**15 Structure of Atoms and Nuclei**

**Multiple Choice Questions**

**1. Which of the following quantities has same units and dimensions as that of plank’s constant?**

(A) Moment of inertia

(B) Angular momentum

(C) Linear momentum

(D) Rate of change of linear momentum

**Ans.** (B) Angular momentum

**2. The energy of an electron in nth Bohr orbit is proportional to …………………**

(A)

(B)

(C)

(D)

**Ans. **(D)

**3. The radius of first Bohr orbit is 0.53 A.U. and radius of **** Bohr orbit is 212 A.U. The value of ‘ **** ‘ is **

(A) 2

(B) 12

(C) 20

(D) 400

**Ans. **(C) 20

**4. The nuclei having same number of protons but different number of neutrons are called…………………**(A) isobars

(B) – particles

(C) isotopes

(D) -particles

**Ans. **(C) isotopes

**5. When electron in hydrogen atom jumps from second orbit to first orbit, the wavelength of radiation emitted is ****. When electron jumps from third orbit to first orbit, the wavelength of emitted radiation would be…………………**

(A)

(B)

(C)

(D)

**Ans.** (A)

**6. Linear momentum of an electron in Bohr orbit of ****-atom (principal quantum number **** ) is proportional to**

(A)

(B)

(C)

(D)

**Ans. **(B)

**7. The ratio of kinetic energy of an electron in Bohr’s orbit to its total energy in the same orbit is…………………**

(A) -1

(B) 2

(C)

(D) -0.5

**Ans.** (A) -1

**8. In hydrogen atom, electron jumps from the **** orbit to the **** orbit. The change in angular momentum is…………………**

(A)

(B)

(C)

(D)

**Ans. **(B)

**9. Balmer series is obtained when all transitions of electron terminate on…………………**

(A) 2nd orbit

(B) 1 st orbit

(C) 3rd orbit

(D) 4th orbit

**Ans.** (A) 2nd orbit

**10. The radius of eighth orbit of electron in H-atom will be more than that of fourth orbit by a factor of…………………**

(A) 2

(B) 4

(C) 8

(D) 16

**Ans.** (B) 4

**Theory Questions**

**15.5 Atomic Spectra**

**State the name of the visible series in hydrogen spectrum.**

**Ans:** Balmer series.

**15.6 Bohr’s Atomic Model**

**State the postulates of Bohr’s theory of hydrogen atom. Write down necessary equations.**

**Ans:** Bohr’s three postulates are:

i. In a hydrogen atom, the electron revolves round the nucleus in a fixed circular orbit with constant speed.

Equation according to Bohr’s first postulate,

ii. The radius of the orbit of an electron can only take certain fixed values such that the angular momentum of the electron in these orbits is an integral multiple of being the Planck’s constant.

Equation according to Bohr’s second postulate,

iii. An electron can make a transition from one of its orbits to another orbit having lower energy. In doing so, it emits a photon of energy equal to the difference in its energies in the two orbits.

Equation according to Bohr’s third postulate,

where,

Energy of electron in higher orbit

Energy of electron in lower orbit

**Derive an expression for the total energy of electron in ‘****‘ th Bohr orbit. Hence show that energy of the electron is inversely proportional to the square of principal quantum number. Also, define binding energy.**

**Ans:** Expression for total energy of electron:

i. Kinetic energy:

Let, mass of electron

According to Bohr’s first postulate,

where, is permittivity of free space.

The revolving electron in the circular orbit has linear speed and hence it possesses kinetic energy.

It is given by,

K.E ….[From equation (1)]

K.E

ii. Potential energy:

Potential energy of electron is given by,

where,

electric potential at any point due to charge on nucleus charge on electron.

In this case,

P.E

P.E

P.E

iii. Total energy:

The total energy of the electron in any orbit is its sum of P.E and K.E.

T.E

…. [From equations (2) and (3)]

T.E

iv. But,

Substituting for in equation (4),

T.E

T.E

This is required expression for energy of electron in orbit of Bohr’s hydrogen atom.

v. The negative sign in equation (5) shows that the electron is bound to the nucleus by an attractive force and hence energy must be supplied to the electron in order to make it free from the influence of the nucleus.

vi. As , and in equation (5) are constant,

constant

Using equation (5),

T.E constant

T.E

Binding energy:

Binding energy of an electron is the minimum energy required to make it free from the nucleus.

**Obtain an expression for the radius of Bohr orbit for****-atom.**

**Ans:** Expression for radius of Bohr orbit in atom:

i. Let, mass of electron,

charge on electron,

radius of Bohr’s orbit,

charge on nucleus,

linear.velocity of electron in orbit,

number of electrons in an atom,

principal quantum number.

ii. From Bohr’s first postulate,

Coulomb’s force Centripetal force

iii. According to Bohr’s second postulate,

iv. From equations (1) and (2),

This is the required expression for radius of orbit.

**State Bohr’s third postulate for hydrogen****atom.**

**Ans:** An electron can make a transition from one of its orbits to another orbit having lower energy. In doing so, it emits a photon of energy equal to the difference in its energies in the two orbits.

**Derive Bohr’s formula for the wave number.**

**Ans:**

i. Let, Energy of electron in higher orbit Energy of electron in lower orbit

ii. According to Bohr’s third postulate,

iii. But

iv. From equations (1), (2) and (3),

where, speed of electromagnetic radiation

v. But, Rydberg’s constant

Equation (4) represents Rydberg’s formula for atomic spectrum.

vi. is called wave number of the line.

**Draw a neat, labelled energy level diagram for****atom showing the transitions.**

**Explain the series of spectral lines for **** atom, whose fixed inner orbit numbers are 3 and 4 respectively.**

**Ans:**

**Paschen series:**

i. The spectral lines which correspond to the transition of an electron from some higher energy state to 3rd orbit are termed as Paschen series.

ii. For Paschen series, and

The wave numbers and the wavelengths of the spectral lines constituting the Paschen series are given by,

iii. Paschen series lies in the infrared region of the spectrum which is invisible and contains infinite number of lines.

iv. Wavelengths for and 5 are and respectively.

**Brackett series:**

i. The spectral lines which correspond to the transition of an electron from a higher energy state to the orbit are termed as Brackett series.

ii. For this series, and

The wave numbers and the wavelengths of the spectral lines constituting the Brackett series are given by,

iii. This series lies in the near infrared region of the spectrum and contains infinite number of lines. Wavelengths for and 6 , are and 26253 Å respectively.

**Draw a neat and labelled energy level diagram and explain Balmer series and Brackett series of spectral lines for hydrogen atom.**

**Ans:** Refer subtopic 15.6: Q. No. 6 for diagram and explanation of Brackett series.

Balmer series:

i. The spectral lines of this series correspond to the transition of an electron from some higher energy state to orbit.

ii. For Balmer series, and ,

The wave numbers and the wavelengths of spectral lines constituting the Balmer series are given by,

iii. There are infinite number lines in this series out of which four lines are seen called .

iv. This series lies in the visible region. Wavelengths for and 4 are and respectively.

**Obtain an expression for energy of an electron in Bohr orbit. Hence obtain the expression for its binding energy.**

**Ans:** Refer subtopic 15.6: Q. No. 2 for expression for energy of an electron in Bohr orbit.

Expression for binding energy of electron in Bohr orbit:

iii. When binding energy is supplied to an electron, the total energy of the system containing nucleus and electron becomes zero.

B.E + T.E

B.E T.E

iv. But T.E

B.

B.E

**What is the expression for minimum angular momentum of electron in hydrogen atom?**

**Ans:** Minimum angular momentum of electron in hydrogen atom is .

**State any two postulates of Bohr’s theory of hydrogen atom.**

**Ans:** Refer subtopic 15.6: Q. No. 1 (Statements only)

[Any two]

**What is the mathematical formula for third postulate of Bohr’s atomic model? [Mar 22]**

**Ans:** Bohr’s third postulate: Where, Energy of electron in higher orbit, Energy of electron in lower orbit.

**Derive an expression for the radius of the****Bohr orbit of the electron in hydrogen atom.**

**Ans:** Refer subtopic 15.6: Q. No. 3

**State the first and second postulate of Bohr’s atomic model.**

**Ans:** Refer subtopic 15.6: Q. No. I[(i) and (ii)] (Statements only)

**State postulates of Bohr’s atomic model.**

**Ans:** Refer subtopic 15.6: Q. No. 1 (Statements only)

**15.8 Nuclear Binding Energy**

**With the help of a neat labelled diagram, describe the Geiger-Marsden experiment. What is mass defect?**

**Ans:** Geiger-Marsden experiment:

i. The experimental arrangement is as shown in the figure.

ii. In this experiment, a narrow beam of -particles from radioactive source was incident on a gold foil.

iii. The scattered particles produced scintillations on the surrounding screen.

iv. The scintillations were observed through a microscope which could be moved to cover different angles with respect to the incident beam. Observations:

i. Most alpha particles passed straight through the foil.

ii. A few were deflected (scattered) through various scattering angles.

iii. Only about of the incident alpha particles were scattered through angles larger than .

iv. About one alpha particle in 8000 was deflected through angle larger than and a fewer still were deflected through angles as large as .

Mass defect:

The difference between the actual mass of the nucleus and the sum of masses of constituent nucleons is called mass defect.

**15.10 Law of Radioactive Decay**

**State the law of radioactive decay. Hence derive the expression****where symbols have their usual meanings.**

**Ans: Statement:**

The number of nuclei undergoing the decay per unit time is proportional to the number of unchanged nuclei present at that instant.

Derivation:

If ‘ ‘ is the number of parent nuclei present at any instant ‘ ‘, ‘ ‘ is the number of nuclei disintegrated in short interval of time ‘ ‘, then,

where, is known as decay constant or disintegration constant.

The negative sign indicates disintegration of atoms.

Integrating both sides of equation,

where, is number of parent atoms at time .

This is the decay law of radioactivity.

**State law of radioactive decay. Hence derive the relation****.**

**Ans:** Refer subtopic 15.10: Q. No. 1 for statement and derivation of law of radioactive decay.

**15.11 Nuclear Energy**

**Write notes on –**

**i. Nuclear fission**

**ii. Nuclear fusion**

**Ans:**

i. Nuclear fission:

a. The process of splitting a heavy nucleus into two lighter nuclei after bombardment with release of energy is called nuclear fission.

b. Example: undergoes fission producing barium and krypton . In the process, it releases 2 neutrons ( ) and energy.

i.e.,

c. Energy generated,

where,

d. The energy produced in the fission is in the form of kinetic energy of the products, i.e., in the form of heat which can be collected and converted to other forms of energy as needed.

ii. Nuclear fusion:

a. The nuclear reaction in which two lighter nuclei are fused to form a heavier nucleus is called nuclear fusion.

b. If any two of the lighter nuclei come sufficiently close, within about one of each other, then they can undergo nuclear reaction and form a heavier nucleus.

c. For two bare nuclei to come into proximity overcoming repulsive force between their positive charges, requires high energy. Hence, fusion process requires very high temperature.

d. The newly formed nucleus has smaller mass than the sum of masses of fused nuclei.

e. Example: At centre of Sun at temperature of four hydrogen nuclei (i.e., protons) fuse to form a helium nucleus. The effective reaction can be written as

neutrinos

f. As mass defect is converted into energy,

energy value,

where,

For given example, ignoring energy taken by neutrinos energy value,

**Numericals**

**15.6 Bohr’s Atomic Model**

**Find the energy of an electron in second Bohr orbit of hydrogen atom.**

**[Energy of an electron in the first Bohr orbit **** ]**

**Solution:**

Given:

To find:

Formula:

Calculation: From formula,

Ans: Energy of an electron in the third orbit is .

**Second member of Balmer series of hydrogen atom has wavelength 4860 A.U. Calculate Rydberg’s constant. Hence calculate energy in****when electron is orbiting in third Bohr orbit. [Plank’s constant****. Speed of light in vacuum****]**

**Solution:**

Given:

To find: i. Rydberg’s constant

ii. Energy in

Formulae: i.

ii.

Calculation: For the line of Balmer series,

and

Using formula (i)

Now, using formulae (ii) and (iii) we get,

Ans: i. The value of Rydberg’s constant is .

ii. Energy in of electron in the third orbit is .

**The velocity of electron in the first Bohr orbit of radius 0.5 A.U. is****. Calculate the period of revolution of electron in the same orbit.**

**Solution:**

Given: ,

To find: Period of revolution (T)

Formula:

Calculation: Using formula (i) we get,

Ans: Period of revolution of electron in the first Bohr orbit is .

**Find the value of energy of electron in****in the third Bohr orbit of hydrogen atom.**

**[Rydberg’s constant ****, Planck’s constant ****, Velocity of light in air ****.]**

**Solution:**

Given:

To find: Energy of electron in orbit

Formula: (in )

Calculation: From Formula,

Ans: The value of energy of electron in the third Bohr orbit of hydrogen atom is .

**Calculate the radius of second Bohr orbit in hydrogen atom from the given data.**

**Mass of electron **

**Charge on the electron **

**Planck’s constant **

**Permittivity of free space **

**Solution:**

Given: ,

To find: Radius of Bohr orbit

Formula:

Calculation: From formula,

Ans: Radius of second Bohr orbit is .

**An electron is orbiting in****Bohr orbit. Calculate ionisation energy for this atom, if the ground state energy is****.**

**Solution:**

Given:

To find: Ionisation energy

Formulae: i. ii. Ionisation energy

Calculation: Using formula (i),

Using formula (ii),

Ionisation energy

Ans: Ionisation energy for given atom is .

**Obtain expressions for longest and shortest wavelength of spectral lines in ultraviolet region for hydrogen atom.**

**Solution:**

Given: For longest wavelength , for shortest wavelength

To find: i. Longest wavelength ii. Shortest wavelength

Formula:

Calculation: The ultraviolet region for hydrogen atom lies in the Lyman series.

For the longest wavelength , from formula,

For Lyman series shortest wavelength , from formula,

Ans: The expression for longest and shortest wavelength of spectral lines in ultraviolet region for hydrogen atom is and respectively

**Find the frequency of revolution of an electron in Bohr’s****orbit; if the radius and speed of electron**

**Solution:**

in that orbit is and respectively.

Given:

To find:

Frequency of revolution

Formula:

Calculation: From formula,

Ans: The frequency of revolution of electron in Bohr orbit is .

**Find the ratio of longest wavelength in Paschen series to shortest wavelength in Balmer series.**

**Solution:**

Let, shortest wavelength

Longest wavelength in Paschen series is obtained when

For longest wavelength,

Shortest wavelength in Balmer series is obtained when

For shortest wavelength,

Ans: The ratio of longest wavelength in Paschen series to shortest wavelength in Balmer series is 5.131.

**The electron in the hydrogen atom is moving with a speed of****in an orbit of radius****.**

**Solution:**

Calculate the period of revolution of electron.

Given:

To find:

Formula:

Calculation: Using formula,

Ans: Period of revolution of electron is .

**The energy of an excited hydrogen atom is****. Find the angular momentum of the electron.**

**Solution:**

Given:

To find:

Formulae:

**Calculation: Using formula (i)**

ii.

Ans: The angular momentum of the electron in the fourth orbit in a hydrogen atom is .

**Calculate the wavelength of****line and series limit for Brackett series.**

**Solution:**

In hydrogen spectrum, line corresponds to line of Balmer series .

Wavelength of line is given by formula,

For series limit of Brackett series,

and ,

Ans: The wavelengths corresponding to line and series limit of Brackett series are and respectively.

**Energy of an electron in the second Bohr orbit is****. Calculate the energy of an electron in the third Bohr orbit.**

**Solution:**

Given:

To find: Energy of electron in third orbit

Formula: .

Calculation: From formula,

Ans: Energy of an electron in the third orbit is .

**Determine the shortest wavelengths of Balmer and Paschen series. Given the limit for Lyman series is 912 À.**

**Solution:**

Given:

To find: Shortest wavelength of

i. Balmer series

ii. Paschen series

Formula:

Calculation: From formula,

For shortest wavelength,

For Lyman series,

i. For Balmer series,

ii. For Paschen series,

Ans: The shortest wavelength of

i. Balmer series is .

ii. Paschen series is .

**Compute the ratio of longest wavelengths of Lyman and Balmer series in hydrogen atom.**

**Solution:**

For hydrogen:

For longest wavelength of Lyman series, and

For longest wavelength of Balmer series, and

**15.9 Radioactive Decays**

**Thorium****is disintegrated into lead****. Find the number of****and****particles emitted in disintegration.**

**Solution:**

When an -particle is emitted by an atom its atomic number decreases by 2 and mass number decreases by 4 .

Whereas, when a – particle is emitted by an atom its atomic number increases by 1 and mass number does not change.

When is disintegrated to , its mass number is changed due to emission of -particles.

Number of – particles emitted

Due to emission of – particles atomic number of has decreased to .

But atomic number of Th is 82 . The difference is due to emission – particles.

Number of – particles emitted

**15.10 Law of Radioactive Decay**

**The decay constant of radioactive substance is****per year. Calculate its half life period.**

**Solution:**

Given: year

To find: Half life period )

Formula:

Calculation: From formula,

Ans: Half-life period of a radioactive element is 1601 years.

**The half-life of a nuclear species is****years. Calculate its decay constant per year.**

**Solution:**

Decay constant per year, per year

**Disintegration rate of a radio-active sample is****per hour at 20 hours from the start. It reduces to****per hour after 30 hours. Calculate the decay constant.**

**Solution:**

Given:

for for

To find:

Formula:

Decay constant

Calculation: From formula (i),

,

10

,

Dividing

Taking natural logarithm on both the sides,

per hour.

Ans: Decay constant is per hour.