Tuesday, January 31, 2023

H C Verma solutions, BOHR'S MODEL AND PHYSICS OF THE ATOM, Chapter-43, OBJECTIVE-I, Concepts of Physics, Part-II

Bohr's Model and Physics of the Atom


OBJECTIVE-I


     1.  The minimum orbital angular momentum of the electron in a hydrogen atom is 

(a) h

(b) h/2

(c) h/2π

(d) h/λ  


ANSWER: (c).   


EXPLANATION: According to Bohr's Postulates, the orbital angular momentum l of the electron about the nucleus is an integral multiple of Planck's constant h divided by 2π. i.e. 

l =n*(h/2π) 

The minimum value of l is for n =1 and it is equal to h/2π. Option (c) is correct.   




     2.  Three photons coming from the excited atomic-hydrogen sample are picked up. Their energies are 12.1 eV, 10.2 eV, and 1.9 eV. These photons must come from

(a) a single atom

(b) two atoms

(c) three atoms

(d) either two atoms or three atoms.  


ANSWER: (d).   


EXPLANATION: When an electron jumps to a lower orbit of an excited hydrogen atom, the energy of the photon released is given as, 

E = 13.6(1/n² -1/n'²). 

For n =1 and n' =3, 

E =13.6(1/1² -1/3²) =12.1 eV


For n =1 and n' =2, 

E =13.6(1 -1/2²) =10.2 eV


For n =2 and n' =3, 

E =13.6(1/2² -1/3²) =1.9 eV 

The three picked-up electrons have these values of energy. So one electron has jumped down from the third energy level to the first in an atom. The other two energies are for an electron jumping from the third to the second energy level and from the second to the first energy level. This might have happened in one atom or in two atoms separately. So the electrons are either from two atoms or three atoms. Option (d) is correct.     



    



     3.  Suppose the electron in a hydrogen atom makes a transition from n =3 to n =2 in 10⁻⁸ s. The order of the torque acting on the electron in this period, using the relation between torque and angular momentum as discussed in the chapter on rotational mechanics is  

(a) 10⁻³⁴ N-m

(b) 10⁻²⁴ N-m

(c) 10⁻⁴² N-m

(d) 10⁻⁸ N-m. 


ANSWER: (b).   


EXPLANATION: The orbital angular momentum of an electron is nh/2π. Change of angular momentum in transition from n =3 to n =2 is 

L =3h/2π -2h/2π 

   =h/2π 

Time in this transition t =10⁻⁸ s. 

The rate of change of angular momentum is equal to the torque.

 Hence torque =L/t

  =h/2πt 

  =6.63x10⁻³⁴/(2*3.14*10⁻⁸) N-m

  =1.05x10⁻²⁶ N-m

  =(1.05/100)x10⁻²⁴ N-m

So it is of the order of 10⁻²⁴ N-m. option (b) is correct.



      



     4.  In which of the following transitions will the wavelength be minimum? 

(a) n =5 to n =4

(b) n =4 to n =3 

(c) n =3 to n =2

(d) n =2 to n =1.  


ANSWER: (d).   


EXPLANATION: Wavelength will be minimum for the highest energy released. 

Since the energy released is proportional to (1/n² -1/m²) 

i.e. =(m²-n²)/m²n² 

  =(m+n)(m-n)/m²n²

In the given options, m-n =1 for each case. So the energy is proportional to (m+n)/m²n². Clearly, the energy will be maximum for the lowest values of m and n, that is m =2 and n =1. Option (d) is correct. 



   



     5.  In which of the following systems will the radius of the first orbit (n =1) be minimum? 

(a) hydrogen atom 

(b) deuterium atom 

(c) singly ionized helium

(d) doubly ionized lithium.    


ANSWER: (d).   


EXPLANATION: The radius of an orbit in a hydrogen-like atom is given by 

rₙ =n²aₒ/Z

where Z is the number of protons in the nucleus.

For the first orbit, n =1, So

r₁ =aₒ/Z

Clearly, it will be the minimum for the maximum Z. Here Z is the maximum for lithium which is 3. So the minimum radius of the first orbit will be for doubly ionized lithium. Option (d) is correct. 



 



     6.  In which of the following systems will the wavelength corresponding to n =2 to n =1 be minimum? 

(a) hydrogen atom 

(b) deuterium atom 

(c) singly ionized helium

(d) doubly ionized lithium.    


ANSWER: (d).   


EXPLANATION: The wavelength corresponding to the transition from n' to n orbit is given as 

1/𝜆 =RZ²{1/n² -1/n'²}

where R is Rydberg's constant and Z is the atomic number. Here n =1 and n' =2, so

1/𝜆 =3RZ²/4

→𝜆 =4/(3RZ²)

So the wavelength will be minimum for maximum Z which is for doubly ionized lithium. Option (d) is correct.    



 



     7.  Which of the following curves may represent the speed of the electron in a hydrogen atom as a function of the principal quantum number n? 
The figure for Q-7


ANSWER: (c).   


EXPLANATION: The speed of an electron in a hydrogen atom is given as, 

v =Ze²/(2εₒhn)

Where Z is the number of protons in the nucleus and n is the principal quantum number. 

Clearly, the speed is inversely proportional to the principal quantum number n. So the curve v versus n will be a rectangular hyperbola that resembles (c) in the picture. So option (c) is correct.



 



     8.  As one considers orbits with higher values of n in a hydrogen atom, the electric potential energy of the atom 

(a) decreases

(b) increases

(c) remains the same

(d) does not increase.  


ANSWER: (b).   


EXPLANATION: The electric potential energy of a hydrogen-like atom is given as, 

V =-mZ²e⁴/(4εₒ²h²n²) 

   =-2*13.6/n² eV

{For a hydrogen atom where Z =1}

So with the higher value of n, the electric potential energy of the atom increases. Option (b) is correct. 



   



     9.  The energy of an atom (or ion) in the ground state is -54.4 eV. It may be 

(a) hydrogen

(b) deuterium

(c) He⁺

(d) Li⁺⁺. 


ANSWER: (c).   


EXPLANATION: The total energy of a hydrogen-like atom is, 

E =-13.6Z²/n² 

Given that, E =-54.4 eV, for the ground state, n =1. So

-54.4 = -13.6Z²/1² 

→Z² =4.06

→Z ≈2 

So the atom/(ion) has two numbers of protons in the nucleus. Hence it must be He⁺ ion. Option (c) is correct. 



     



     10.  The radius of the shortest orbit in a one-electron system is 18 pm. It may be

(a) hydrogen

(b) deuterium

(c) He⁺

(d) Li⁺⁺. 


ANSWER: (d).   


EXPLANATION: For a one-electron system ion, the radius of an orbit is given as, 

rₙ =n²aₒ/Z

Where aₒ is the radius of the shortest orbit in a hydrogen atom and its value is 53 pm.

Given for the atom, r₁ =18 pm, n =1, hence,

18 =1²*53/Z

→Z =53/18 ≈ 3.

So it is a Li⁺⁺ ion. Option (d) is correct.




 



     11.  A hydrogen atom in the ground state absorbs 10.2 eV of energy. The orbital angular momentum of the electron is increased by

(a) 1.05x10⁻³⁴ J-s

(b) 2.11x10⁻³⁴ J-s

(c) 3.16x10⁻³⁴ J-s

(d) 4.22x10⁻³⁴ J-s.  


ANSWER: (a).   


EXPLANATION: In the ground state energy of a hydrogen atom, 

E =-13.6/n² eV =-13.6 eV 

In the next excited state (for n =2) the energy,

E' =-13.6/2² eV =-3.4 eV 

The difference in the energies of the two states =E'-E

→ =-3.4 -(-13.6) eV =10.2 eV.

So after absorbing 10.2 eV of energy, the hydrogen atom in the ground state is excited to the next energy state and the electron jumps to the 2nd orbit i.e. n =2. The orbital angular momentum of an electron corresponding to the orbits are

=nh/2π

Since the electron jumps from n =1 to n =2, the orbital angular momentum increases by {2h/2π -h/2π}

=h/2π

=6.63x10⁻³⁴/2π J-s

=1.05x10⁻³⁴ J-s

Option (a) is correct. 



 



     12.  Which of the following parameters are the same for all hydrogen-like atoms and ions in their ground states? 

(a) radius and orbit 

(b) speed of the electron

(c) the energy of the atom

(d) orbital angular momentum of the electron.     


ANSWER: (d).   


EXPLANATION: The radius of an orbit in a hydrogen-like atom in the ground state is 

rₙ =n²aₒ/Z

→r₁ =aₒ/Z. 

It depends on the number of protons in the nucleus. Option (a) is not correct. 


Speed of an electron in the ground state, 

v =Ze²/(2εₒh)

it also depends on Z, so option (b) is not correct.


The energy of the atom in the ground state is =-13.6Z² eV

Again it depends on Z, so option (c) is not correct.


The orbital angular momentum in the ground state is =nh/2π =h/2π. It is a constant. So Option (d) is correct. 


  


   



     13.  In a laser tube, all the photons

(a) have the same wavelength

(b) have same energy

(c) move in the same direction

(d) move with the same speed.  


ANSWER: (d).   


EXPLANATION: The photons in a LASER tube have a very small difference in wavelengths that are called spread Δλ. So their wavelengths and energies are not exactly the same. Options (a) and (b) are not correct.

The directions of the photons are almost parallel but not exactly parallel. So option (c) is also not correct.

Since the photons in the LASER tube are electromagnetic radiation, they all travel with the speed of light. Hence option (d) is correct.    



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Links to the Chapters




CHAPTER- 43- Bohr's Model and Physics of Atom



CHAPTER- 34- Magnetic Field

CHAPTER- 29- Electric Field and Potential











CHAPTER- 28- Heat Transfer

OBJECTIVE -I







EXERCISES - Q51 to Q55


CHAPTER- 27-Specific Heat Capacities of Gases

CHAPTER- 26-Laws of Thermodynamics


CHAPTER- 25-CALORIMETRY

Questions for Short Answer

OBJECTIVE-I

OBJECTIVE-II


EXERCISES - Q-11 to Q-18


CHAPTER- 24-Kinetic Theory of Gases







CHAPTER- 23 - Heat and Temperature






CHAPTER- 17 - Light Waves




CHAPTER- 14 - Fluid Mechanics




CHAPTER- 13 - Fluid Mechanics


CHAPTER- 12 - Simple Harmonic Motion









CHAPTER- 11 - Gravitation





CHAPTER- 10 - Rotational Mechanics






CHAPTER- 9 - Center of Mass, Linear Momentum, Collision


CHAPTER- 8 - Work and Energy

Click here for → Question for Short Answers

Click here for → OBJECTIVE-I

Click here for → OBJECTIVE-II

Click here for → Exercises (1-10)

Click here for → Exercises (11-20)

Click here for → Exercises (21-30)

Click here for → Exercises (31-42)

Click here for → Exercise(43-54)

CHAPTER- 7 - Circular Motion

Click here for → Questions for Short Answer 

Click here for → OBJECTIVE-I

Click here for → OBJECTIVE-II

Click here for → EXERCISES (1-10)

Click here for → EXERCISES (11-20)

Click here for → EXERCISES (21-30)

CHAPTER- 6 - Friction

Click here for → Questions for Short Answer

Click here for → OBJECTIVE-I

Click here for → Friction - OBJECTIVE-II

Click here for → EXERCISES (1-10)

Click here for → Exercises (11-20)

Click here for → EXERCISES (21-31)

For more practice on problems on friction solve these- "New Questions on Friction".

---------------------------------------------------------------------------------

CHAPTER- 5 - Newton's Laws of Motion


Click here for → QUESTIONS FOR SHORT ANSWER

Click here for→Newton's Laws of Motion,Exercises(Q.No. 13 to 27)

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CHAPTER- 4 - The Forces

The Forces-

"Questions for short Answers"    


Click here for "The Forces" - OBJECTIVE-I


Click here for "The Forces" - OBJECTIVE-II


Click here for "The Forces" - Exercises


--------------------------------------------------------------------------------------------------------------

CHAPTER- 3 - Kinematics - Rest and Motion

Click here for "Questions for short Answers"


Click here for "OBJECTIVE-I"


Click here for EXERCISES (Question number 1 to 10)


Click here for EXERCISES (Question number 11 to 20)


Click here for EXERCISES (Question number 21 to 30)


Click here for EXERCISES (Question number 31 to 40)


Click here for EXERCISES (Question number 41 to 52)


CHAPTER- 2 - "Physics and Mathematics"

Click here for "Questions for Short Answers"


Click here for "OBJECTIVE-II"

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