Question

When a certain metal was irradiated with light of frequency \[{\text{3}}{\text{.2}} \times {\text{1}}{{\text{0}}^{{\text{16}}}}{\text{Hz}}\], the photoelectrons emitted has twice the kinetic energy as did photoelectrons emitted when the same metal was irradiated with light of frequency \[{\text{2}}{\text{.0}} \times {\text{1}}{{\text{0}}^{{\text{16}}}}{\text{Hz}}\]. Calculate V0 for the metal.

Answer 1

Hint: As we know that metals are good conductors of heat and electricity. The energy of the matter is divided into two types. This classification of the energy depends on the position of the matter. There is potential energy and kinetic energy. The energy present in initial matter or energy of rest matter is called potential energy. The energy of the matter at the motion or movement is called potential energy. The kinetic energy of the matter depends on the mass and velocity of the matter. The kinetic energy of the matter is equal to half of the product of the mass and square velocity of the matter.
Formula used:
The photoelectric equation is,
\[{\text{KE = hv - h}}{{\text{v}}_{\text{0}}}\]
\[{\text{KE = h(v - }}{{\text{v}}_{\text{0}}})\]
\[\dfrac{{{\text{KE}}}}{{\text{h}}}{\text{ = v - }}{{\text{v}}_{\text{0}}}\]
Here,
\[{\text{KE}}\] is the kinetic energy.
\[{\text{h}}\] is Planck’s constant.
\[{\text{v}}\] is the final frequency of light.
\[{{\text{v}}_{\text{0}}}\] is the initial frequency of light.

Complete answer:
Metal was irradiated with light of frequency, \[{{\text{v}}_2}\] is \[{\text{3}}{\text{.2}} \times {\text{1}}{{\text{0}}^{{\text{16}}}}{\text{Hz}}\].
Metal was irradiated with light of frequency, \[{{\text{v}}_1}\] is \[{\text{2}}{\text{.0}} \times {\text{1}}{{\text{0}}^{{\text{16}}}}{\text{Hz}}\].
The photoelectrons emitted has twice the kinetic energy as did photoelectrons emitted when the same metal was irradiated with light of frequency \[{\text{2}}{\text{.0}} \times {\text{1}}{{\text{0}}^{{\text{16}}}}{\text{Hz}}\].
\[{\text{K}}{{\text{E}}_{\text{2}}}{\text{ = 2K}}{{\text{E}}_{\text{1}}}\]
For the first kinetic energy is,
\[\dfrac{{{\text{K}}{{\text{E}}_{\text{1}}}}}{{\text{h}}}{\text{ = }}{{\text{v}}_{\text{1}}}{\text{ - }}{{\text{v}}_{\text{0}}}\]
For the second kinetic energy is,
\[\dfrac{{{\text{K}}{{\text{E}}_2}}}{{\text{h}}}{\text{ = }}{{\text{v}}_2}{\text{ - }}{{\text{v}}_{\text{0}}}\]
Dividing the first kinetic energy equation with the second kinetic energy equation.
\[\dfrac{{{\text{K}}{{\text{E}}_{\text{1}}}}}{{\text{h}}}{\text{ = }}{{\text{v}}_{\text{1}}}{\text{ - }}{{\text{v}}_{\text{0}}}\]
\[\dfrac{{{\text{K}}{{\text{E}}_2}}}{{\text{h}}}{\text{ = }}{{\text{v}}_2}{\text{ - }}{{\text{v}}_{\text{0}}}\]
\[{\text{K}}{{\text{E}}_{\text{1}}}\] is the first kinetic energy.
\[{\text{K}}{{\text{E}}_{\text{2}}}\] is the second kinetic energy.
\[{{\text{v}}_{\text{1}}}\] is the first frequency of light.
\[{{\text{v}}_{\text{2}}}\] is the second frequency of light.
\[\dfrac{{{\text{K}}{{\text{E}}_2}}}{{{\text{K}}{{\text{E}}_1}}}{\text{ = }}\dfrac{{{{\text{v}}_2}{\text{ - }}{{\text{v}}_{\text{0}}}}}{{{{\text{v}}_1}{\text{ - }}{{\text{v}}_{\text{0}}}}}\]
\[{\text{K}}{{\text{E}}_{\text{2}}}{\text{ = 2K}}{{\text{E}}_{\text{1}}}\]
\[\dfrac{{{\text{2K}}{{\text{E}}_1}}}{{{\text{K}}{{\text{E}}_1}}}{\text{ = }}\dfrac{{{{\text{v}}_2}{\text{ - }}{{\text{v}}_{\text{0}}}}}{{{{\text{v}}_1}{\text{ - }}{{\text{v}}_{\text{0}}}}}\]
\[\dfrac{{{{\text{v}}_2}{\text{ - }}{{\text{v}}_{\text{0}}}}}{{{{\text{v}}_1}{\text{ - }}{{\text{v}}_{\text{0}}}}} = 2\]
\[{{\text{v}}_2}{\text{ - }}{{\text{v}}_{\text{0}}} = 2({{\text{v}}_1}{\text{ - }}{{\text{v}}_{\text{0}}})\]
\[{{\text{v}}_2}{\text{ - }}{{\text{v}}_{\text{0}}} = 2{{\text{v}}_1}{\text{ - 2}}{{\text{v}}_{\text{0}}}\]
\[ \Rightarrow {{\text{v}}_{\text{0}}} = 2{{\text{v}}_1}{\text{ - }}{{\text{v}}_2}\]
\[{{\text{v}}_{\text{0}}} = 2({\text{2}}{\text{.0}} \times {\text{1}}{{\text{0}}^{{\text{16}}}}{\text{Hz}}){\text{ - 3}}{\text{.2}} \times {\text{1}}{{\text{0}}^{{\text{16}}}}{\text{Hz}}\]
\[ \Rightarrow {{\text{v}}_{\text{0}}} = \;8 \times {\text{1}}{{\text{0}}^{{\text{1}}5}}{\text{Hz}}\]
The initial frequency of light for the metal \[{{\text{v}}_{\text{0}}}\] is \[8 \times {\text{1}}{{\text{0}}^{{\text{1}}5}}{\text{Hz}}\].

Note:
In matters are divided into three types. There are solids, liquids and gas. Solids are divided into types. They are crystalline and amorphous. The crystalline is further divided into ionic solids, covalent solids, molecular solids and metallic solids. In metal solids play a major role in chemistry. The SI unit of frequency is hertz. The symbol of frequency in hertz is \[{\text{Hz}}\].