Q5: The ground state energy of the hydrogen atom is –13.6 eV. What are the kinetic and potential energies of the electron in this state?
Ground state energy of hydrogen atom, E = − 13.6 eV
The total energy of the hydrogen atom is -13.6 eV. The kinetic energy is equal to the negative of the total energy.
Kinetic energy = − E = − (− 13.6) = 13.6 eV
Potential energy = negative of two times of kinetic energy.
Potential energy = −2 × ( 13.6 ) = −27.2 eV.
Q6: A hydrogen atom initially in the ground level absorbs a photon, which excites it to the n = 4 level. Determine the wavelength and frequency of the photon.
For ground level, n1= 1
Let E1 be the energy of this level. It is known that E1 is related to n1 as:
The atom is excited to a higher level, n2= 4
Let E2 be the energy of this level:
Following is the amount of energy absorbed by the photon:
E = E2− E1
= 2.04 × 10-18 J
For a photon of wavelength λ, the expression of energy is written as:
h = Planck’s constant = 6.6 × 10−34 Js
c = Speed of light = 3 × 108 m/s
v = c/λ
= 3.1 × 1015 Hz
Therefore, 97 nm and 3.1×1015 Hz is the wavelength and frequency of the photon.
Q.7: (a) Using Bohr’s model, calculate the speed of the electron in a hydrogen atom in the n = 1, 2, and 3 levels. (b) Calculate the orbital period in each of these levels.
(a) Let v1 be the orbital speed of the electron in a hydrogen atom in the ground state level, n1= 1. For charge (e) of an electron, v1 is given by the relation,
Where, e = 1.6 × 10−19 C
h = Planck’s constant = 6.62 × 10−34 J s
For level n2= 2, we can write the relation for the corresponding orbital speed as:
For level n3= 3, we can write the relation for the corresponding orbital speed as:
Therefore, in a hydrogen atom, the speed of the electron at different levels, that is, n = 1, n = 2, and n = 3, is 2.18×106m/s, 1.09×106m/s and 7.27 × 105 m/s.
(b) Let T1 be the orbital period of the electron when it is in level n1= 1.
The orbital period is related to orbital speed as:
h = Planck’s constant = 6.62 × 10−34 J s
e = Charge on an electron = 1.6 × 10−19 C
ε0 = Permittivity of free space = 8.85 × 10−12 N−1 C2 m−2
m = Mass of an electron = 9.1 × 10−31 kg
For level n2 = 2, we can write the period as:
For level n3 = 3, we can write the period as:
Therefore, 1.52×10-16s, 1.22×10-15s and 4.12 × 10-15s are the orbital periods in each level.
Q8: The radius of the innermost electron orbit of a hydrogen atom is 5.3×10–11 m. What are the radii of the n = 2 and n = 3 orbits?
The radius of the innermost orbit of a hydrogen atom, r1 = 5.3 × 10−11 m.
Let r2 be the radius of the orbit at n = 2. It is related to the radius of the innermost orbit as:
For n = 3, we can write the corresponding electron radius as:
Therefore, 2.12 × 10−10 m and 4.77 × 10−10 m are the radii of an electron for n = 2 and n = 3 orbits, respectively.
Q9: A 12.5 eV electron beam is used to bombard gaseous hydrogen at room temperature. What series of wavelengths will be emitted?
It is given that the energy of the electron beam used to bombard gaseous hydrogen at room temperature is 12.5 eV. Also, the energy of the gaseous hydrogen in its ground state at room temperature is −13.6 eV.
When gaseous hydrogen is bombarded with an electron beam, the energy of the gaseous hydrogen becomes −13.6 + 12.5 eV, i.e., −1.1 eV.
Orbital energy is related to orbit level (n) as:
For n = 3, E = -13.6 / 9 = -1.5 eV
This energy is approximately equal to the energy of gaseous hydrogen.
It can be concluded that the electron has jumped from n = 1 to the n = 3 level.
During its de-excitation, the electrons can jump from n = 3 to n = 1 directly, which forms a line of the Lyman series of the hydrogen spectrum.
We have the relation for wave number for the Lyman series as:
Where, Ry= Rydberg constant = 1.097 × 107 m−1
λ= Wavelength of radiation emitted by the transition of the electron
For n = 3, we can obtain λ as:
If the electron jumps from n = 2 to n = 1, then the wavelength of the radiation is given as:
If the transition takes place from n = 3 to n = 2, then the wavelength of the radiation is given as:
This radiation corresponds to the Balmer series of the hydrogen spectrum.
Therefore, there are two wavelengths that are emitted in the Lyman series, which are approximately 102.5 nm and 121.5 nm, and one wavelength in the Balmer series, which is 656.33 nm.
Q10: In accordance with Bohr’s model, find the quantum number that characterises the Earth’s revolution around the Sun in an orbit of radius 3 × 1011 m with orbital speed 3 × 104 m/s. (Mass of earth = 6.0 × 1024 kg.)
The radius of the orbit of the Earth around the Sun, r = 1.5 × 1011 m
The orbital speed of the Earth, ν = 3 × 104 m/s
Mass of the Earth, m = 6.0 × 1024 kg
According to Bohr’s model, angular momentum is quantised and given as:
h = Planck’s constant = 6.62 × 10−34 J s
n = Quantum number
Therefore, the Earth revolution can be characterised by the quantum number 2.6 × 1074.
Q11: Choose a suitable solution to the given statements which justify the difference between Thomson’s model and Rutherford’s model
(a) In the case of scattering of alpha particles by a gold foil, the average angle of deflection of alpha particles stated by Rutherford’s model is (less than, almost the same as, or much greater than) stated by Thomson’s model.
(b) Is the likelihood of reverse scattering (i.e., dispersing of α-particles at points more prominent than 90°) anticipated by Thomson’s model (considerably less, about the same, or much more prominent) than that anticipated by Rutherford’s model?
(c) For a small thickness T, keeping other factors constant, it has been found that the amount of alpha particles scattered at direct angles is proportional to T. What does this linear dependence imply?
(d) To calculate the average angle of scattering of alpha particles by thin gold foil, which model states that it’s wrong to skip multiple scattering?
(a) almost the same
The normal point of diversion of alpha particles by a thin gold film anticipated by Thomson’s model is about the same as anticipated by Rutherford’s model. This is on the grounds that the average angle was taken in both models.
(b) considerably less
The likelihood of scattering of alpha particles at points more than 90° anticipated by Thomson’s model is considerably less than that anticipated by Rutherford’s model.
(c) Dispersing is predominantly because of single collisions. The odds of a single collision increment linearly with the amount of target molecules. Since the number of target particles increments with an expansion in thickness, the impact likelihood depends straightly on the thickness of the objective.
(d) Thomson’s model
It isn’t right to disregard multiple scattering in Thomson’s model for figuring out the average angle of scattering of alpha particles by a thin gold film. This is on the grounds that a solitary collision causes almost no deflection in this model. Subsequently, the watched normal scattering edge can be clarified just by considering multiple scattering.
Q12: The gravitational attraction between proton and electron in a hydrogen atom is weaker than the coulomb attraction by a component of around 10−40. Another option method for taking a gander at this case is to assess the span of the first Bohr circle of a hydrogen particle if the electron and proton were bound by gravitational attraction. You will discover the appropriate response fascinating.
The radius of the first Bohr orbit is given by the relation:
∈0 = Permittivity of free space
h = Planck’s constant = 6.63 × 10−34 J s
me = Mass of an electron = 9.1 × 10−31 kg
e = Charge of an electron = 1.9 × 10−19 C
mp = Mass of a proton = 1.67 × 10−27 kg
r = Distance between the electron and the proton
Coulomb attraction between an electron and a proton is:
The gravitational force of attraction between an electron and a proton is:
G = Gravitational constant = 6.67 × 10−11 N m2/kg2
Considering electrostatic force and the gravitational force between an electron and a proton to be equal, we get:
∴ FG = FC
Putting the value of equation (4) in equation (1), we get:
It is known that the universe is 156 billion light-years wide or 1.5 × 1027 m wide. Hence, we can conclude that the radius of the first Bohr orbit is much greater than the estimated size of the whole universe.
Q13: Obtain an expression for the frequency of radiation emitted when a hydrogen atom de-excites from level n to level (n–1). For large n, show that this frequency equals the classical frequency of the revolution of the electron in orbit.
It is given that the hydrogen atom de-excites from level n to level (n-1).
The equation for energy (E1) of the radiation at the level n is
ν1 is the frequency at level n
h = Planck’s constant
m = mass of the hydrogen atom
e = Electron charge
ε0 = Permittivity of free space
The energy (E2) of the radiation at level (n-1) is
ν2 is the frequency at level (n – 1)
Energy (E) is released as a result of de-excitation
E = E2 – E1
hν = E2 – E1——– (3)
ν = Frequency of the radiation emitted
Substituting (1) and (2) in (3), we get
For large n, we can write (2n -1 ) ≈ 2n and (n-1) ≈ n
The classical relation of the frequency of revolution of an electron is given as
νc = v/2πr ——-(5)
In the nth orbit, the velocity of the electron is
The radius of the nth orbit r is given as
Putting equations (6) and (7) in equation (5), we get
Therefore, the frequency of radiation emitted by the hydrogen atom is equal to the classical orbital frequency.
Q 14: Classically, an electron can be in any orbit around the nucleus of an atom. Then what determines the typical atomic size? Why is an atom not, say, a thousand times bigger than its typical size? The question had greatly puzzled Bohr before he arrived at his famous model of the atom that you have learnt in the textbook. To simulate what he might well have done before his discovery, let us play as follows with the basic constants of nature and see if we can get a quantity with the dimensions of length that is roughly equal to the known size of an atom (~ 10–10m).
(a) Construct a quantity with the dimensions of length from the fundamental constants e, me, and c. Determine its numerical value.
(b) You will find that the length obtained in (a) is many orders of magnitude smaller than the atomic dimensions. Further, it involves c. But the energies of atoms are mostly in a non-relativistic domain where c is not expected to play any role. This is what may have suggested Bohr discard c and look for ‘something else’ to get the right atomic size. Now, the Planck’s constant h had already made its appearance elsewhere. Bohr’s great insight lay in recognising that h, me, and e will yield the right atomic size. Construct a quantity with the dimension of length from h, me, and e and confirm that its numerical value has, indeed, the correct order of magnitude.
Charge of an electron, e = 1.6 x 10-19 C
Mass of the electron, me = 9.1 x 10-31 kg
Speed of light, c = 3 x 108 m/s
The equation from the given quantities is given as,
ε0 is the permittivity of free space
The numerical value of the quantity is
= 9 x 109 x [ (1.6 x 10 -19)2/ 9.1 x 10-31 x (3 x 108)2]
= 2.81 x 10-15 m
The numerical value of the equation taken is much lesser than the size of the atom.
Charge of an electron, e = 1.6 x 10-19 C
Mass of the electron, me = 9.1 x 10-31 kg
Planck’s constant , h = 6.63 x 10-34 Js
Considering a quantity involving all these values as
Where, ε0 = Permittivity of free space
The numerical value of the above equation is
= 0.53 x 10-10 m
The numerical value of the quantity is the order of the atomic size.
Q 15: The total energy of an electron in the first excited state of the hydrogen atom is about –3.4 eV.
(a) What is the kinetic energy of the electron in this state?
(b) What is the potential energy of the electron in this state?
(c) Which of the answers above would change if the choice of the zero of potential energy is changed?
(a) Total energy of the electron, E = -3.4 eV
The kinetic energy of the electron is equal to the negative of the total energy.
K.E = – E
= – (- 3.4 ) = + 3.4 eV
Kinetic energy = + 3.4 eV
(b) Potential energy (U) of the electron is equal to the negative of twice the kinetic energy,
P.E = -2 (K.E)
= – 2 x 3.4 = -6.8 eV
Potential energy = -6.8 eV
(c) The potential energy of the system depends on the reference point. If the reference point is changed from zero, then the potential energy will change. The total energy is given by the sum of the potential energy and kinetic energy. Therefore, the total energy will also change.
Q16: If Bohr’s quantisation postulate (angular momentum = nh/2π) is a basic law of nature, it should be equally valid for the case of planetary motion as well. Why, then, do we never speak of the quantisation of the orbits of planets around the Sun?
The quantum level for a planetary motion is considered to be continuous. This is because the angular momentum associated with planetary motion is largely relative to the value of Planck’s constant (h). 1070h is the order of the angular momentum of the Earth in its orbit. As the values of n increase, the angular momenta decreases. So, planetary motion is considered to be continuous.
Q17: Obtain the first Bohr’s radius and the ground state energy of a muonic hydrogen atom [i.e., an atom in which a negatively charged muon (μ−) of mass about 207me orbits around a proton].
Mass of a negatively charged muon, mμ = 207 me
According to Bohr’s model:
And, the energy of a ground state electronic hydrogen atom, Ee
Also, the energy of a ground state muonic hydrogen atom, Eμ α mμ
We have the value of the first Bohr orbit, re = 0.53 A = 0.53 × 10-10 m
Let rμ be the radius of the muonic hydrogen atom.
Following is the relation at equilibrium:
mμrμ= me re
207 me × rμ = me re
rμ= ( 0.53 × 10-10 )/ 207 = 2.56 × 10-13 m
Hence, for a muonic hydrogen atom, 2.56 × 10−13 m is the value of the first Bohr radius.
Ee= − 13.6 eV
Considering the ratio of the energies, we get
Eμ= 207 Ee = 207 × ( – 13.6 ) = – 2.81 k eV
−2.81 k eV is the ground state energy of a muonic hydrogen atom.
Class 12 Physics NCERT Solutions for Chapter 12 Atoms
The NCERT Solutions for Class 12 Physics Chapter 12 Atoms is formulated as per the latest CBSE Syllabus 2023-24. The NCERT Class 12 Solutions for Physics Chapter 12 Atoms given here are very simple and easy to understand. These solutions can help the students easily clear their doubts and have a proper understanding of the fundamental concepts covered in this chapter.
Topics covered in Class 12 Physics Chapter 12 Atoms are as follows:
|12.2||Alpha-Particle Scattering and Rutherford’s Nuclear Model of Atom|
|12.4||Bohr Model of the Hydrogen Atom|
|12.5||The Line Spectra of the Hydrogen Atom|
|12.6||De Broglie’s Explanation of Bohr’s Second Postulate of Quantisation|
Define Rutherford Atomic Model
Everything in the universe is composed of atoms. An atom is the fundamental building block of all matter. The Rutherford atomic model was proposed by Ernest Rutherford. In this model, the atom is described as a minute, dense, positively charged core called a nucleus, around which negatively charged constituents called electrons revolve, much like the planets revolving around the sun.
|NCERT Exemplar for Class 12 Physics Chapter 12|
|CBSE Notes for Class 12 Physics Chapter 12|
This chapter is an important one as it contains crucial topics of Atomic Physics. All the topics in the chapter are covered in the NCERT Solutions, which makes it easy for students to revise them for the board exam preparation. Download the free PDF and take a printout to keep handy during the preparation for the board exam.
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Dropped Topics –
12.3.1 Spectral Series
12.4 Bohr Model of the Hydrogen Atom (retain only the expression for the radius of nth possible orbit but delete its derivation)
12.5 The Line Spectra of the Hydrogen Atom (retain only qualitative treatment)
Exercises 12.3, 12.11–12.17
Frequently Asked Questions on NCERT Solutions for Class 12 Physics Chapter 12
Why should the students download the NCERT Solutions for Class 12 Physics Chapter 12 PDF?
List out the topics covered in Chapter 12 of NCERT Solutions for Class 12 Physics.
2. Alpha-Particle Scattering and Rutherford’s Nuclear Model of Atom
3. Alpha-particle Trajectory
4. Electron Orbits
5. Atomic Spectra
6. Spectral Series
7. Bohr Model of the Hydrogen Atom
8. Energy levels
9. The Line Spectra of The Hydrogen Atom
10. De Broglie’s Explanation of Bohr’s Second Postulate of Quantisation