Question

The peak emission from a black body at a certain temperature occurs at a wavelength of 9000˚A. On increasing its temperature, the total radiation emitted is increased 81 times. At the initial temperature when the peak radiation from the black body is incident on a metal surface, it does not cause any photoemission from the surface. After the increase of temperature, the peak radiation from the black body caused photoemission. To bring these photoelectrons to rest, a potential equivalent to the excitation energy between n=2 and n=3 Bohr levels of hydrogen atoms is required. Find the work function of the metal.

Answer 1

The work function of the metal is found by relating the stopping potential needed to bring photoelectrons to a halt, which is equivalent to the energy difference between n=2 and n=3 Bohr levels in hydrogen, to the energy of incident photons from the heated black body. After calculating this energy and knowing the peak radiation increases with temperature, the work function can be determined.

Step-by-step explanation:

To find the work function of the metal, we must use Planck's equation relating energy to frequency (E = hν) and the photoelectric effect. The work function (φ) is the minimum energy required to remove an electron from the surface of the metal. According to the problem, the stopping potential required to bring photoelectrons to rest is equivalent to the excitation energy between the n=2 and n=3 levels in the hydrogen atom. Using the Bohr model, this energy can be calculated using the formula E = -13.6 eV * (1/n²1 - 1/n²2), where n1 and n2 are the principal quantum numbers of the initial and final states, respectively.

In this case, E equals the energy difference between n=2 (first excited state) and n=3 (second excited state), hence E = -13.6 eV * (1/2² - 1/3²) = 1.89 eV. Now that we know the energy needed to bring the photoelectrons to a halt, and since the intensity of the radiation increased 81 times (which implies that the temperature is increased), we need to find the new peak wavelength emitted using Wien's displacement law, λT=constant, with the initial wavelength (λ) given as 9000Å, and hence calculate the frequency (ν) associated with the peak radiation at the increased temperature.

Since the energy E of the incident photons is responsible for overcoming the work function and providing the kinetic energy to the electrons (E=hν=φ+K.E.), and we know the energy E (1.89 eV from the stopping potential), we can solve for the work function φ, considering hν = E and thus find the work function of the metal.