stopping potential is the voltage required to stop electrons, emitted from a metal due to the photoelectric effect, from reaching the other side of a setup in an experiment. This concept allows us to measure the maximum kinetic energy of the ejected electrons.Using the stopping potential \( V \), you can calculate the kinetic energy of an electron as: - \( KE_{max} = eV \) - Here, \( e \) is the charge of an electron, which is approximately \(1.602 \times 10^{-19} \text{ C}\).In the provided exercise, the stopping potential is given as \(0.5 \text{ V}\). Therefore, the maximum kinetic energy of the photoelectron is \(0.5 \text{ eV}\). When calculating the work function, the stopping potential's impact on the electron’s kinetic energy is subtracted from the total photon energy to find the remaining energy needed to overcome the work function.

Planck's Constant

Planck's constant is a pivotal element of quantum mechanics, representing the quantization of energy. It is used to relate the energy of photons to their frequency. This constant is key in all calculations involving the photoelectric effect.The value of Planck's constant is: - \( h = 6.626 \times 10^{-34} \text{ J s}\)In the context of the photoelectric effect, Planck’s constant helps in calculating the energy of the photons using the formula: - \( E = hu \)Using Planck's constant, along with the frequency of incident radiation, you'll determine the energy of the incoming photons. This calculation is crucial since it must be compared against the work function and any kinetic energy to solve problems regarding electron ejection, as shown in the exercise.