NCERT Solutions for Class 12 Physics Chapter 13 Nuclei

NCERT Solutions For Class 12 Physics Chapter 13 PDF Free Download

NCERT Solutions for Class 12 Physics Chapter 13 Nuclei is provided here with an aim to help students in their CBSE class 12 board exam preparations. All the solutions present in these NCERT solutions are prepared by our subject experts under the guidelines of NCERT and as per the latest CBSE syllabus 2019-20.

NCERT solutions for class 12 physics Chapter 13 Nuclei provided here has answers to the textbook questions along with nuclei class 12 important questions, exemplary problems, worksheets and exercises which will help you gain complete knowledge on the topics Nuclei.

Class 12 Physics NCERT Solutions for Chapter 13 Nuclei

One of the best methods to prepare physics is by taking notes and writing down all the important formulas and points in a copy. The notes will also help students during the time of revision. Once you get an idea about the type of question being asked in examination it is easy for you to collect important points and prepare class 12 physics chapter 13 notes.

Subtopics of class 12  physics chapter 13 Nuclei

  1. Introduction
  2. Atomic Masses And Composition Of Nucleus
  3. Size Of The Nucleus
  4. Mass-energy And Nuclear Binding Energy
    1. Mass – Energy
    2. Nuclear binding energy
  5. Nuclear Force
  6. Radioactivity
    1. Law of radioactive decay
    2. Alpha decay
    3. Beta-decay
    4. Gamma decay
  7. Nuclear Energy
    1. Fission
    2. Nuclear reactor
    3. Nuclear fusion – energy generation in stars
    4. Controlled thermonuclear fusion.

 

Class 12 Physics NCERT Solutions for chapter 13 Nuclei Important Questions


Q 13.1 (a) Lithium has two stable isotopes 36Li_{3}^{6}\textrm{Li} and 37Li_{3}^{7}\textrm{Li} have respective abundances of 7.5% and 92.5%. These isotopes have masses 6.01512 u and 7.01600 u, respectively. Find the atomic mass of lithium.

(b) Two stable isotopes of Boron , 510B_{5}^{10}\textrm{B} and 511B_{5}^{11}\textrm{B} . Their respective masses are 10.01294 u and 11.00931 u, and the atomic mass of boron is 10.811 u. Find the abundances of 510B_{5}^{10}\textrm{B} and 511B_{5}^{11}\textrm{B}.

A1 :

 

(a) Mass of 36Li_{3}^{6}\textrm{Li} lithium isotope , m1 = 6.01512 u

Mass of 37Li_{3}^{7}\textrm{Li} lithium isotope , m = 7.01600 u

Abundance of 36Li_{3}^{6}\textrm{Li} , η1= 7.5%

Abundance of 37Li_{3}^{7}\textrm{Li} , η2= 92.5%

The atomic mass of lithium atom is given as:

 

m=m1n1+m2n2n1+n2m =\frac{ m_1 n_1 + m_2 n_2 }{ n_1 + n_2 } m=6.01512×7.5+7.1600×92.592.5+7.5m =\frac{ 6.01512 \times 7.5 + 7.1600 \times 92.5 }{ 92.5 + 7.5 }

= 6.940934 u

(b) Mass of boron isotope 510B_{5}^{10}\textrm{B} m = 10.01294 u

Mass of boron isotope 511B_{5}^{11}\textrm{B} m = 11.00931 u

Abundance of 510B_{5}^{10}\textrm{B}, η1 = x%

Abundance of 511B_{5}^{11}\textrm{B}, η2= (100 − x)%

Atomic mass of boron, m = 10.811 u The atomic mass of boron atom is given as:

m=m1n1+m2n2n1+n2m =\frac{ m_1 n_1 + m_2 n_2 }{ n_1 + n_2 } 10.811=10.01294×x+11.00931×(100x)x+100x10.811 =\frac{ 10.01294 \times x + 11.00931 \times ( 100 – x ) }{ x + 100 -x }

1081.11 = 10.01294x + 1100.931 – 11.00931 x

x = 19.821/0.99637 = 19.89 %

And 100 − x = 80.11%

Hence, the abundance of 510B_{5}^{10}\textrm{B} is 19.89% and that of 511B_{5}^{11}\textrm{B} is 80.11%.

 

Q 13.2 :The three stable isotopes of neon: 1020Ne_{10}^{20}\textrm{Ne}, 1021Ne_{10}^{21}\textrm{Ne} and 1022Ne_{10}^{22}\textrm{Ne} have respective abundances of 90.51%, 0.27% and 9.22%. The atomic masses of the three isotopes are 19.99 u, 20.99 u and 21.99 u, respectively. Obtain the average atomic mass of neon.

Answer

Atomic mass of 1020Ne_{10}^{20}\textrm{Ne}, m1= 19.99 u

Abundance of 1020Ne_{10}^{20}\textrm{Ne}, η1 = 90.51%

Atomic mass of 1021Ne_{10}^{21}\textrm{Ne}, m2 = 20.99 u

Abundance of 1021Ne_{10}^{21}\textrm{Ne}, η2 = 0.27%

Atomic mass of 1022Ne_{10}^{22}\textrm{Ne}, m3 = 21.99 u

Abundance of 1022Ne_{10}^{22}\textrm{Ne}, η3 = 9.22%

The average atomic mass of neon is given as:

m=m1n1+m2n2+m3n3n1+n2+n3m=\frac{ m_1 n_1 + m_2 n_2 + m_3 n_3 }{ n_1 + n_2 + n_3 }

= 19.99×90.51+20.99×0.27+21.99×9.2290.51+0.27+9.22\frac{ 19.99 \times 90.51 + 20.99 \times 0.27 + 21.99 \times 9.22 }{ 90.51 + 0.27 + 9.22 }

= 20.1771 u

 

Q 13.3: Obtain the binding energy in MeV of a nitrogen nucleus 714N_{7}^{14}\textrm{N}, given m(714N_{7}^{14}\textrm{N})=14.00307 u

Ans:

Atomic mass of nitrogen 714N_{7}^{14}\textrm{N}, m = 14.00307 u

A nucleus of 714N_{7}^{14}\textrm{N}nitrogen contains 7 neutrons and 7 protons.

Hence, the mass defect of this nucleus, ∆m = 7mH + 7mn − m

Where,

Mass of a proton, mH = 1.007825 u

Mass of a neutron, mn = 1.008665 u

∆m = 7 × 1.007825 + 7 × 1.008665 − 14.00307

= 7.054775 + 7.06055 − 14.00307

= 0.11236 u

But 1 u = 931.5 MeV/c2

∆m = 0.11236 × 931.5 MeV/c2

Hence, the binding energy of the nucleus is given as:

Eb = ∆mc2

Where, c =Speed of light

Eb=0.11236×931.5(MeVc2)×c2E_b = 0.11236 \times 931.5 \left ( \frac{MeV}{c^{2}} \right ) \times c^{2}

= 104.66334 Mev

Hence, the binding energy of a nitrogen nucleus is 104.66334 MeV.

 

Q 13.4: Find the binding energy of the nuclei 2356Fe_{23}^{56}\textrm{Fe} and 83209Bi_{83}^{209}\textrm{Bi} in units of MeV from the following

data:

m (2356Fe_{23}^{56}\textrm{Fe}) = 55.934939 u m(83209Bi_{83}^{209}\textrm{Bi}) = 208.980388 u

Ans :Atomic mass of 2356Fe_{23}^{56}\textrm{Fe}, m1 = 55.934939 u

2356Fe_{23}^{56}\textrm{Fe} nucleus has 26 protons and (56 − 26) = 30 neutrons

Hence, the mass defect of the nucleus, ∆m = 26 × mH + 30 × mn − m1

Where,

Mass of a proton, mH = 1.007825 u

Mass of a neutron, mn = 1.008665 u

∆m = 26 × 1.007825 + 30 × 1.008665 − 55.934939

= 26.20345 + 30.25995 − 55.934939

= 0.528461 u

But 1 u = 931.5 MeV/c2

∆m = 0.528461 × 931.5 MeV/c2

The binding energy of this nucleus is given as:

Eb1 = ∆mc2

Where, c = Speed of light

Eb=0.528461×931.5(MeVc2)×c2E_b = 0.528461 \times 931.5 \left ( \frac{MeV}{c^{2}} \right ) \times c^{2}

= 492.26 MeV

Average binding energy per nucleon = 492.2656=8.79MeV\frac{492.26}{56} = 8.79 MeV

Atomic mass of 83209Bi_{83}^{209}\textrm{Bi}, m2 = 208.980388 u

83209Bi_{83}^{209}\textrm{Bi} nucleus has 83 protons and (209 − 83) 126 neutrons.

Hence, the mass defect of this nucleus is given as:

∆m’ = 83 × mH + 126 × mn − m2

Where,

Mass of a proton, mH = 1.007825 u

Mass of a neutron, mn = 1.008665 u

∆m’ = 83 × 1.007825 + 126 × 1.008665 − 208.980388

= 83.649475 + 127.091790 − 208.980388

= 1.760877 u

But 1 u = 931.5 MeV/c2

∆m’ = 1.760877 × 931.5 MeV/c2

Hence, the binding energy of this nucleus is given as:

Eb2 = ∆m’c2

= 1.760877 × 931.5 (MeVc2)×c2 \left ( \frac{MeV}{c^{2}} \right ) \times c^{2}

= 1640.26 MeV

Average binding energy per nucleon = 1640.26/209 = 7.848 MeV

 

Q 13.5: A coin of mass 3.0 g. Find the amount of nuclear energy to separate the neutrons and protons from each other, assuming that the coin has atoms of 2963Cu_{29}^{63}\textrm{Cu} with mass = 62.92960 u.

Ans :

Mass of a copper coin, m’ = 3 g

Atomic mass of 2963Cu_{29}^{63}\textrm{Cu} atom, m = 62.92960 u

The total number of 2963Cu_{29}^{63}\textrm{Cu} atoms in the coin, N=NA×mMass  numberN = \frac{N_A \times m’}{Mass \; number}

Where,

NA = Avogadro’s number = 6.023 × 1023 atoms /g

Mass number = 63 g

N=6.023×1023×363N = \frac{6.023 \times 10^{23}\times 3}{63} = 2.868 x 1022 atoms

2963Cu_{29}^{63}\textrm{Cu} nucleus has 29 protons and (63 − 29) 34 neutrons

Mass defect of this nucleus, ∆m’ = 29 × mH + 34 × mn − m

Where,

Mass of a proton, mH = 1.007825 u

Mass of a neutron, mn = 1.008665 u

∆m’ = 29 × 1.007825 + 34 × 1.008665 − 62.9296

= 0.591935 u

Mass defect of all the atoms present in the coin, ∆m = 0.591935 × 2.868 × 1022

= 1.69766958 × 1022 u

But 1 u = 931.5 MeV/c2

∆m = 1.69766958 × 1022 × 931.5 MeV/c2

Hence, the binding energy of the nuclei of the coin is given as:

Eb = ∆mc2

= 1.69766958 × 1022 × 931.5

= 1.581 × 1025 MeV

But 1 MeV = 1.6 × 10−13 J

Eb = 1.581 × 1025 × 1.6 × 10−13

= 2.5296 × 1012 J

This much energy is required to separate all the neutrons and protons from the given

coin.

 

Q 13.6: Write nuclear reaction equations for

(i) α-decay of88226Ra_{88}^{226}\textrm{Ra}  (ii) α-decay of94242Pu_{94}^{242}\textrm{Pu}

(iii) β−-decay of 1532P_{15}^{32}\textrm{P} (iv) β -decay of83210Bi_{83}^{210}\textrm{Bi}

(v) β+-decay of611C_{6}^{11}\textrm{C} (vi) β -decay of4397Tc_{43}^{97}\textrm{Tc}

(vii) Electron capture of54120Xe_{54}^{120}\textrm{Xe}

Ans:

α is a nucleus of helium 24He_{2}^{4}\textrm{He} and β is an electron (e− for β− and e+ for β+). In every α decay, there is a loss of 2 protons and 4 neutrons. In every β + decay, there is a loss of 1 proton and a neutrino is emitted from the nucleus. In every β – decay, there is a gain of 1 proton and an anti-neutrino is emitted from the nucleus.

For the given cases, the various nuclear reactions can be written as:

(i) 88226Ra  86222Rn  +  24He_{88}^{226}\textrm{Ra} \rightarrow\; _{86}^{222}\textrm{Rn} \; + \; _{2}^{4}\textrm{He}

(ii) 94242 Pu   92238 U   +  24 He _{ 94 }^{ 242 }\textrm{ Pu } \rightarrow\; _{ 92 }^{ 238 }\textrm{ U } \; + \; _{ 2 }^{ 4 }\textrm{ He }

(iii) 1532P  1632 S   +  e+vˉ_{15}^{ 32 }\textrm{P} \rightarrow\; _{ 16 }^{ 32 }\textrm{ S } \; + \; e^- +\bar{v}

(iv) 83210B  84210 PO   +  e+vˉ_{83}^{ 210 }\textrm{B} \rightarrow\; _{ 84 }^{ 210 }\textrm{ PO } \; + \; e^- +\bar{v}

(v) 611C  511 B   +  e++  v_{6}^{ 11 }\textrm{C} \rightarrow\; _{ 5 }^{ 11 }\textrm{ B } \; + \; e^+ + \; v

(vi) 4397Tc  4297 MO   +  e++  v_{43}^{ 97 }\textrm{Tc} \rightarrow\; _{ 42 }^{ 97 }\textrm{ MO } \; + \; e^+ + \; v

(vii) 54120Xe+  e+  53120 I   +  v_{54}^{ 120 }\textrm{Xe}+ \; e^+ \rightarrow\; _{ 53 }^{ 120 }\textrm{ I } \; + \; v

 

Q 13.7: How much time will it take for the activity of a radioactive isotope of half life T years to come down to:

 (a) 3.125% of its original value

(b) 1% of its original value?

Ans:

Half-life of the radioactive isotope = T years

Original amount of the radioactive isotope = N0

(a) After decay, the amount of the radioactive isotope = N

It is given that only 3.125% of N0 remains after decay. Hence, we can write:

NN0=3.125%=3.125100=132\frac{N}{N_0} = 3.125 \% = \frac{3.125}{100}=\frac{1}{32}

But,  NN0=eλt\frac{N}{N_0} = e^{-\lambda t}

Where, λ = Decay

Constant t = Time

eλt=132e^{-\lambda t} = \frac{1}{32} λt=ln1ln32-\lambda t = ln 1 – ln 32 λt=03.4657-\lambda t = 0 – 3.4657

Since, λ=0.693T-\lambda = \frac{0.693}{T}

t = 3.4660.693T  5Tyears\frac{3.466}{\frac{0.693}{T}} \approx \; 5 T years

Hence, the isotope will take about 5T years to reduce to 3.125% of its original value.

(b) After decay, the amount of the radioactive isotope = N

It is given that only 1% of N0 remains after decay. Hence, we can write:

NN0=1%=1100\frac{N}{N_0} = 1 \% = \frac{1}{100} NN0=eλt\frac{N}{N_0} = e^{-\lambda t} eλt=1100e^{-\lambda t} = \frac{1}{100} λt=ln1ln100-\lambda t = ln1 – ln100 eλt=e^{-\lambda t} = 0 – 4.6052

t = 4.6052/λ\lambda

Since, λ = 0.693/T

t=4.60520.693Tt = \frac{4.6052}{\frac{0.693}{T}} = 6.645 T years

Hence, the isotope will take about 6.645T years to reduce to 1% of its original value.

 

Q 13.8: The normal activity of living carbon-containing matter is found to be about 15 decays per minute for every gram of carbon. This activity arises from the small proportion of radioactive614C_{6}^{14}\textrm{C}  present with the stable carbon isotope 612C_{6}^{12}\textrm{C} . When the organism is dead, its interaction with the atmosphere (which maintains the above equilibrium activity) ceases and its activity begins to drop. From the known half-life (5730 years) of614C_{6}^{14}\textrm{C} , and the measured activity, the age of the specimen can be approximately estimated. This is the principle of 614C_{6}^{14}\textrm{C} dating used in archaeology. Suppose a specimen from Mohenjodaro gives an activity of 9 decays per minute per gram of carbon. Estimate the approximate age of the Indus-Valley civilisation.

Ans :

Decay rate of living carbon-containing matter, R = 15 decay/min

Let N be the number of radioactive atoms present in a normal carbon- containing matter.

Half life of 614C,T12_{6}^{14}\textrm{C},T_{\frac{1}{2}} = 5730 years

The decay rate of the specimen obtained from the Mohenjo-Daro site:

R’ = 9 decays/min

Let N’ be the number of radioactive atoms present in the specimen during the Mohenjo-Daro period.

Therefore, we can relate the decay constant, λ and time, t as:

NN=RR=eλt\frac{N}{N’}=\frac{R}{R’}= e^{-\lambda t} eλt=915=35e^{-\lambda t} = \frac{9}{15}=\frac{3}{5} λt=loge35=0.5108-\lambda t = log_e\frac{3}{5} = -0.5108 t=0.5108λt = \frac{0.5108}{\lambda}

But λ=0.693T12=0.6935730{\lambda} = \frac{0.693}{T_\frac{1}{2}}=\frac{0.693}{5730} t=0.51080.6935730=4223.5yearst = \frac{0.5108}{\frac{0.693}{5730}}= 4223.5 years

Hence, the approximate age of the Indus-Valley civilization is 4223.5 years.

 

Q 13.9: Calculate the amount of 2760Co_{27}^{60}\textrm{Co} required to create a radioactive source of strength 8.0 mCi considering that the half-life of 2760Co_{27}^{60}\textrm{Co}  is 5.3 years.

Ans:

The strength of the radioactive source is given as:

dNdt=8.0mCi\frac{dN}{dt} = 8.0 mCi

= 8 x 10-3 x 3.7 x 1010

= 29.6 x 107 decay/s

Where,

N = Required number of atoms

Half-life of 2760 Co _{ 27 }^{ 60 }\textrm{ Co } = 5.3 years

= 5.3 × 365 × 24 × 60 × 60

= 1.67 × 108 s

For decay constant λ, we have the rate of decay as:

dNdt=λN\frac{dN}{dt} = \lambda N

Where, λ=0.693T12=0.6931.67×108s1\lambda = \frac{0.693}{T_\frac{1}{2}} = \frac{0.693}{1.67 \times 10^{8}}s^{-1} N=1λdNdtN = \frac{1}{\lambda} \frac{dN}{dt} =29.6×1070.6931.67×108= \frac{29.6 \times 10^{ 7}}{\frac{ 0.693 }{ 1.67 \times 10^{ 8 }}}

= 7.133 x 1016 atoms

For 2760Co_{27}^{60}\textrm{Co} :

Mass of 6.023 × 1023 (Avogadro’s number) atoms = 60 g

Mass of atoms 7.133 x 1016 atoms = =60×7.133×10166.023×1023= \frac{60\times7.133\times10^{16}}{6.023\times10^{23}}

= 7.106 x 10-6 g

Hence, the amount of 2760Co_{27}^{60}\textrm{Co} necessary for the purpose is 7.106 × 10−6 g.

 

Q 13.10 :If 28 years is the half life of 3890 Sr _{ 38 }^{ 90 }\textrm{ Sr }. What will be the disintegration rate of a quantity of 15 mg of said 3890 Sr _{ 38 }^{ 90 }\textrm{ Sr }isotope ?

Ans:r

Half life of 3890 Sr _{ 38 }^{ 90 }\textrm{ Sr }, t12t_{\frac{1}{2}} = 28 years

= 28 × 365 × 24 × 60 × 60

= 8.83 × 108 s

Mass of the isotope, m = 15 mg

90 g of 3890 Sr _{ 38 }^{ 90 }\textrm{ Sr } atom contains 6.023 × 1023 (Avogadro’s number) atoms.

Therefore, 15 mg of 3890 Sr _{ 38 }^{ 90 }\textrm{ Sr } contains :

6.023×1023×15×10×10390\frac{6.023 \times 10^{23} \times 15 \times 10 \times 10^{-3}}{90}

i.e 1.038 x 1020 number of atoms

Rate of disintegration, dNdt=λN\frac{dN}{dt}= \lambda N

Where,

λ\lambda = decay constant = 0.6938.83×108s1\frac{0.693}{8.83 \times 10^{8}}s^{-1} dNdt=0.693×1.0038×10208.83×108\frac{dN}{dt}=\frac{0.693\times 1.0038 \times 10^{20}}{8.83 \times 10^{8}}

= 7.878 x 1010 atoms/s

Hence, the disintegration rate of 15 mg of 3890 Sr _{ 38 }^{ 90 }\textrm{ Sr } is

7.878 × 1010 atoms/s.

 

Q 13.11: Obtain approximately the ratio of the nuclear radii of the gold isotope 79197Au_{ 79 }^{ 197 }\textrm{Au} and the 47107Ag_{ 47 }^{ 107 }\textrm{Ag}  silver isotope .

Ans:

Nuclear radius of the gold isotope 79197Au_{ 79 }^{ 197 }\textrm{Au} = RAu

Nuclear radius of the silver isotope 47107Ag_{ 47 }^{ 107 }\textrm{Ag} = RAg

Mass number of gold, AAu = 197

Mass number of silver, AAg = 107

The ratio of the radii of the two nuclei is related with their mass numbers as :

RAuRAg=(RAuRAg)13\frac{ R_{ Au }}{ R_{ Ag }}=\left (\frac{ R_{ Au }}{ R_{ Ag }} \right )^{\frac{ 1 }{ 3 } } =(197107)13=\left (\frac{ 197 }{ 107 } \right )^{\frac{ 1 }{ 3 } }

= 1.2256

Hence, the ratio of the nuclear radii of the gold and silver isotopes is about 1.23.

 

Q 13.12: Find the Q-value and the kinetic energy of the emitted α-particle in the α-decay of

(a) 88226Ra_{88}^{226}\textrm{Ra}

(b) 86220Rn_{86}^{220}\textrm{Rn}

m(88226 Ra )m ( _{ 88 }^{ 226 }\textrm{ Ra })= 226.02540 u, m(86222 Rn )m ( _{ 86 }^{ 222 }\textrm{ Rn }) = 222.01750 u,

m(86220 Rn )m ( _{ 86 }^{ 220 }\textrm{ Rn })= 220.01137 u, m(84216 Po )m ( _{ 84 }^{ 216 }\textrm{ Po }) = 216.00189 u.

Answer

(a) Alpha particle decay of 88226Ra_{88}^{226}\textrm{Ra} emits a helium nucleus. As a result, its mass number reduces to 222 (226 − 4) and its atomic number reduces to 86 (88 − 2). This is shown in the following nuclear reaction.

88226 Ra 86222 Ra +24 He _{ 88 }^{ 226 }\textrm{ Ra }\rightarrow _{ 86 }^{ 222 }\textrm{ Ra }+_{ 2 }^{ 4 }\textrm{ He }

Q-value of emitted α-particle = (Sum of initial mass − Sum of final mass) c2

Where, c = Speed of light

It is given that :

m(88226 Ra )m ( _{ 88 }^{ 226 }\textrm{ Ra })= 226.02540 u

m(86222 Rn )m ( _{ 86 }^{ 222 }\textrm{ Rn }) = 222.01750 u

m(24 He )m ( _{ 2 }^{ 4 }\textrm{ He }) = 4.002603 u

Q-value = [226.02540 − (222.01750 + 4.002603)] u c2

= 0.005297 u c2

But 1 u = 931.5 MeV/c2

Q = 0.005297 × 931.5 ≈ 4.94 MeV

Kinetic energy of the α-particle = (Mass  number  after  decayMass  number  before  decay)×Q\left ( \frac{Mass \; number \; after \; decay }{Mass \; number \; before \; decay } \right ) \times Q =222226×4.94= \frac{222}{226} \times 4.94 = 4.85 MeV

(b) Alpha particle decay of 86220Rn_{86}^{220}\textrm{Rn}

It is given that:

Mass of 86220Rn_{86}^{220}\textrm{Rn} = 220.01137 u

Mass of 84216Po_{84}^{216}\textrm{Po} = 216.00189 u

Q-value =[ 220.01137 – ( 216.00189 + 4.00260 ) ] x 931.5

≈ 641 MeV

Kinetic energy of the α-particle = =(2204220)×6.41= \left( \frac{ 220 – 4 }{220} \right ) \times 6.41

= 6.29 MeV

 

Q 13.13: The radionuclide 11C decays according to 611C511B+e++v  :T12  =  20.3  min_{6}^{11}\textrm{C}\rightarrow _{5}^{11}\textrm{B} + e^{+} + v \; : T_{\frac{1}{2}} \; = \; 20.3 \; min. The maximum energy of the emitted positron is 0.960 MeV. Given the mass values:

m(611C)m(_{6}^{11}\textrm{C}) = 11.011434 u and m(511B)m(_{5}^{11}\textrm{B}) = 11.009305 u.

Calculate Q and compare it with the maximum energy of the positron emitted

Ans:

The given nuclear reaction is:

611C511B+e++v_{6}^{11}\textrm{C}\rightarrow _{5}^{11}\textrm{B} + e^{+} + v

Half life of 611C_{6}^{11}\textrm{C} nuclei, T1/2 = 20.3 min

Atomic mass of m(611C)m(_{6}^{11}\textrm{C}) = 11.011434 u

Atomic mass of m(511B)m(_{5}^{11}\textrm{B}) = 11.009305 u

Maximum energy possessed by the emitted positron = 0.960 MeV

The change in the Q-value (∆Q) of the nuclear masses of the 611C_{6}^{11}\textrm{C} ΔQ=[m(611C)[m(511B+me)]]c2\Delta Q = [m(_{6}^{11}\textrm{C}) – [m'(_{5}^{11}\textrm{B}+m_e)]]c^2               —–(1)

Where,

me = Mass of an electron or positron = 0.000548 u

c = Speed of light

m’ = Respective nuclear masses

If atomic masses are used instead of nuclear masses, then we have to add 6 me in the

case of 11C_{}^{11}\textrm{C} and 5 me in the case of 11B_{}^{11}\textrm{B}.

= 11.011434 u

Hence, equation (1) reduces to:

ΔQ=[m(611rmC)m(511rmB)2me]c2\Delta Q = [m(_{6}^{11}\text rm{C}) – m(_{5}^{11}\text rm{B})- 2m_e]c^{2}

Here, m(611rmC)m(_{6}^{11}\text rm{C}) and m(511rmB)m(_{5}^{11}\text rm{B}) are the atomic masses.

∆Q = [11.011434 − 11.009305 − 2 × 0.000548] c2

= (0.001033 c2 ) u

But 1 u = 931.5 Mev/c2

∆Q = 0.001033 × 931.5 ≈ 0.962 MeV

The value of Q is almost comparable to the maximum energy of the emitted positron.

 

Q 13.14: The nucleus 1023Ne_{10}^{23}\textrm{Ne} decays β\beta ^- by emission. Write down theβ\beta  decay equation and determine the maximum kinetic energy of the electrons emitted. Given that:

m(1023Ne)m(_{10}^{23}\textrm{Ne}) = 22.994466 u

m(1123Na)m(_{11}^{23}\textrm{Na}) = 22.989770 u.

Ans:

In β\beta ^- emission, the number of protons increases by 1, and one electron and an antineutrino is emitted from the parent nucleus.

β\beta ^- emission from the nucleus 1023Ne_{10}^{23}\textrm{Ne}.

1023 Ne 1123 Na +e+vˉ+Q_{ 10 }^{ 23 }\textrm{ Ne }\rightarrow _{ 11 }^{ 23 }\textrm{ Na } + e^{ – } + \bar{ v } + Q

It is given that:

Atomic mass of m(1023Ne)m(_{10}^{23}\textrm{Ne})= 22.994466 u

Atomic mass of m(1123Na)m(_{11}^{23}\textrm{Na}) = 22.989770 u

Mass of an electron, me = 0.000548 u

Q-value of the given reaction is given as:

Q=[m(1023Ne)[m(1123Na)+me]]c2Q = [m(_{ 10 }^{ 23 }\textrm{Ne}) – [m(_{ 11 }^{ 23 }\textrm{Na}) + m_e]]c^{ 2 }

There are 10 electrons and 11 electrons in 1023Ne_{ 10 }^{ 23 }\textrm{Ne} and 1123Na_{ 11 }^{ 23 }\textrm{Na} respectively. Hence, the mass of the electron is cancelled in the Q-value equation.

Q  = [ 22.994466 – 22.989770 ] c2

= (0.004696 c2) u

But 1 u = 931.5 Mev/c2

Q = 0.004696 x 931.5 = 4.374 MeV

The daughter nucleus is too heavy as compared to e and vˉ\bar{v}. Hence, it carries negligible energy. The kinetic energy of the antineutrino is nearly zero. Hence, the maximum kinetic energy of the emitted electrons is almost equal to the Q-value, i.e., 4.374 MeV.

 

Q 13.15: The Q value of a nuclear reaction A + b → C + d is defined by

Q = [ mA+ mb− mC− md]c2 where the masses refer to the respective nuclei. Determine

from the given data the Q-value of the following reactions and state whether the reactions

are exothermic or endothermic.

(i) 11H+13H12 H +12H_{ 1 }^{ 1 }\textrm{H} + _{ 1 }^{ 3 }\textrm{H}\rightarrow _{ 1 }^{ 2 }\textrm{ H } + _{ 1 }^{ 2 }\textrm{H}

(ii) 612C+612C1020 Ne +24H_{ 6 }^{ 12 }\textrm{C} + _{ 6 }^{ 12 }\textrm{C}\rightarrow _{ 10 }^{ 20 }\textrm{ Ne } + _{ 2 }^{ 4 }\textrm{H}

Atomic masses are given to be

m(12H)=  2.014102  um( _{ 1 }^{ 2 }\textrm{H} ) = \; 2.014102 \; u

m(13H)=  3.016049  um( _{ 1 }^{ 3 }\textrm{H} ) = \; 3.016049 \; u

m(612C)=  12.00000  um( _{ 6 }^{ 12 }\textrm{C} ) = \; 12.00000 \; u

m(1020Ne)=  19.992439  um( _{ 10 }^{ 20 }\textrm{Ne} ) = \; 19.992439 \; u

Ans

(i) The given nuclear reaction is:

11H+13H12 H +12H_{ 1 }^{ 1 }\textrm{H} + _{ 1 }^{ 3 }\textrm{H}\rightarrow _{ 1 }^{ 2 }\textrm{ H } + _{ 1 }^{ 2 }\textrm{H}

It is given that:

Atomic mass m(11H)=  1.007825  um( _{ 1 }^{ 1 }\textrm{H} ) = \; 1.007825 \; u

Atomic mass m(13H)=  3.016049  um( _{ 1 }^{ 3 }\textrm{H} ) = \; 3.016049 \; u

Atomic mass m(12H)=  1.007825  um( _{ 1 }^{ 2 }\textrm{H} ) = \; 1.007825 \; u

According to the question, the Q-value of the reaction can be written as :

Q = [m(11H)m( _{ 1 }^{ 1 }\textrm{H} ) + m(13H)m( _{ 1 }^{ 3 }\textrm{H} ) – 2mm(12H)m( _{ 1 }^{ 2 }\textrm{H} ) ] c2

                        = [ 1.007825 + 3.016049 – 2 x 2.014102] c2

Q = ( – 0.00433 c2) u

But 1 u = 931.5 MeV/c2

Q = -0.00433 x 931.5 = – 4.0334 MeV

The negative Q-value of the reaction shows that the reaction is endothermic.

(ii) The given nuclear reaction is:

612C+612C1020 Ne +24H_{ 6 }^{ 12 }\textrm{C} + _{ 6 }^{ 12 }\textrm{C}\rightarrow _{ 10 }^{ 20 }\textrm{ Ne } + _{ 2 }^{ 4 }\textrm{H}

It is given that:

Atomic mass of m(612C)=  12.00000  um( _{ 6 }^{ 12 }\textrm{C} ) = \; 12.00000 \; u

Atomic mass of m(1020Ne)=  19.992439  um( _{ 10 }^{ 20 }\textrm{Ne} ) = \; 19.992439 \; u

Atomic mass of m(24He)=  4.002603  um( _{ 2 }^{ 4 }\textrm{He} ) = \; 4.002603 \; u

The Q-value of this reaction is given as :

Q = [ m(612C)m( _{ 6 }^{ 12 }\textrm{C} ) m(1020Ne)m( _{ 10 }^{ 20 }\textrm{Ne} )m(24He)=  4.002603  um( _{ 2 }^{ 4 }\textrm{He} ) = \; 4.002603 \; u ] c2

= [ 2 x 12.0 – 19.992439 – 4.002603 ] c2

                        = [ 0.004958 c2] u

= 0.004958 x 931.5 = 4.618377 Mev

The positive Q-value of the reaction shows that the reaction is exothermic.

 

Q 13.16: Suppose, we think of fission of a2656Fe_{ 26 }^{ 56 }\textrm{Fe} nucleus into two equal fragments, 1328Al_{ 13 }^{ 28 }\textrm{Al} . Is the fission energetically possible? Argue by working out Q of the process. Given

m(2656Fe)=  55.93494  um( _{ 26 }^{ 56 }\textrm{Fe} ) = \; 55.93494 \; u

m(1328Al)=  27.98191  um( _{ 13 }^{ 28 }\textrm{Al} ) = \; 27.98191 \; u

Ans:

The fission of can 2656Fe_{ 26 }^{ 56 }\textrm{Fe} be given as :

2656Fe_{ 26 }^{ 56 }\textrm{Fe} -> 2 1328Al_{ 13 }^{ 28 }\textrm{Al}

It is given that:

Atomic mass of m(2656Fe)=  55.93494  um( _{ 26 }^{ 56 }\textrm{Fe} ) = \; 55.93494 \; u

Atomic mass of m(1328Al)=  27.98191  um( _{ 13 }^{ 28 }\textrm{Al} ) = \; 27.98191 \; u

The Q-value of this nuclear reaction is given as :

Q = [m(2656rmFe)2m(1328rmAl)m( _{ 26 }^{ 56 }\text rm{Fe} ) – 2m( _{ 13 }^{ 28 }\text rm{Al} ) ] c2

= [ 55.93494 – 2 x 27.98191 ]c2

= ( -0.02888 c2) u

But 1 u = 931.5 MeV/c2

Q = – 0.02888 x 931.5 = -26.902 MeV

The Q-value of the fission is negative. Therefore, the fission is not possible energetically.

For an energetically-possible fission reaction, the Q-value must be positive.

 

Q 13.17: The fission properties94239 Pu _{ 94 }^{ 239 }\textrm{ Pu } of are very similar to those of92235 U _{ 92 }^{ 235 }\textrm{ U } .The average energy released per fission is 180 MeV. How much energy is released if all the atoms in 1 kg of pure 94239 Pu _{ 94 }^{ 239 }\textrm{ Pu } undergo fission ?

Ans:

Average energy released per fission of 94239 Pu _{ 94 }^{ 239 }\textrm{ Pu } , Eav  = 180Mev

Amount of pure 94239 Pu _{ 94 }^{ 239 }\textrm{ Pu }, m = 1 kg = 1000 g

NA= Avogadro number = 6.023 × 1023

Mass number of 94239 Pu _{ 94 }^{ 239 }\textrm{ Pu } = 239 g

1 mole of 94239 Pu _{ 94 }^{ 239 }\textrm{ Pu } contains NA atoms.

Therefore, mg of 94239 Pu _{ 94 }^{ 239 }\textrm{ Pu } contains (NAMass  Number×m)\left ( \frac{N_A}{Mass \; Number}\times m \right ) atoms

6.023×1023239×1000=2.52×1024\frac{6.023 \times 10^{23}}{239}\times 1000 = 2.52 \times 10^{ 24 } atoms

Total energy released during the fission of 1 kg of 94239 Pu _{ 94 }^{ 239 }\textrm{ Pu } is calculated as:

E = Eav x 2.52 x 1024

= 180 x 2.52 x 1024

                         = 4.536 x 1026 MeV

Hence, 4.536 x 1026 MeV is released if all the atoms in 1 kg of pure 94239 Pu _{ 94 }^{ 239 }\textrm{ Pu } undergo fission.

 

Q 13.18: A 1000 MW fission reactor consumes half of its fuel in 5.00 y. How much 92235 U _{ 92 }^{ 235 }\textrm{ U }  did it contain initially? Assume that the reactor operates 80% of the time, that all the energy generated arises from the fission of 92235 U _{ 92 }^{ 235 }\textrm{ U }  and that this nuclide is consumed only by the fission process.

Answer

Half life of the fuel of the fission reactor, t12=5t_{\frac{1}{2}} = 5 years

= 5 × 365 × 24 × 60 × 60 s

We know that in the fission of 1 g of 92235 U _{ 92 }^{ 235 }\textrm{ U } nucleus, the energy released is equal to 200MeV.

1 mole, i.e., 235 g of 92235 U _{ 92 }^{ 235 }\textrm{ U } contains 6.023 × 1023 atoms.

1 g contains 92235 U _{ 92 }^{ 235 }\textrm{ U } => 6.023×1023235\frac{6.023 \times 10^{23}}{235} atoms contains

The total energy generated per gram of 92235 U _{ 92 }^{ 235 }\textrm{ U } is calculated as:

E=6.023×1023235×200MeV/gE = \frac{6.023 \times 10^{23}}{235} \times 200 MeV/g MeV/g

=200×6.023×1023×1.6×1019×106235= \frac{200 \times 6.023 \times 10^{ 23 }\times 1.6 \times 10^{-19}\times 10^{6}}{235}

= 8.20 x 1016 J/g

The reactor operates only 80% of the time.

Hence, the amount of 92235 U _{ 92 }^{ 235 }\textrm{ U } consumed in 5 years by the 1000 MW fission reactor is

calculated as:

=5×80×60×60×365×24×1000×106100×8.20×1010= \frac{5 \times 80 \times 60 \times 60 \times 365\times 24 \times 1000 \times 10^6}{100\times 8.20 \times 10^{10}} 1538\approx 1538 Kg

Initial amount of 92235 U _{ 92 }^{ 235 }\textrm{ U } = 2 × 1538 = 3076 kg

 

Q 13.19: How long can an electric lamp of 100W be kept glowing by fusion of 2.0 kg of deuterium?

Take the fusion reaction as

12 H +12H13 He +n+3.27MeV_{ 1 }^{ 2 }\textrm{ H } + _{ 1 }^{ 2 }\textrm{H} \rightarrow _{ 1 }^{ 3 }\textrm{ He } + n + 3.27 MeV

Ans :

The given fusion reaction is:

12 H +12H13 He +n+3.27MeV_{ 1 }^{ 2 }\textrm{ H } + _{ 1 }^{ 2 }\textrm{H} \rightarrow _{ 1 }^{ 3 }\textrm{ He } + n + 3.27 MeV

Amount of deuterium, m = 2 kg

1 mole, i.e., 2 g of deuterium contains 6.023 × 1023 atoms.

2.0 kg of deuterium contains =6.023×10232×2000=6.023×1023= \frac{6.023 \times 10^{23}}{2}\times 2000 = 6.023 \times 10^{23}

It can be inferred from the given reaction that when two atoms of deuterium fuse, 3.27 MeV energy is released.

Total energy per nucleus released in the fusion reaction:

E=3.272×6.023×1026MeVE = \frac{3.27}{2}\times 6.023 \times 10^{26} MeV E=3.272×6.023×1026×1.6×1019×106E = \frac{3.27}{2}\times 6.023 \times 10^{26} \times 1.6 \times 10^{-19} \times 10^{6}

= 1.576 x 1014 J

Power of the electric lamp, P = 100 W = 100 J/s

Hence, the energy consumed by the lamp per second = 100 J

The total time for which the electric lamp will glow is calculated as:

= 1.576×1014100s\frac{1.576 \times 10^{14}}{100} s

= 1.576×1014100s\frac{1.576 \times 10^{14}}{100} s years

 

Q 13.20: Calculate the height of the potential barrier for a head on collision of two deuterons. (Hint: The height of the potential barrier is given by the Coulomb repulsion between the two deuterons when they just touch each other. Assume that they can be taken as hard spheres of radius 2.0 fm.)

Ans :

When two deuterons collide head-on, the distance between their centers, d is given as:

Radius of 1st deuteron + Radius of 2nd deuteron

Radius of a deuteron nucleus = 2 fm = 2 × 10−15 m

d = 2 × 10−15 + 2 × 10−15 = 4 × 10−15 m

Charge on a deuteron nucleus = Charge on an electron = e = 1.6 × 10−19 C

Potential energy of the two-deuteron system:

v=e24πϵ0dv = \frac{ e^{ 2 } }{ 4 \pi \epsilon_0 d }

Where,

ϵ0\epsilon_0= Permittivity of free space

14πϵ0=9×109Nm2C2\frac{1}{4 \pi \epsilon_0} = 9 \times 10^{9} Nm^2C^{-2}

Therefore,

V=9×109×(1.6×1019)24×1015JV = \frac{9 \times 10^{9} \times (1.6 \times 10^{-19})^2}{4 \times 10^{-15} } J V=9×109×(1.6×1019)24×1015×(1.6×1019)eVV = \frac{9 \times 10^{9} \times (1.6 \times 10^{-19})^2}{4 \times 10^{-15} \times (1.6 \times 10^{-19})}eV

= 360 keV

Hence, the height of the potential barrier of the two-deuteron system is 360 keV.

 

Q 13.21: From the relation R = R0A1/3, where R0 is a constant and A is the mass number of a nucleus, show that the nuclear matter density is nearly constant (i.e. independent of A).

Ans :

We have the expression for nuclear radius as:

R = R0A1/3

Where,

R0 = Constant.

A = Mass number of the nucleus

Nuclear matter density, ρ=Mass  of  the  nucleusVolume  of  the  nucleus\rho = \frac{Mass \; of \; the \; nucleus}{Volume \; of \; the \; nucleus}

Let m be the average mass of the nucleus.

Hence, mass of the nucleus = mA

Therefore , ρ=mA43πR3\rho = \frac{ mA }{\frac{ 4 }{ 3 }\pi R^{ 3 }}

= 3mA4π(R0A13)3\frac{ 3mA }{4 \pi( R_0 A^{ \frac{1}{3} })^{3}}

= 3mA4πR03A\frac{ 3mA }{4 \pi R_0^3 A}

= 3m4πR03\frac{ 3m }{4 \pi R_0^3 }

Hence, the nuclear matter density is independent of A. It is nearly constant.

 

Q 13.22: For the β+\beta^{+} (positron) emission from a nucleus, there is another competing process known as electron capture (electron from an inner orbit, say, the K−shell, is captured by the nucleus and a neutrino is emitted).

e++ZAXZ1AY+ve^+ + _{Z}^{A}\textrm{X} \rightarrow _{ Z-1 }^{A}\textrm{Y}+v

 Show that ifβ+\beta^{+} emission is energetically allowed, electron capture is necessarily allowed but not vice−versa.

Ans :

Let the amount of energy released during the electron capture process be Q1. The

nuclear reaction can be written as:

e++ZAXZ1AY+v+Q1e^+ + _{Z}^{A}\textrm{X} \rightarrow _{ Z-1 }^{A}\textrm{Y} + v +Q_1      —–(1)

Let the amount of energy released during the positron capture process be Q2. The nuclear reaction can be written as:

e++ZAXZ1AY+v+Q2e^+ + _{Z}^{A}\textrm{X} \rightarrow _{ Z-1 }^{A}\textrm{Y} + v +Q_2      —–(2)

mn(ZAX)=Nuclear  mass  of  ZAXm_n(_{Z}^{A}\textrm{X}) = Nuclear \; mass \; of \;_{Z}^{A}\textrm{X} mn(Z1AX)=Nuclear  mass  of  Z1AYm_n(_{Z-1}^{A}\textrm{X}) = Nuclear \; mass \; of \;_{Z-1}^{A}\textrm{Y} m(ZAX)=Atomic  mass  of  ZAXm(_{Z}^{A}\textrm{X}) = Atomic \; mass \; of \;_{Z}^{A}\textrm{X} m(Z1AX)=Atomic  mass  of  Z1AYm(_{Z-1}^{A}\textrm{X}) = Atomic \; mass \; of \;_{Z-1}^{A}\textrm{Y}

me = Mass of an electron

c = Speed of light

Q-value of the electron capture reaction is given as:

Q1 = [mn(ZAX)m_n(_{Z}^{A}\textrm{X}) + memn(Z1AY)m_n(_{Z-1}^{A}\textrm{Y}) ] c2

= [ mn(ZAX)m_n(_{Z}^{A}\textrm{X}) – Zme + mn(Z1AY)m_n(_{Z-1}^{A}\textrm{Y}) – ( Z – 1 )me ] c2

                        = [mn(ZAX)m_n(_{Z}^{A}\textrm{X})mn(Z1AY)m_n(_{Z-1}^{A}\textrm{Y}) ] c2           —–(3)

Q – value of the positron capture reaction is given as :

Q2 = [ mn(ZAX)m_n(_{Z}^{A}\textrm{X})mn(Z1AY)m_n(_{Z-1}^{A}\textrm{Y}) -me ] c2

= [ mn(ZAX)m_n(_{Z}^{A}\textrm{X}) – Zme mn(Z1AY)m_n(_{Z-1}^{A}\textrm{Y}) +( Z – 1 )me – me ] c2

                        = [mn(ZAX)m_n(_{Z}^{A}\textrm{X})mn(Z1AY)m_n(_{Z-1}^{A}\textrm{Y}) -2me  ] c2              —–(4)

It can be inferred that if Q2 > 0, then Q1 > 0; Also, if Q1> 0, it does not necessarily mean that Q2 > 0.

In other words, this means that if β+\beta ^+ emission is energetically allowed, then the electron capture process is necessarily allowed, but not vice-versa. This is because the Q-value must be positive for an energetically allowed nuclear reaction.

 

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