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Violet light of wavelength \({\rm{400 nm}}\) ejects electrons with a maximum kinetic energy of \({\rm{0}}{\rm{.860 eV}}\) from sodium metal. What is the binding energy of electrons to sodium metal?

Short Answer

Expert verified

The binding energy of electrons to sodium metal \(2.24\,{\rm{eV}}\)

Step by step solution

01

Given data

Given,

Wavelength is,\(\lambda = 400\,{\rm{nm}} = 400 \times {10^{ - 9}}{\rm{m}}\).

Kinetic energy is,\({\rm{KE}} = 0.860\,{\rm{eV}}\).

We also know that: Planks constant\(h = 4.13 \times {10^{ - 15}}\,{\rm{eV}}{\rm{.s}}\)

Speed of light \(c = 3 \times {10^8}\,{\rm{m/s}}\)

02

The longest-wavelength EM radiation can eject an electron

The kinetic energy of the electron is given by

\(K{E_e} = hf - BE\) ...(1)

Here\(K{E_e}\)is the kinetic energy,\(h\)is the plank constant,\(f\)is the frequency of the EM radiation and\(BE\)is the binding energy.

Now we know that the wavelength of EM radiation is given by

\(\lambda = \frac{c}{f}\) ...(2)

Where\(c\)is the speed of light.

So equation becomes,

\(K{E_e} = \frac{{hc}}{\lambda } - BE\) ...(3)

03

Calculate the binding energy of electrons to sodium metal

Hence the binding energy is expressed as,

\(BE = \frac{{hc}}{\lambda } - KE\)

Substitute all the value in the above equation

\(\begin{aligned}{}BE &= \frac{{\left( {4.13 \times {{10}^{ - 15}}\,{\rm{eV}}{\rm{.s}}} \right)\left( {3.00 \times {{10}^8}\,{\rm{m}}{{\rm{s}}^{{\rm{ - 1}}}}} \right)}}{{400 \times {{10}^{ - 9}}\,{\rm{m}}}} - (0.860\,{\rm{eV}})\\ &= 2.24\,{\rm{eV}}\end{aligned}\)

Therefore, the binding energy of electrons to sodium metal \(2.24\,{\rm{eV}}\)

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Most popular questions from this chapter

Question: Photoelectrons from a material with a binding energy of \({\rm{2}}{\rm{.71 eV}}\) are ejected by \({\rm{420 - nm}}\) photons. Once ejected, how long does it take these electrons to travel \({\rm{2}}{\rm{.50 cm}}\) to a detection device?

(a) If the power output of a \({\rm{650 - kHz}}\) radio station is \({\rm{50}}{\rm{.0 kW}}\), how many photons per second are produced?

(b) If the radio waves are broadcast uniformly in all directions, find the number of photons per second per square meter at a distance of \({\rm{100 km}}\). Assume no reflection from the ground or absorption by the air.

A certain heat lamp emits 200 W of mostly IR radiation averaging 1500 nm in wavelength.

(a) What is the average photon energy in joules?

(b) How many of these photons are required to increase the temperature of a person's shoulder by 2.0oC, assuming the affected mass is 4.0 kg with a specific heat of 0.83 kcal/kgoC.

{\vphantom {{\;{\bf{kcal}}} {{\bf{kg}}^\circ {\bf{C}}}}} \right.

\kern-\nulldelimiterspace} {{\bf{kg}}^\circ {\bf{C}}}}\]. Also assume no other significant heat transfer.

(c) How long does this take?

What is the binding energy in eV of electrons in magnesium, if the longest-wavelength photon that can eject electrons is 337nm?

Some satellites use nuclear power.

(a) If such a satellite emits a \({\rm{1}}{\rm{.00 - W}}\) flux of \(\gamma \) rays having an average energy of \({\rm{0}}{\rm{.500 MeV}}\), how many are emitted per second?

(b) These \(\gamma \) rays affect other satellites. How far away must another satellite be to only receive one \(\gamma \) ray per second per square meter?

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