Question

The threshold wavelength for the photoelectric effect of a metal is \[6500\mathop A\limits^ \circ \]. The work function of the metal is approximately
A. 2 eV
B. 1 eV
C. 0.1 eV
D. 3 eV

Answer 1

Hint:If the energy of the photon exceeds the minimum energy needed to eject the electron then the rest of the energy is transferred as kinetic energy of the ejected electrons.

Formula used:
\[K = \dfrac{{hc}}{\lambda } - \phi \]
where K is the kinetic energy of the emitted electron, h is the planck's constant, c is the speed of light, \[\lambda \] is the wavelength of the photon and \[\phi \] is the work function of the metal.

Complete step by step solution:
The metal's threshold wavelength is the wavelength that corresponds to the minimal energy required to overcome the attractive attraction that binds the valence electron to the shell of the metal's atom. Because photon energy is inversely related to wavelength, the wavelength should be the highest permissible wavelength for a minimal value of energy.

For minimum conditions, the electron is just knocked out of the metal, i.e. the speed of the emitted electron is zero. Hence, the kinetic energy of the emitted electron is zero.
When the kinetic energy is zero, then the corresponding energy of the photon is called the work function of the metal.

The threshold wavelength is given as 6500 angstrom.
\[{\lambda _0} = 6500\mathop A\limits^ \circ \]
\[\Rightarrow {\lambda _0} = 6.5 \times {10^{ - 7}}m\]
Then the work function is,
\[\phi = \dfrac{{hc}}{{{\lambda _0}}} \\ \]
\[\Rightarrow \phi = \dfrac{{\left( {6.63 \times {{10}^{ - 34}}} \right)\left( {3 \times {{10}^8}} \right)}}{{\left( {6.5 \times {{10}^{ - 7}}} \right)}}J \\ \]
\[\Rightarrow \phi = 3.1 \times {10^{ - 19}}J\]
As 1 eV is equal to \[1.6 \times {10^{ - 19}}J\]. So,
\[\phi = \dfrac{{3.1 \times {{10}^{ - 19}}}}{{1.6 \times {{10}^{ - 19}}}}eV \\ \]
\[\therefore \phi = 1.9\,eV \approx 2\,eV\]
Hence, the work function of the photoelectric metal is 2 eV.

Therefore, the correct option is A.

Note: The work function is inversely proportional to the threshold wavelength. The wavelength of the incident photon should be less than the threshold wavelength for the photoelectric event to occur.