Calculate the de Broglie wavelengths of an electron, proton and uranium atom, all having the same kinetic energy 100 eV . Mass of the electron is m = 9 . 1 × 10 - 31 kg , mass of proton is 1 . 67 × 10 - 27 kg , mass of uranium atom is 3 . 95 × 10 - 25 kg , Planck's constant h = 6 . 626 × 10 - 34 J Hz - 1 .

From the question, the kinetic energy T = 100 eV = 100 × 1 . 6 × 10 - 19 J . Now, we know that kinetic energy of electron is, T = 1 2 m e v e 2 , where symbols have their usual meaning. Now, the relation between kinetic energy and momentum of electron is, p e = m e v e ⇒ v e = p e m e T = 1 2 × p e 2 m e Now, same as for proton and uranium, the formula for proton is, T = 1 2 × p p 2 m p and that for uranium is, T = 1 2 × p ur 2 m ur For electron, λ e = h p e = h 2 m e T = 6 . 67 × 10 - 34 2 × 9 . 1 × 10 - 31 × 100 × 1 . 6 × 10 - 19 = 122 . 6 × 10 - 12 m = 122 . 6 pm For proton, λ p = h p p = h 2 m p T = 6 . 67 × 10 - 34 2 × 1 . 67 × 10 - 27 × 100 × 1 . 6 × 10 - 19 = 2 . 86 × 10 - 12 m = 2 . 86 pm For uranium, λ ur = h p ur = h 2 m ur T = 6 . 67 × 10 - 34 2 × 238 × 1 . 67 × 10 - 27 × 100 × 1 . 6 × 10 - 19 = 0 . 185 × 10 - 12 m = 0 . 185 pm using T = 1 . 602 × 10 - 17 J (where we have used a mass number of 238 for the U nucleus).

What amount of energy should be added to an electron to reduce its de Broglie wavelength from 100 to 50 pm ? Mass of the electron is m = 9 . 1 × 10 - 31 kg , Planck's constant h = 6 . 626 × 10 - 34 J Hz - 1 .

Given in the question, the wavelengths are, λ 1 = 100 pm = 100 × 10 - 12 m and λ 2 = 50 pm = 50 × 10 - 12 m Now, we know that kinetic energy of electron is, T = 1 2 m e v e 2 , where symbols have their usual meaning. Now, the relation between kinetic energy and momentum of electron is, p e = m e v e ⇒ v e = p e m e T = 1 2 × p e 2 m e From de-Broglie hypothesis, the wavelength of electron is, λ e = h p e = h 2 m e T Now, we find T = h 2 2 m e λ e 2 Now, use this formula for two different wavelengths, ⇒ T 1 = h 2 2 m e λ 1 2 and T 2 = h 2 2 m e λ 2 2 Thus, T 2 - T 1 = h 2 2 m e 1 λ 2 2 - 1 λ 1 2 = 6 . 67 × 10 - 34 2 2 × 9 . 1 × 10 - 31 1 50 × 10 - 12 2 - 1 100 × 10 - 12 2 = 721 . 6 × 10 - 19 J Substitution gives Δ T = 451 eV = 0 · 451 keV

HARD

JEE Main

IMPORTANT

Earn 100

A stationary ${He}^{+}$ ion emitted a photon corresponding to the first line of the Lyman series. That photon liberated a photoelectron from a stationary hydrogen atom in the ground state. Find the velocity of the photoelectron.
Mass of the electron is $m = 9.1 \times 10^{- 31} kg$ , $R$ is the Rydberg's constant ( $R = 1.097 \times 10^{7} m^{- 1}$ ), Planck's constant $h = 6.626 \times 10^{- 34} {JHz}^{- 1}$ , speed of light $c = 3 \times 10^{8} {ms}^{- 1}$ .

Important Questions on ATOMIC AND NUCLEAR PHYSICS

HARD

JEE Main

IMPORTANT

Problems in General Physics>Chapter 6 - ATOMIC AND NUCLEAR PHYSICS>SCATTERING OF PARTICLES. RUTHERFORD-BOHR AATOM>Q 6.46.

Calculate the separation between the particles of a system in the ground state, the corresponding binding energy, and the wavelength of the first line of the Lyman series, if such a system is
(a) a mesonic hydrogen atom whose nucleus is a proton (in a mesonic atom, an electron is replaced by a meson whose charge is the same and mass is

207

that of an electron);
(b) a positronium consisting of an electron and a positron revolving around their common centre of masses.

Mass of the electron is $m_{e} = 9.1 \times 10^{- 31} kg$ , mass of proton is $1.67 \times 10^{- 27} kg$ , charge on electron is $e = 1.6 \times 10^{- 19} C$ , Planck's constant $h = 6.626 \times 10^{- 34} {JHz}^{- 1}$ .

HARD

JEE Main

IMPORTANT

Problems in General Physics>Chapter 6 - ATOMIC AND NUCLEAR PHYSICS>SCATTERING OF PARTICLES. RUTHERFORD-BOHR AATOM>Q 6.47.

For atoms of light and heavy hydrogen ( $H$ and $D$ ), find the difference
(a) between the binding energies of their electrons in the ground state;
(b) between the wavelengths of first lines of the Lyman series.

Mass of the electron is $m = 9.1 \times 10^{- 31} kg$ , charge on electron is $e = 1.6 \times 10^{- 19} C$ , $R$ is the Rydberg's constant ( $R = 1.097 \times 10^{7} m^{- 1}$ ), Planck's constant $h = 6.626 \times 10^{- 34} {JHz}^{- 1}$ , speed of light $c = 3 \times 10^{8} {ms}^{- 1}$ .

HARD

JEE Main

IMPORTANT

Problems in General Physics>Chapter 6 - ATOMIC AND NUCLEAR PHYSICS>SCATTERING OF PARTICLES. RUTHERFORD-BOHR AATOM>Q 6.48.

Taking into account the motion of the nucleus of a hydrogen atom, find the expressions for the electron's binding energy in the ground state and for the Rydberg's constant. How much (in per cent) do the binding energy and the Rydberg's constant, obtained without taking into account the motion of the nucleus, differ from the more accurate corresponding values of these quantities?

MEDIUM

JEE Main

IMPORTANT

Problems in General Physics>Chapter 6 - ATOMIC AND NUCLEAR PHYSICS>WAVE PROPERTIES OF PARTICLES. SCHRODINGER EQUATION>Q 6.49.

Calculate the de Broglie wavelengths of an electron, proton and uranium atom, all having the same kinetic energy

100 eV .

Mass of the electron is $m = 9.1 \times 10^{- 31} kg$ , mass of proton is $1.67 \times 10^{- 27} kg$ , mass of uranium atom is $3.95 \times 10^{- 25} kg$ , Planck's constant $h = 6.626 \times 10^{- 34} J {Hz}^{- 1}$ .

HARD

JEE Main

IMPORTANT

Problems in General Physics>Chapter 6 - ATOMIC AND NUCLEAR PHYSICS>WAVE PROPERTIES OF PARTICLES. SCHRODINGER EQUATION>Q 6.50.

What amount of energy should be added to an electron to reduce its de Broglie wavelength from

100

50 pm

Mass of the electron is $m = 9.1 \times 10^{- 31} kg$ , Planck's constant $h = 6.626 \times 10^{- 34} J {Hz}^{- 1}$ .

HARD

JEE Main

IMPORTANT

Problems in General Physics>Chapter 6 - ATOMIC AND NUCLEAR PHYSICS>WAVE PROPERTIES OF PARTICLES. SCHRODINGER EQUATION>Q 6.51.

A neutron with kinetic energy

T = 25 eV

strikes a stationary deuteron (heavy hydrogen nucleus). Find the de Broglie wavelengths of both the particles in the frame of their centre of inertia.

Mass of the neutron is $m_{n} = 1.67 \times 10^{- 27} kg$ , Planck's constant $h = 6.626 \times 10^{- 34} J {Hz}^{- 1}$ .

HARD

JEE Main

IMPORTANT

Problems in General Physics>Chapter 6 - ATOMIC AND NUCLEAR PHYSICS>WAVE PROPERTIES OF PARTICLES. SCHRODINGER EQUATION>Q 6.52.

Two identical non-relativistic particles move at right angles to each other, possessing de Broglie wavelengths

λ_{1}

and

λ_{2} .

Find the de Broglie wavelengths of each particle in the frame of their centre of inertia.

HARD

JEE Main

IMPORTANT

Problems in General Physics>Chapter 6 - ATOMIC AND NUCLEAR PHYSICS>WAVE PROPERTIES OF PARTICLES. SCHRODINGER EQUATION>Q 6.53.

Find the de Broglie wavelength of hydrogen molecules, which corresponds to their most probable velocity at room temperature.

Mass of the hydrogen atom is $m_{H} = 1.67 \times 10^{- 27} kg$ , Boltzmann constant $k = 1.38 \times 10^{- 23} {JK}^{- 1}$ , Planck's constant $h = 6.626 \times 10^{- 34} {JHz}^{- 1}$ .

Important Questions on ATOMIC AND NUCLEAR PHYSICS

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