# NCERT Solutions for Class 12 Physics Chapter 13 Nuclei

NCERT solutions for class 12 Physics chapter 13 Nuclei: Do you know that the size of an atom is 10000 times the size of a nucleus? But the nucleus contains 99.9% of the mass of an atom. We know that an atom has a structure. Does the nucleus also have a structure? If so what are the constituents and how they are arranged? All these questions are answered in solutions of NCERT class 12 physics chapter 13 nuclei. This chapter comes under modern physics and you can expect at least one question for board exam from CBSE NCERT solutions for class 12 physics chapter 13 nuclei. Learning the solutions of NCERT is important to score well in the board exam. Certain important formulas required for NCERT solutions for class 12 physics chapter 13 nuclei are listed below.

## Important formulas:

• Radii of the nuclei, $R=R_0A^{\frac{1}{3}}$

Where A is the mass number and $R_0=1.2fm$

• Mass defect: $\Delta M=(Zm_p+(A-Z)m_n)-M$

Here Z is the atomic number, M is the mass of the nucleus and A is the mass number.    This equation tells that the mass of the nucleus is always less than the mass of their constituents.

• Another important relation which helps in CBSE NCERT solutions for class 12 physics chapter 13 nuclei is Einstein's mass-energy relation.

$E_b=\Delta Mc^2$

Where c is the speed of light.

• Another main concept of NCERT grade 12 chapter 13 nuclei is the law of radioactive decay. This is given by $N=N_0e^{-\lambda t}$

Where N is the number of nuclei at any time t, $N_0$ is the number of nuclei at any time $t_0$ and lambda is disintegration constant.

• The half-life of a radionuclide is given by $T_{\frac{1}{2}}=\frac{ln2}{\lambda}$

Where lambda is the disintegration constant.

After completing all these topics try to do NCERT class 12 chapter 13 exercises. If you are unable to solve or have any doubts refer to the solutions of NCERT class 12 physics chapter 13 nuclei provided below.

## NCERT solutions for class 12 physics chapter 13 nuclei exercises

Mass of the two stable isotopes and their respective abundances are $6.01512 \; u$ and $7.01600 \; u$ and $7.5\; ^{o}/_{o}$ and $92.5\; ^{o}/_{o}$.

$m=\frac{6.01512\times7.5+7.01600\times92.5}{100}$

m=6.940934 u

The atomic mass of boron is 10.811 u

Mass of the two stable isotopes are  $10.01294 \; u$ and $11.00931\; u$ respectively

Let the two isotopes have abundances x% and (100-x)%

$\\10.811=\frac{10.01294\times x+11.00931\times(100-x)}{100} \\x=19.89\\ 100-x=80.11$

Therefore the abundance of  $_{5}^{10}\textrm{B}$  is 19.89% and that of  $_{5}^{11}\textrm{B}$ is 80.11%

The atomic masses of the three isotopes are 19.99 u(m1), 20.99 u(m2) and 21.99u(m3)

Their respective abundances are 90.51%(p1), 0.27%(p2) and 9.22%(p3)

$\\m= \frac{19.99\times 90.51+20.99\times 0.27+21.99\times 9.22}{100}\\m=20.1771u$

The average atomic mass of neon is 20.1771 u.

mn = 1.00866 u

mp= 1.00727 u

Atomic mass of Nitrogen m= 14.00307 u

Mass defect $\Delta$m=7$\times$mn+7$\times$m- m

$\Delta$m=7$\times$1.00866+7$\times$1.00727 - 14.00307

$\Delta$m=0.10844

Now 1u is equivalent to 931.5 MeV

Eb=0.10844$\times$931.5

Eb=101.01186 MeV

Therefore binding energy of a Nitrogen nucleus is 101.01186 MeV.

$(i)m (_{26}^{56}\textrm{Fe})=55.934939\; \; u$

mH = 1.007825 u

mn = 1.008665 u

The atomic mass of $_{26}^{56}\textrm{Fe}$ is m=55.934939 u

Mass defect

$\Delta m=(56-26)\times$$m_H+26\times m_p - m$

$\Delta m=30\times1.008665+26\times1.007825 - 55.934939$

$\Delta$m=0.528461

Now 1u is equivalent to 931.5 MeV

Eb=0.528461$\times$931.5

Eb=492.2614215 MeV

Therefore the binding energy of a $_{26}^{56}\textrm{Fe}$ nucleus is 492.2614215 MeV.

Average binding energy

$=\frac{492.26}{56}MeV=8.79 MeV$

## Q. 13.4 (ii) Obtain the binding energy of the nuclei $_{26}^{56}\textrm{Fe}$ and $_{83}^{209}\textrm{Bi}$ in units of MeV from the following data:

$(ii)m(_{83}^{209}\textrm{Bi})=208.980388\; \; u$

mH = 1.007825 u

mn = 1.008665 u

The atomic mass of $_{83}^{209}\textrm{Bi}$ is m=208.980388 u

Mass defect

$\Delta m=(209-83)\timesmn+83\times m_H - m$

$\Delta$m=126$\times$1.008665+83$\times$1.007825 - 208.980388

$\Delta$m=1.760877 u

Now 1u is equivalent to 931.5 MeV

Eb=1.760877 $\times$931.5

Eb=1640.2569255 MeV

Therefore the binding energy of a $_{83}^{209}\textrm{Bi}$ nucleus is 1640.2569255 MeV.

$Average\ binding\ energy=\frac{1640.25}{208.98}=7.84MeV$

Mass of the coin is w = 3g

Total number of Cu atoms in the coin is n

$\\n=\frac{w\times N_{A}}{Atomic\ Mass}\\ n=\frac{3\times 6.023\times 10^{23}}{62.92960}$

n=2.871$\times$1022

mH = 1.007825 u

mn = 1.008665 u

Atomic mass of $_{29}^{63}\textrm{Cu}$ is m=62.92960 u

Mass defect $\Delta$m=(63-29)$\times$mn+29$\times$m- m

$\Delta$m=34$\times$1.008665+29$\times$1.007825 - 62.92960

$\Delta$m=0.591935 u

Now 1u is equivalent to 931.5 MeV

Eb=0.591935$\times$931.5

Eb=551.38745 MeV

Therefore binding energy of a $_{29}^{63}\textrm{Cu}$ nucleus is 551.38745 MeV.

The nuclear energy that would be required to separate all the neutrons and protons from each other is

n$\times$Eb=2.871$\times$1022$\times$551.38745

=1.5832$\times$1025 MeV

=1.5832$\times$1025$\times$1.6$\times$10-19$\times$106J

=2.5331$\times$109 kJ

Q.13.6 (i)   Write nuclear reaction equations for

$(i)\; \alpha -decay\; of\; _{88}^{226}\textrm{Ra}$

The nuclear reaction equations for the given alpha decay

$_{88}^{226}\textrm{Ra}\rightarrow _{86}^{222}\textrm{Rn}+_{2}^{4}\textrm{He}$

Q.13.6 (ii) Write nuclear reaction equations for

$(ii)\; \alpha -decay\; of\; _{94}^{242}\textrm{Pu}$

The nuclear reaction equations for the given alpha decay is

$_{94}^{242}\textrm{Pu}\rightarrow _{92}^{238}\textrm{U}+_{2}^{4}\textrm{He}$

Q.13.6 (iii) Write nuclear reaction equations for

$(iii)\; \beta ^{-} -\: decay\; of\; _{15}^{32}\textrm{P}$

The nuclear reaction equations for the given beta minus decay is

$_{15}^{32}\textrm{P}\rightarrow _{16}^{32}\textrm{S}+e^{-}+\bar{\nu }$

Q.13.6 (iv) Write nuclear reaction equations for

$(iv)\; \beta ^{-} -\: decay\; of\; _{83}^{210}\textrm{Bi}$

The nuclear reaction equation for the given beta minus decay is

$_{83}^{210}\textrm{Bi}\rightarrow _{84}^{210}\textrm{Po}+e^{-}+\bar{\nu }$

Q.13.6 (v) Write nuclear reaction equations for

$(v)\; \beta ^{+} -\: decay\; of\; _{6}^{11}\textrm{C}$

The nuclear reaction for the given beta plus decay will be

$_{6}^{11}\textrm{C}\rightarrow _{5}^{11}\textrm{P}+e^{+}+\nu$

Q.13.6 (vi)  Write nuclear reaction equations for

$(vi)\; \beta ^{+} -\: decay\; of\; _{43}^{97}\textrm{Tc}$

nuclear reaction equations for

$\beta ^{+} -\: decay\; of\; _{43}^{97}\textrm{Tc}\ is$

$_{43}^{97}\textrm{Tc}\rightarrow _{42}^{97}\textrm{Mo}+e^{+}+\nu$

Q.13.6 (vii) Write nuclear reaction equations for

Electron capture of $_{54}^{120}\textrm{Xe}$

The nuclear reaction  for electron capture of   $_{54}^{120}\textrm{Xe}$  is

$_{54}^{120}\textrm{Xe}+e^{-}\rightarrow _{53}^{120}\textrm{I}+\nu$

(a) The activity is proportional to the number of radioactive isotopes present

The number of half years in which the number of radioactive isotopes reduces to x% of its original value is n.

$n=log_{2}(\frac{100}{x})$

In this case

$n=log_{2}(\frac{100}{3.125})=log_{2}32=5$

It will take 5T years to reach 3.125% of the original activity.

(b) In this case

$n=log_{2}(\frac{100}{1})=log_{2}100=6.64$

It will take 6.64T years to reach 1% of the original activity.

Since we know that activity is proportional to the number of radioactive isotopes present in the sample.

$\frac{R}{R_{0}}=\frac{N}{N_{0}}=\frac{9}{15}=0.6$

Also

$N=N_{0}e^{-\lambda t}$

$\\t=-\frac{1}{\lambda }ln\frac{N}{N_{0}}\\ t=-\frac{1}{\lambda }ln0.6\\ t=\frac{0.51}{\lambda }$

but $\lambda = \frac{0.693}{T_{1/2}}$

Therefore

$\\t=0.51\times \frac{T_{1/2}}{0.693}\\ \\t=0.735T_{1/2}$

$\\t\approx 4217$

The age of the Indus-Valley civilisation calculated using the given specimen is approximately 4217 years.

Required activity=8.0 mCi

1 Ci=3.7$\times$1010 decay s-1

8.0 mCi=8$\times$10-3$\times$3.7$\times$1010 =2.96$\times$108 decay s-1

T1/2=5.3 years

$\lambda =\frac{0.693}{T_{1/2}}$

$\lambda =\frac{0.693}{5.3\times 365\times 24\times 3600}$

$\lambda =4.14\times 10^{-9}\ s^{-1}$

$\\\frac{\mathrm{d} N}{\mathrm{d} t}=-N\lambda \\ N=-\frac{\mathrm{d} N}{\mathrm{d} t}\times \frac{1}{\lambda }\\ N=-(-2.96\times 10^{8})\times \frac{1}{4.14\times 10^{-9}}\\ N=7.15\times 10^{16}\ atoms$

Mass of those many atoms of Cu will be

$w=\frac{7.15\times 10^{16}\times 60}{6.023\times 10^{23}}$

$w=7.12\times10^{-6} g$

7.12$\times$10-6  g of $_{27}^{60}\textrm{Co}$  is necessary to provide a radioactive source of 8.0 mCi strength.

T1/2=28 years

$\\\lambda =\frac{0.693}{28\times 365\times 24\times 3600}\\ \lambda =7.85\times 10^{-10} \ decay\ s^{-1}$

The number of atoms in 15 mg of  $_{38}^{90}\textrm{Sr}$ is

$N=\frac{15\times 10^{-3}\times 6.023\times 10^{23}}{90}$

N=1.0038$\times$1020

The disintegration rate will be

$\frac{\mathrm{d} N}{\mathrm{d} t}=-N\lambda$

=-1.0038$\times$1020$\times$7.85$\times$10-10

=-7.88$\times$1010 s-1

The disintegration rate is therefore 7.88$\times$1010 decay s-1.

The nuclear radii are directly proportional to the cube root of the mass number.

The ratio of the radii of the given isotopes is therefore

$\left ( \frac{197}{107} \right )^{1/3}=1.23$

$(a)\; _{88}^{226}\textrm{Ra}$

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

$m(_{86}^{222}\textrm{Rn})=220.01137\; u,$     $m(_{84}^{216}\textrm{Po})=216.00189\; u,$

Mass defect is $\Delta$m

$\Delta m=m(_{88}^{226}\textrm{Ra})-m(_{86}^{222}\textrm{Rn})-m(_{2}^{4}\textrm{He})$

$\Delta$m=226.02540-222.0175-4.002603

$\Delta$m=0.005297 u

1 u = 931.5 MeV/c2

Q-value=$\Delta$m$\times$931.5

=4.934515 MeV

By using Linear Momentum Conservation and Energy Conservation

The kinetic energy of alpha particle =

$\frac{mass\ of\ nucleus\ after\ decay}{mass\ of\ nucleus\ before\ decay}\times Q-value$

=$\frac{222.01750}{226.0254}\times 4.934515$

=4.847 MeV

## Q.13.12 (b)  Find the Q-value and the kinetic energy of the emitted $\alpha$ -particle in the $\alpha$-decay of

$(b)\; _{86}^{220}\textrm{Rn}$

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

$m(_{86}^{222}\textrm{Rn})=220.01137\; u,$     $m(_{84}^{216}\textrm{Po})=216.00189\; u,$

Mass defect is $\Delta$m

$\Delta m=m(_{86}^{222}\textrm{Rn})-m(_{84}^{216}\textrm{Po})-m(_{2}^{4}\textrm{He})$

$\Delta$m=220.01137-216.00189-4.002603

$\Delta$m=0.006877 u

1 u = 931.5 MeV/c2

Q-value=$\Delta$m$\times$931.5

=6.406 MeV

By using Linear Momentum Conservation and Energy Conservation

The kinetic energy of alpha particle =

$\frac{mass\ of\ nucleus\ after\ decay}{mass\ of\ nucleus\ before\ decay}\times Q-value$

=$\frac{216.00189}{220.01138}\times 6.406$

=6.289 MeV

$_{6}^{11}\textrm{C}\rightarrow B+e^{+}+v:T_{1/2}=20.3\; min$
The maximum energy of the emitted positron is $0.960\; MeV.$.

Given the mass values:

$m(_{6}^{11}\textrm{C})=11.011434\; u$ and $m(_{6}^{11}\textrm{B})=11.009305\; u$
calculate Q and compare it with the maximum energy of the positron emitted.

If we use atomic masses

$\\\Delta m=m(_{6}^{11}\textrm{C})-m(_{5}^{11}\textrm{B})-2m_{e}\\ \Delta m=11.011434-11.009305-2\times 0.000548\\ \Delta m=0.001033u$

Q-value= 0.001033$\times$931.5=0.9622 MeV which is comparable with a maximum energy of the emitted positron.

$(i) m (_{10}^{23}\textrm{Ne} ) = 22.994466 \; u$

The $\beta$ decay equation is

$_{10}^{23}\textrm{Ne}\rightarrow _{11}^{23}\textrm{Na}+e^{-}+\bar{\nu }+Q$

$\\\Delta m=m(_{10}^{23}\textrm{Ne})-_{11}^{23}\textrm{Na}-m_{e}\\ \Delta m=22.994466-22.989770\\ \Delta m=0.004696u$

(we did not subtract the mass of the electron as it is cancelled because of the presence of one more electron in the sodium atom)

Q=0.004696$\times$931.5

Q=4.3743 eV

The emitted nucleus is way heavier than the $\beta$ particle and the energy of the antineutrino is also negligible and therefore the maximum energy of the emitted electron is equal to the Q value.

## Q. 13.15 (i) The Q value of a nuclear reaction $A + b \rightarrow C + d$ is defined by                                                                    $Q=[m_{A}+m_{b}-m_{c}-m_{d}]c^{2}$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.

$(i) _{1}^{1}\textrm{H}+_{1}^{3}\textrm{H}\rightarrow _{1}^{2}\textrm{H}+_{1}^{2}\textrm{H}$the following

Atomic masses are given to be

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

$m(_{1}^{3}\textrm{H})=3.0016049\; u$

$m(_{6}^{12}\textrm{H})=12.000000\; u$

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

$\\\Delta m=m(_{1}^{1}\textrm{H})+m(_{1}^{3}\textrm{H})-2m(_{1}^{2}\textrm{H})\\ \Delta m=1.007825+3.0016049-2\times 2.014102\\ \Delta m=-0.00433$

The above negative value of mass defect implies there will be a negative Q value and therefore the reaction is endothermic

## Q. 13.15 (ii) The Q value of a nuclear reaction $A + b \rightarrow C + d$ is defined by                                                                    $Q=[m_{A}+m_{b}-m_{c}-m_{d}]c^{2}$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.

$(ii) _{6}^{12}\textrm{C}+_{6}^{12}\textrm{C}\rightarrow _{10}^{20}\textrm{Ne}+_{2}^{4}\textrm{He}$

Atomic masses are given to be

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

$m(_{1}^{3}\textrm{H})=3.0016049\; u$

$m(_{6}^{12}\textrm{H})=12.000000\; u$

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

$\\\Delta m=2m(_{6}^{12}\textrm{C})-m(_{10}^{20}\textrm{Ne})-m(_{2}^{4}\textrm{He})\\ \Delta m=2\times 12.00000-19.992439-4.002603\\ \Delta m=0.004958$

The above positive value of mass defect implies Q value would be positive and therefore the reaction is exothermic

The reaction will be $_{26}^{56}\textrm{Fe}\rightarrow _{13}^{28}\textrm{Al}+_{13}^{28}\textrm{Al}$

The mass defect of the reaction will be

$\\\Delta m=m(_{26}^{56}\textrm{Fe})-2m( _{13}^{28}\textrm{Al})\\ \Delta m=55.93494-2\times 27.98191\\ \Delta m=-0.02888u$

Since the mass defect is negative the Q value will also negative and therefore the fission is not energetically possible

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

Number of atoms present in 1 kg(w) of  $_{94}^{239}\textrm{Pu}$ =n

$\\n=\frac{w\times N_{A}}{mass\ number\ of\ Pu}\\ n=\frac{1000\times 6.023\times 10^{23}}{239}\\n=2.52\times 10^{24}$

Energy per fission (E)=180 MeV

Total Energy released if all the atoms in 1 kg  $_{94}^{239}\textrm{Pu}$ undergo fission = E $\times$ n

=180 $\times$ 2.52$\times$1024

=4.536$\times$1026 MeV

The amount of energy liberated on fission of 1  $_{92}^{235}\textrm{U}$ atom is 200 MeV.

The amount of energy liberated on fission of 1g  $_{92}^{235}\textrm{U}$

$\\=\frac{200\times 10^{6} \times 1.6\times 10^{-19}\times 6.023\times 10^{23}}{235}\\=8.2\times 10^{10}\ Jg^{-1}$

Total Energy produced in the reactor in 5 years

$\\=1000\times 10^{6}\times 0.8\times 5\times 365\times 24\times 3600\\ =1.261\times 10^{17}\ J$

Mass  of $_{92}^{235}\textrm{U}$ which underwent fission, m

$=\frac{1.261\times 10^{17}}{8.2\times 10^{10}}$

=1537.8 kg

The amount present initially in the reactor = 2m

=2$\times$1537.8

=3075.6 kg

$_{1}^{2}\textrm{H}+_{1}^{2}\textrm{H}\rightarrow _{2}^{3}\textrm{He}+n+3.27\; MeV$

The energy liberated on the fusion of two atoms of deuterium= 3.27 MeV

Number of fusion reactions in 2 kg of deuterium = NA$\times$500

The energy liberated by fusion of 2.0 kg of deuterium atoms E

$\\=3.27\times 10^{6}\times 1.6\times 10^{-19}\times 6.023\times 10^{23}\times 500\\=1.576\times 10^{14}\ J$

Power of lamp (P)= 100 W

Time the lamp would glow using E amount of energy is T=

$\\=\frac{E}{P}\\ =\frac{1.576\times 10^{14}}{100\times 3600\times 24\times 365}$

=4.99$\times$104 years

For a head-on collision of two deuterons, the closest  distances between their centres will be d=2$\times$r

d=2$\times$2.0

d=4.0 fm

d=4$\times$10-15 m

charge on each deuteron = charge of one proton=q =1.6$\times$10-19 C

The maximum electrostatic potential energy of the system during the head-on collision will be E

$\\=\frac{q^{2}}{4\pi \epsilon _{0}d}\\ =\frac{9\times 10^{9}\times (1.6\times 10^{-19})^{2}}{4\times 10^{-15}}\ J\\ =\frac{9\times 10^{9}\times (1.6\times 10^{-19})^{2}}{4\times 10^{-15}\times 1.6\times 10^{-19}}\ eV\\=360\ keV$

The above basically means to bring two deuterons from infinity to each other would require 360 keV of work to be done or would require 360 keV of energy to be spent.

Mass of an element with mass number A will be about A u. The density of its nucleus, therefore, would be

$\\d=\frac{m}{v}\\ d=\frac{A}{\frac{4\pi }{3}R^{3}}\\d=\frac{A}{\frac{4\pi }{3}(R_{0}A^{1/3})^{3}}\\d=\frac{3}{4\pi R{_{0}}^{3}}$

As we can see the above density comes out to be independent of mass number A and R0 is constant, so  matter density is nearly constant

## Q. 13.22  For the $\beta ^{+}$ (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).

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

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

For the electron capture, the reaction would be

$_{Z}^{A}\textrm{X}+e^{-}\rightarrow _{Z-1}^{A}\textrm{Y}+\nu +Q_{1}$

The mass defect and q value of the above reaction would be

$\\\Delta m_{1}=m(_{Z}^{A}\textrm{X})+m_{e}-m(_{Z-1}^{A}\textrm{Y})\\ Q_{1}=([m(_{Z}^{A}\textrm{X})-m(_{Z-1}^{A}\textrm{Y})]+m_{e})c^{2}$

where mN$(_{Z}^{A}\textrm{X})$ and mN$(_{Z-1}^{A}\textrm{Y})$ are the nuclear masses of elements X and Y respectively

For positron emission, the reaction would be

$_{Z}^{A}\textrm{X}\rightarrow _{Z-1}^{A}\textrm{Y}+e^{+}+\bar{\nu }+Q_{2}$

The mass defect and q value for the above reaction would be

$\\\Delta m_{2}=m(_{Z}^{A}\textrm{X})-m(_{Z-1}^{A}\textrm{Y})-m_{e}\\ Q_{2}=([m(_{Z}^{A}\textrm{X})-m(_{Z-1}^{A}\textrm{Y})]-m_{e})c^{2}$

From the above values, we can see that if Q2 is positive Q1 will also be positive but Q1 being positive does not imply that Q2 will also have to positive.

NCERT solutions for class 12 physics chapter 13 nuclei additional exercises

Let the abundances of $_{12}^{25}\textrm{Mg}$ and $_{12}^{26}\textrm{Mg}$ be x and y respectively.

x+y+78.99=100

y=21.01-x

The average atomic mass of Mg is 24.312 u

$\\24.312=\frac{78.99\times 23.98504+x\times 24.98584+(100-x)\times 25.98259}{100}\\ x\approx 9.3\\ y=21.01-x\\ y=21.01-9.3\\ y=11.71$

The abundances of $_{12}^{25}\textrm{Mg}$ and $_{12}^{26}\textrm{Mg}$  are 9.3% and 11.71% respectively

$m(_{20}^{40}\textrm{Ca})=39.962591\; u$

$m(_{20}^{41}\textrm{Ca})=40.962278 \; u$

$m(_{13}^{26}\textrm{Al})=25.986895 \; u$

$m(_{13}^{27}\textrm{Al})=26.981541 \; u$

The reaction showing the neutron separation is

$_{20}^{41}\textrm{Ca}+E\rightarrow _{20}^{40}\textrm{Ca}+_{0}^{1}\textrm{n}$

$\\E=(m(_{20}^{40}\textrm{Ca})+m(_{0}^{1}\textrm{n})-m(_{20}^{41}\textrm{Ca}))c^{2}\\ E=(39.962591+1.008665-40.962278)c^{2}\\ E=(0.008978)u\times c^{2}$

But 1u=931.5 MeV/c2

Therefore E=(0.008978)$\times$931.5

E=8.363007 MeV

Therefore to remove a neutron from the $_{20}^{41}\textrm{Ca}$ nucleus 8.363007 MeV of energy is required

$m(_{20}^{40}\textrm{Ca})=39.962591\; u$

$m(_{20}^{41}\textrm{Ca})=40.962278 \; u$

$m(_{13}^{26}\textrm{Al})=25.986895 \; u$

$m(_{13}^{27}\textrm{Al})=26.981541 \; u$

The reaction showing the neutron separation is

$_{13}^{27}\textrm{Al}+E\rightarrow _{13}^{26}\textrm{Al}+_{0}^{1}\textrm{n}$

$\\E=(m(_{13}^{26}\textrm{Ca})+m(_{0}^{1}\textrm{n})-m(_{13}^{27}\textrm{Ca}))c^{2}\\ E=(25.986895+1.008665-26.981541)c^{2}\\ E=(0.014019)u\times c^{2}$

But 1u=931.5 MeV/c2

Therefore E=(0.014019)$\times$931.5

E=13.059 MeV

Therefore to remove a neutron from the $_{13}^{27}\textrm{Al}$ nucleus 13.059 MeV of energy is required

Let initially there be N1 atoms of $_{15}^{32}\textrm{P}$ and N2 atoms of $_{15}^{33}\textrm{P}$ and let their  decay constants be $\lambda _{1}$ and $\lambda _{2}$ respectively

Since initially the activity of  $_{15}^{33}\textrm{P}$ is 1/9 times that of $_{15}^{32}\textrm{P}$ we have

$N_{1 } \lambda_{1}=\frac{N_{2}\lambda _{2}}{9}$      (i)

Let after time t the activity of  $_{15}^{33}\textrm{P}$ be 9 times that of $_{15}^{32}\textrm{P}$

$N_{1 } \lambda_{1}e^{-\lambda _{1}t}=9N_{2}\lambda _{2}e^{-\lambda _{2}t}$    (ii)

Dividing equation (ii) by (i) and taking the natural log of both sides we get

$\\-\lambda _{1}t=ln81-\lambda _{2}t \\t=\frac{ln81}{\lambda _{2}-\lambda _{1}}$

where $\lambda _{2}=0.048/ day$ and$\lambda _{1}=0.027/ day$

t comes out to be 208.5 days

$_{88}^{223}\textrm{Ra}\rightarrow _{82}^{209}\textrm{Pb}+_{6}^{14}\textrm{C}$

$_{88}^{223}\textrm{Ra}\rightarrow _{86}^{219}\textrm{Rn}+_{2}^{4}\textrm{He}$

Calculate the Q-values for these decays and determine that both are energetically allowed.

$_{88}^{223}\textrm{Ra}\rightarrow _{82}^{209}\textrm{Pb}+_{6}^{14}\textrm{C}$

$\\\Delta m=m(_{88}^{223}\textrm{Ra})-m(_{82}^{209}\textrm{Pb})-m(_{6}^{14}\textrm{C})\\ =223.01850-208.98107-14.00324 \\=0.03419u$

1 u = 931.5 MeV/c2

Q=0.03419$\times$931.5

=31.848 MeV

As the Q value is positive the reaction is energetically allowed

$_{88}^{223}\textrm{Ra}\rightarrow _{86}^{219}\textrm{Rn}+_{2}^{4}\textrm{He}$

$\\\Delta m=m(_{88}^{223}\textrm{Ra})-m(_{86}^{219}\textrm{Rn})-m(_{2}^{4}\textrm{He})\\ =223.01850-219.00948-4.00260 \\=0.00642u$

1 u = 931.5 MeV/c2

Q=0.00642$\times$931.5

=5.98 MeV

As the Q value is positive the reaction is energetically allowed

$m(_{92}^{238}\textrm{U})=238.05079\; u$

$m(_{58}^{140}\textrm{Ce})=139.90543\; u$

$m(_{44}^{99}\textrm{Ru})= 98.90594\; u$

The fission reaction given in the question can be written as

$_{92}^{238}\textrm{U}+_{0}^{1}\textrm{n}\rightarrow _{58}^{140}\textrm{Ce}+_{44}^{99}\textrm{Ru}+10e^{-}$

The mass defect for the above reaction would be

$\Delta m=m_{N}(_{92}^{238}\textrm{U})+m(_{0}^{1}\textrm{n})-m_{N}(_{58}^{140}\textrm{Ce})-m_{N}(_{44}^{99}\textrm{Ce})-10m_{e}$

In the above equation, mN represents nuclear masses

$\\\Delta m=m(_{92}^{238}\textrm{U})-92m_{e}+m(_{0}^{1}\textrm{n})-m(_{58}^{140}\textrm{Ce})+58m_{e}-m(_{44}^{99}\textrm{Ru})+44m_{e}-10m_{e} \\\Delta m=m(_{92}^{238}\textrm{U})+m(_{0}^{1}\textrm{n})-m(_{58}^{140}\textrm{Ce})-m(_{44}^{99}\textrm{Ru})\\ \Delta m=238.05079+1.008665-139.90543-98.90594\\ \Delta m=0.247995u$

but 1u =931.5 MeV/c2

Q=0.247995$\times$931.5

Q=231.007 MeV

Q value of the fission process is 231.007 MeV

$_{1}^{2}\textrm{H}+_{1}^{3}\textrm{H}\rightarrow _{2}^{4}\textrm{He}+n$

(a) Calculate the energy released in MeV in this reaction from the data:

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

The mass defect of the reaction is

$\\\Delta m=m(_{1}^{2}\textrm{H})+m(_{1}^{3}\textrm{H})-m(_{2}^{4}\textrm{He})-m(_{0}^{1}\textrm{n})\\ \Delta m=2.014102+3.016049-4.002603-1.008665\\ \Delta m=0.018883u$

1u = 931.5 MeV/c2

Q=0.018883$\times$931.5=17.59 MeV

## Q.13.28 (b) Consider the D–T reaction (deuterium–tritium fusion)

$_{1}^{2}\textrm{H}+_{1}^{3}\textrm{H}\rightarrow _{2}^{4}\textrm{He}+n$

(b) Consider the radius of both deuterium and tritium to be approximately 2.0 fm. What is the kinetic energy needed to overcome the coulomb repulsion between the two nuclei? To what temperature must the gas be heated to initiate the reaction? (Hint: Kinetic energy required for one fusion event =average thermal kinetic energy available with the interacting particles $= 2(3kT/2)$; k = Boltzman’s constant, T = absolute temperature.)

To initiate the reaction both the nuclei would have to come in contact with each other.

Just before the reaction the distance between their centres would be 4.0 fm.

The electrostatic potential energy of the system at that point would be

$\\U=\frac{q^{2}}{4\pi \epsilon _{0}d}\\ U=\frac{9\times 10^{9}(1.6\times 10^{-19})^{2}}{4\times 10^{-15}}\\U=5.76\times 10^{-14}J$

The same amount of Kinetic Energy K would be required to overcome the electrostatic forces of repulsion to initiate the reaction

It is given that $K=2\times \frac{3kT}{2}$

Therefore the temperature required to initiate the reaction is

$\\T=\frac{K}{3k}\\ =\frac{5.76\times 10^{-14}}{3\times 1.38\times 10^{-23}}\\=1.39\times 10^{9}\ K$

$m(^{198}Au)=197.968233\; u$

$m(^{198}Hg)=197.966760 \; u$

$\gamma _{1}$ decays from 1.088 MeV to 0 V

Frequency of $\gamma _{1}$ is

$\\\nu _{1}=\frac{1.088\times 10^{6}\times 1.6\times 10^{-19}}{6.62\times 10^{-34}}\\ \nu _{1}=2.637\times 10^{20}\ Hz$ Plank's constant, h=6.62$\times$10-34 Js $E=h\nu$

Similarly, we can calculate frequencies of $\gamma _{2}$ and $\gamma _{3}$

$\\\nu _{2}=9.988\times 10^{19}\ Hz\\ \nu _{3}=1.639\times 10^{20}\ Hz$

The energy of the highest level would be equal to the energy released after the decay

Mass defect is

$\\\Delta m=m(_{79}^{196}\textrm{U})-m(_{80}^{196}\textrm{Hg})\\ \Delta m=197.968233-197.966760\\\Delta m=0.001473u$

We know 1u = 931.5 MeV/c2

Q value= 0.001473$\times$931.5=1.3721 MeV

The maximum Kinetic energy of $\beta _{1}^{-}$ would be 1.3721-1.088=0.2841 MeV

The maximum Kinetic energy of $\beta _{2}^{-}$ would be 1.3721-0.412=0.9601 MeV

(a) $_{1}^{1}\textrm{H}$ $_{1}^{1}\textrm{H}+_{1}^{1}\textrm{H}+_{1}^{1}\textrm{H}+_{1}^{1}\textrm{H}\rightarrow _{2}^{4}\textrm{He}$

The above fusion reaction releases the energy of 26 MeV

Number of Hydrogen atoms in 1.0 kg of Hydrogen is 1000NA

Therefore 250NA such reactions would take place

The energy released in the whole process is E1

$\\=250\times 6.023\times 10^{23}\times 26\times 10^{6}\times 1.6\times 10^{-19}\\=6.2639\times 10^{14}\ J$

(b) The energy released in fission of one $_{92}^{235}\textrm{U}$ atom is 200 MeV

Number of $_{92}^{235}\textrm{U}$ atoms present in 1 kg of $_{92}^{235}\textrm{U}$ is N

$\\N=\frac{1000\times 6.023\times 10^{23}}{235}\\ N=2.562\times 10^{24}$

The energy released on fission of N atoms is E2

$\\E=2.562\times 10^{24}\times 200\times 10^{6}\times 1.6\times 10^{-19}\\ E=8.198\times 10^{13}J$

$\frac{E_{1}}{E_{2}}=\frac{6.2639\times 10^{14}}{8.198\times 10^{13}}\approx 8$

Let the amount of energy to be produced using nuclear power per year in 2020 is E

$E=\frac{200000\times 10^{6}\times 0.1\times 365\times 24\times 3600}{0.25}\ J$

(Only 10% of the required electrical energy is to be produced by Nuclear power and only 25% of thermo-nuclear is successfully converted into electrical energy)

Amount of Uranium required to produce this much energy is M

$=\frac{200000\times 10^{6}\times 0.1\times 365\times 24\times 3600\times 235}{0.25\times 200\times 10^{6}\times 1.6\times 10^{-19}\times 6.023\times 10^{23}\times 1000}$    (NA=6.023$\times$1023, Atomic mass of Uranium is 235 g)

=3.076$\times$104 kg

## Chapter list for class 12 physics is given below-

 NCERT solutions for class 12 physics chapter 1 Electric Charges and Fields Solutions of NCERT class 12 physics chapter 2 Electrostatic Potential and Capacitance CBSE NCERT solutions for class 12 physics chapter 3 Current Electricity NCERT solutions for class 12 physics chapter 4 Moving Charges and Magnetism Solutions of NCERT class 12 physics chapter 5 Magnetism and Matter CBSE NCERT solutions for class 12 physics chapter 6 Electromagnetic Induction NCERT solutions for class 12 physics chapter 7 Alternating Current Solutions of NCERT class 12 physics chapter 8 Electromagnetic Waves CBSE NCERT solutions for class 12 physics chapter 9 Ray Optics and Optical Instruments NCERT solutions for class 12 physics chapter 10 Wave Optics Solutions Solutions of NCERT class 12 physics chapter 11 Dual nature of radiation and matter CBSE NCERT solutions for class 12 physics chapter 12 Atoms NCERT solutions for class 12 physics chapter 13 Nuclei Solutions of NCERT class 12 physics chapter 14 Semiconductor Electronics Materials Devices and Simple Circuits

## Significance of NCERT solutions for class 12 physics chapter 13 nuclei:

• About 10% of questions are expected from the chapters atoms and nuclei for CBSE board exams.
• The NCERT solutions for class 12 physics chapter 13 nuclei will help to score well in this chapter.
• The topic of radioactive decay and half-life is important for the board and competitive exams like NEET and JEE Main.
• Sometimes same questions which are discussed in the solutions of NCERT class 12 physics chapter 13 nuclei come in CBSE board exam. Good Luck!