Question

Sodium and copper have work functions 2.3 eV and 4.5 eV respectively. Then the ratio of threshold wavelengths is nearest to :
A. \[1:2\]
B. \[4:1\]
C. \[2:1\]
D. \[1:4\]

Answer 1

Hint: When the photon is incident on metal then it must contain the minimum energy to overcome the attractive force with which the valence electron is bound to the shell of the atom of the metal. Here, we are going to apply Einstein's photoelectric effect equation to obtain the answer.

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 threshold wavelength of the metal is the wavelength corresponding to the minimum energy which is sufficient to overcome the attractive force which causes the valence electron to be bound to the shell of the atom of the metal. As the energy of the photon is inversely proportional to the wavelength, so for a minimum value of the energy, the wavelength should be the maximum allowed wavelength.

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 work function of the Sodium is given as 2.3 eV and that of Copper is 4.5 eV, i.e. the minimum energy of the photon should be 2.3 eV to eject electrons from Sodium and 4.5 eV to eject electrons from Copper.
\[\dfrac{{hc}}{{{\lambda _{Na}}}} = 2.5eV \ldots \left( i \right)\]
\[\Rightarrow \dfrac{{hc}}{{{\lambda _{Cu}}}} = 4.5eV \ldots \left( {ii} \right)\]
On dividing 2nd equation with 1st equation, we get
\[\dfrac{{\left( {\dfrac{{hc}}{{{\lambda _{Cu}}}}} \right)}}{{\left( {\dfrac{{hc}}{{{\lambda _{Cu}}}}} \right)}} = \dfrac{{4.5}}{{2.5}} \\ \]
\[\therefore \dfrac{{{\lambda _{Na}}}}{{{\lambda _{Cu}}}} \approx 2\]
Hence, the nearest ratio of the threshold wavelength of Sodium to Copper is \[2:1\].

Therefore, the correct option is C.

Note: As we know that the frequency is inversely proportional to the wavelength. So for threshold frequency the minimum value of the frequency of the photon which contains sufficient energy to eject the electron from metal.