Question

If the threshold wavelength for sodium is \[5420\mathop A\limits^ \circ \] then the work function of sodium is
A. 4.58eV
B. 2.28eV
C. 1.14eV
D. 0.23eV

Answer 1

Hint: When a photon strikes a metal, it must have the least amount of energy to overcome the attractive attraction that holds the valence electron to the shell of the metal's atom. The photon is the light wave's qualitative energy unit. It varies with the frequency of the light pulse.

Formula used:
\[E = h\nu \],
Here $h$ is the Plank’s constant and E is the energy of the photon with frequency equals to \[\nu \].
\[c = \nu \lambda \],
Here c is the speed of light, \[\nu \] is the frequency of the photon and \[\lambda \] is the wavelength of the light wave.

Complete step by step solution:
The minimum energy is called the work function of the metal. The wavelength of the photon corresponding to the energy equal to the work function of the metal is called the threshold wavelength. The threshold wavelength of the Sodium is given as \[5420\mathop A\limits^ \circ \].
\[{\lambda _0} = 5.420 \times {10^{ - 7}}m\]

The photon is the qualitative unit of energy of the light wave. It is proportional to the frequency of the light wave. The frequency corresponding to the energy of the work function is called the threshold frequency.
\[\phi = h{\nu _0}\]
\[\Rightarrow {\nu _0} = \dfrac{\phi }{h}\]

Using the relation between the speed of light, frequency and the wavelength, we get
\[\dfrac{c}{{{\lambda _0}}} = \dfrac{\phi }{h}\]
\[\Rightarrow \phi = \dfrac{{hc}}{{{\lambda _0}}}\]
\[\Rightarrow \phi = \dfrac{{\left( {6.626 \times {{10}^{ - 34}}} \right) \times \left( {3 \times {{10}^8}} \right)}}{{5.420 \times {{10}^{ - 7}}}}J\]
\[\Rightarrow \phi = 3.668 \times {10^{ - 19}}J\]

One electron volt is the energy equivalent to the work done to move one electron in a potential difference of one volt. So, the work function in terms of eV will be,
\[\therefore \phi = \dfrac{{3.668 \times {{10}^{ - 19}}}}{{1.6 \times {{10}^{ - 19}}}}eV = 2.28eV\]
Hence, the work function of the sodium is 2.28 eV.

Therefore, the correct option is B.

Note: As we all know, frequency is inversely related to wavelength. As a result, the threshold frequency is the lowest value of the photon's frequency that contains enough energy to evict the electron from metal.