Advertisements
Advertisements
प्रश्न
The wavelength of light from the spectral emission line of sodium is 589 nm. Find the kinetic energy at which
(a) an electron, and
(b) a neutron, would have the same de Broglie wavelength.
Advertisements
उत्तर
Wavelength of light of a sodium line, λ = 589 nm = 589 × 10−9 m
Mass of an electron, me = 9.1 × 10−31 kg
Mass of a neutron, mn = 1.66 × 10−27 kg
Planck’s constant, h = 6.6 × 10−34 Js
(a) For the kinetic energy K, of an electron accelerating with a velocity v, we have the relation:
`"K" = 1/2 "m"_"e""v"^2` ...............(1)
We have the relation for de Broglie wavelength as:
`lambda = "h"/("m"_"e""v")`
∴ `"v"^2 = "h"^2/(lambda^2"m"_"e"^2)` ........(2)
Substituting equation (2) in equation (1), we get the relation:
`"K" = 1/2 ("m"_"e""h"^2)/(lambda^2"m"_"e"^2) = "h"^2/(2lambda^2"m"_"e")` ..........(3)
= `(6.6 xx 10^(-34))^2/(2 xx (589 xx 10^(-9))^2 xx 9.1 xx 10^(-31))`
≈ 6.9 × 10−25 J
= `(6.9 xx 10^(-25))/(1.6 xx 10^(-19))`
= 4.31 × 10−6 eV
= 4.31 μeV
Hence, the kinetic energy of the electron is 6.9 × 10−25 J or 4.31 μeV.
(b) Using equation (3), we can write the relation for the kinetic energy of the neutron as:
`"h"^2/(2lambda^2 "m"_"n")`
= `(6.6 xx 10^(-34))^2/(2 xx (589 xx 10^(-9))^2 xx 1.66 xx 10^(-27))`
= 3.78 × 10−28 J
= `(3.78 xx 10^(-28))/(1.6 xx 10^(-19))`
= 2.36 × 10−9 eV
= 2.36 neV
Hence, the kinetic energy of the neutron is 3.78 × 10−28 J or 2.36 neV.
APPEARS IN
संबंधित प्रश्न
A proton and an α-particle have the same de-Broglie wavelength Determine the ratio of their speeds.
For what kinetic energy of a neutron will the associated de Broglie wavelength be 1.40 × 10−10 m?
Compute the typical de Broglie wavelength of an electron in a metal at 27°C and compare it with the mean separation between two electrons in a metal which is given to be about 2 × 10−10 m.
The energy and momentum of an electron are related to the frequency and wavelength of the associated matter wave by the relations:
E = hv, p = `"h"/lambda`
But while the value of λ is physically significant, the value of v (and therefore, the value of the phase speed vλ) has no physical significance. Why?
The wavelength λ of a photon and the de-Broglie wavelength of an electron have the same value. Show that energy of a photon in (2λmc/h) times the kinetic energy of electron; where m, c and h have their usual meaning.
Sodium and copper have work function 2.3 eV and 4.5 eV respectively. Then, the ratio of the wavelengths is nearest to ______.
70 cal of heat is required to raise the temperature of 2 moles of an ideal gas at constant pressure from 30°C to 35°C. The amount of heat required to raise the temperature of the gas through the same range at constant volume will be (assume R = 2 cal/mol-K).
An electromagnetic wave of wavelength ‘λ’ is incident on a photosensitive surface of negligible work function. If ‘m’ mass is of photoelectron emitted from the surface has de-Broglie wavelength λd, then ______.
An electron (mass m) with an initial velocity `v = v_0hati (v_0 > 0)` is in an electric field `E = - E_0hati `(E0 = constant > 0). It’s de Broglie wavelength at time t is given by ______.
An electron (mass m) with an initial velocity `v = v_0hati` is in an electric field `E = E_0hatj`. If λ0 = h/mv0, it’s de Broglie wavelength at time t is given by ______.
A particle moves in a closed orbit around the origin, due to a force which is directed towards the origin. The de Broglie wavelength of the particle varies cyclically between two values λ1, λ2 with λ1 > λ2. Which of the following statement are true?
- The particle could be moving in a circular orbit with origin as centre.
- The particle could be moving in an elliptic orbit with origin as its focus.
- When the de Broglie wavelength is λ1, the particle is nearer the origin than when its value is λ2.
- When the de Broglie wavelength is λ2, the particle is nearer the origin than when its value is λ1.
A proton and an α-particle are accelerated, using the same potential difference. How are the de-Broglie wavelengths λp and λa related to each other?
An electron is accelerated from rest through a potential difference of 100 V. Find:
- the wavelength associated with
- the momentum and
- the velocity required by the electron.
Given below are two statements:
Statement - I: Two photons having equal linear momenta have equal wavelengths.
Statement - II: If the wavelength of photon is decreased, then the momentum and energy of a photon will also decrease.
In the light of the above statements, choose the correct answer from the options given below.
An electron of mass me, and a proton of mass mp = 1836 me are moving with the same speed. The ratio of the de Broglie wavelength `lambda_"electron"/lambda_"proton"` will be:
In a Frank-Hertz experiment, an electron of energy 5.6 eV passes through mercury vapour and emerges with an energy 0.7 eV. The minimum wavelength of photons emitted by mercury atoms is close to ______.
How will the de-Broglie wavelength associated with an electron be affected when the velocity of the electron decreases? Justify your answer.
How will the de-Broglie wavelength associated with an electron be affected when the accelerating potential is increased? Justify your answer.
