Question 1
When a nucleus in an atom undergoes a radioactive decay, the electronic energy levels of the atom
(a) do not change for any type of radioactivity .
(b) change for alpha and beta radioactivity but not for gamma-radioactivity.
(c) change for alpha-radioactivity but not for others.
(d) change for beta-radioactivity but not for others. Solution
(b)
Question 2
What is the relationship between mean life and half life of the radioactive substance
(a) $t_m= 1.443 T_{1/2}$
(b) $T_{1/2}= 1.443 t_m$
(c) $T_{1/2}= 1.413 t_m$
(d) None of the these Solution
(a)
Question 3
How is the radius of nucleus of related to Mass Number?
(a) $R= R_0 A^3$
(b) $R= R_0 A^{2/3}$
(c) $R= R_0 A^{1/3}$
(d) None of the these Solution
(c)
Question 4
Heavy stable nuclei have more neutrons than protons. This is because of the fact that
(a) neutrons are heavier than protons.
(b) nuclear forces between neutrons are weaker than that between protons.
(c) neutrons decay into protons through beta decay.
(d) electrostatic force between protons are repulsive. Solution
(d)
Question 5
x and y denote the atomic masses of the parent and the daughter nuclei respectively in a radioactive decay. The Q-value for a $\beta ^{�}$
decay is $Q_1$ and that for a $\beta ^{+}$ decay is $Q_2$. If z denotes the mass of an electron, then which of the following statements is correct?
(a) $Q_1 = (x - y) c^2$ and $Q_2 = (x - y - 2z)c^2$
(b) $Q_1 = (x - y) c^2$ and $Q_2 = (x - y )c^2$
(c) $Q_1 = (x - y - 2z)c^2$ and $Q_2 = (x - y +2z)c^2$
(d) $Q_1 = (x - y + 2z)c^2$ and $Q_2 = (x - y +2z)c^2$ Solution
(a)
Question 6
Which of following is false
a. 1 Bq=1 disintegration/seconds
b. 1 curie = $3 \times 10^7$ decays/sec
c. 1 Rutherford= $10^6$ decays/sec
d. None of these Solution
(d)
Question 7
In a sample of radioactive material, what percentage of the initial number of active nuclei will decay during one mean life (a) 63%
(b) 69%
(c) 50%
(d) 33% Solution
Number of atoms remains undecayed $N=N_0 e^{-\lambda t}$
Number of atoms decayed is given by = Initial Atoms - Atoms remaining
=$N_0(1-e^{-\lambda t}) =N_0(1-e^{-\lambda \times 1/\lambda})=N_0(1-\frac {1}{e})=0.63N_0$
Therefore 63% is decayed
Hence (a) is correct
Question 8
A nucleus of $^{210}_{84}Po$ originally at rest emits $\alpha$ particle with speed $v$. What will be the recoil speed of the daughter nucleus ?
(a) v/210
(b) 2v/103
(c) 2v/105
(d) v/206 Solution
Using conservation of linear moments
$206v_1+4v=0$
$v_1=-\frac {4v}{206}=-\frac {2v}{103}$
Hence (b) is correct
Question 9
Nuclear matter density is given by
(a) $ \frac {m}{4 \pi R_0^3}$
(b) $ \frac {3m}{4 \pi R_0^3}$
(c) $ \frac {m}{2 \pi R_0^3}$
(d) $ \frac {3mA}{4 \pi R_0^3}$ Solution
(b)
Question 10
A radioactive substance decays to 1/32th of its initial activity in 25 days. Find its half life
a. 6 days
b. 2 days
c. 5 days
d. 7 days Solution
Here
$R= \frac {R_0}{16}$
Now
$\frac {R}{R_0} = (\frac {1}{2})^n$
Therefore
$(\frac {1}{32})= (\frac {1}{2})^n$
n= 4
Half Life Period= Time of disintegration/Number of Half Lives = 25/5 = 5 days
Hence (c) is correct
Question 11
A radioactive substance has a half life period of 30 days. What is the disintegration constants?
a. .0231
b. .3
c. .54
d. None of these Solution
(a)
$\lambda = \frac {.693}{T_{1/2}}= .0231$
Question 12
How is rutherford and curie related?
a. $1 \; curie= 3.7 \times 10^4 \; rutherford$
b. $1 \; curie= 1.7 \times 10^4 \; rutherford$
c. $1 \; curie= 3.7 \times 10^5 \; rutherford$
d. $1 \; curie= 1 \times 10^4 \; rutherford$ Solution
(a)
Question 13
which of these is not deflected by electric Field?
a. $\alpha$ rays
b. $\beta$ rays
c. $\gamma$ rays
d. None of these Solution
(c)
Question 14
In the below nuclear reaction
$^1_1H \rightarrow ^1_0n + ^P_QX$
which of these are the values of P and Q
a. (0,1)
b. (1,1)
c. (1,0)
d. None of these Solution
Using laws of conservation of mass and charge,we get
P+1=1 or P=0
Q+0=1 or Q=1
Hence (a) is correct
Question 15
A radioactive nucleus undergoes a series of decays according to the sequence
$A \overset{\beta}{\rightarrow} A_1 \overset{\alpha}{\rightarrow} A_2 \overset{\alpha}{\rightarrow} A_3$
If the mass number and atomic number of $A_3$ are 172 and 69 respectively then which of the following is true
(a) Atomic Number and Mass Number of A are 72 and 180
(b) Atomic Number and Mass Number of $A_2$ are 71 and 176
(c) Atomic Number and Mass Number of $A_1$ are 73 and 180
(d) All the above Solution
$(^{180}_{72}A) \xrightarrow[^0_{-1}e]{\beta} (^{180}_{73}A_1) \xrightarrow[^4_{2}He]{\alpha} (^{176}_{71}A_2) \xrightarrow[^4_{2}He]{\alpha} (^{172}_{79}A_3)$
Therefore all the above is correct
Question 16
In alpha decay, the neutron to proton ratio
a. Decreases
b. Increases
c. Remains same
d. Can decrease or increase Solution
Consider the alpha decay
$^{238}_{92}U \overset{\alpha}{\rightarrow} ^{234}_{90}Th$
neutron to proton ratio before decay
$=\frac {238-92}{92}= \frac {146}{92}$
neutron to proton ratio After decay
$=\frac {234-90}{90}= \frac {144}{90}$
Clearly $\frac {144}{90} > \frac {146}{92}$
Thus neutron to proton ratio increased during the alpha decay
Hence (b) is correct
Question 17
In beta decay, the neutron to proton ratio
a. Decreases
b. Increases
c. Remains same
d. Can decrease or increase Solution
Consider the beta decay
$^{210}_{83}Bi \overset{\beta}{\rightarrow} ^{210}_{84}Po + ^{0}_{-1}e$
neutron to proton ratio before decay
$=\frac {210-83}{83}= \frac {127}{83}$
neutron to proton ratio After decay
$=\frac {210-84}{84}= \frac {126}{84}$
Clearly $\frac {127}{83} > \frac {126}{84}$
Thus neutron to proton ratio decreases during the beta decay
Hence (a) is correct
Question 18
Number and type of nucleons in the nucleus of Deuterium ( $^2_1H$) will be
(a) 2 Protons
(b) 1 proton and 1 neutron
(c) 2 neutrons
(d) 1 proton and 1 electron Solution
(b)
Question 19
In the radioactive series below
$^A_ZX \rightarrow ^A_{Z+1}Y \rightarrow ^{A-4}_{Z-1} X \rightarrow ^{A-4}_{Z-1} X$
(a) $\alpha$, $\beta$, $\gamma$
(b) $\beta$,$\alpha$, $\gamma$
(c) $\gamma$,$\alpha$,$\beta$
(d) $\gamma$,$\beta$,$\alpha$ Solution
(b)
Question 20
if $\frac {1 \; amu}{1 \; KWH} =x$,then
(a) x =1
(b) x > 1
(c) x < 1
(d) none of the above Solution
(c)
Question 21
In a radioactive decay, neither the atomic number nor the mass number changes. Which of the following would be emitted in the decay process
(a) Proton
(b) Neutron
(c) Electron
(d) Photon Solution
(d)
Question 22
Decay constant of radium is $\lambda$. By a suitable process its compound radium bromide is obtained. The decay constant of radium bromide will be
(a) $\lambda$
(b) More than $\lambda$
(d) Less than $\lambda$
(d) Zero Solution
(a)
Question 23
After 1 $\alpha$ and 2$\beta$ emissions
(a) Mass number reduces by 3
(b) Mass number reduces by 4
(c) Mass number reduces by 6
(d) Atomic number remains unchanged Solution
(b)
Question 24
A radioactive substance emits
(a) alpha-rays
(b) beta-rays
(c) gamma-rays
(d) All of these Solution
(d)
Text Based Questions(long)
Question 1
Draw a graph showing the variation of binding energy per nucleon versus the mass number A. Explain with the help of this graph, the release of energy in the process of
nuclear fission and fusion Question 2
(i)The mass of the nucleus in its ground state is always less than the total mass of its constituents- neutrons and Protons. Explain
(ii) Plot a graph showing the variation of the potential energy of a pair of nucleons as a function of their separation .Mark the region where nuclear force is attractive and repulsive Question 3
(i)Draw the plot of binding energy per nucleon as a function of mass Number A. Write two important conclusions that can be drawn regarding the nature of nuclear force
(ii) Use this graph to explain the release of energy in both the processes of nuclear fission and fusion
(iii) Write the basic nuclear process of neutron undergoing beta decay. Why is the detection of neutrinos found very difficult
Question 4
Write characteristics of the Nuclear forces
Question 5
Write the basic nuclear processes underlying $\beta ^{+}$ and $\beta ^{-}$ decays.
Question 6
Distinguish between Nuclear Fission and Fusion. Show how in both these processes energy is released.
Question 7
Show that nuclear density in a given nucleus is independent of the mass Number A
Question 8
In a typical nuclear reaction although the number of nucleons is conserved,yet energy is released. How ?. explain
Question 9
Explain the below terms with respect to radioactive substance
(a) Half -life
(b) Mean -life time.
(c) decay constant
Write the relationship between them.
Question 10
State the law of Radioactive disintegration. Using this law show that radioactive decay is exponential in nature
Short Answer type
Question 1
What is nuclear Fusion? Give One example Question 2
Calculate the energy equivalent in (MeV) of 1/12 of the mass of one atom of C-12? Solution
1/12 of the mass of one atom of C-12 = 1 amu=$1.66 \times 10^{-27}$kg
Energy equivalent of this mass = $mc^2$
$=1.66 \times 10^{-27} \times (3 \times 10^8)^2=933.75$ MeV
Question 3
Write the nuclear reactions for the following
(i) $\alpha$ - decay of $^{204}_{84}Po$
(ii) $\beta$ -decay of $^{32}_{15}P$
(iii) $\beta ^{+}$ - decay of $^{11}_{6}C$ Solution
Question 4
Give two points of difference between nuclear fission and nuclear fusion Question 5
Identify X in the below nuclear reactions
(i) $^{27}_{13}Al + X \rightarrow ^{30}_{15}P + ^{1}_{0}n$
(ii) $_{232}^{90}Th \rightarrow ^{228}_{88}Ra + X$
(iii) $^{234}_{90}Th \rightarrow ^{234}_{91}Pa + X + \overline{\nu}$ Solution
Question 6
Why is heavy water used as a moderator in a thermal nuclear reaction? Question 7
How are protons, which are positively charged, held together inside a nucleus? Explain the variation of potential energy of a pair of nucleons as a function of their separation. State the significance of negative potential energy in this region? Solution
Protons are held together because of the strong attractive nuclear forces. These attractive forces overcome the repulsion forces between the protons. The negative potential energy signifies that
Question 8
What is the value of a and b in the below nuclear reaction
$^1_0n + ^{235}_{92}U \rightarrow ^{a}_{54}Xe + ^{94}_{b}Sr + 2 ^1_0n$ Solution
In nuclear reaction, mass number and atomic number should be same on both the sides
so, 1 + 235 =a+94 +2 => a=140
0+92=54+b+0 => b=38
Numerical Questions
Question 1
The following table shows some measurements of the decay rate of a radionuclide sample. Find the disintegration constant. Solution
$R =R_o e^{-\lambda t}$
Taking ln
$ln R= ln R_o - \lambda t$
$ln R = -\lambda t + ln R_o $
slope of ln R v/s t is $-\lambda$
$- \lambda =\frac {1 - 1.52}{218-164}$
$\lambda= 0.028 \; minute^{-1}$
Question 2
Show that $^{238}_{92}U$ cannot spontaneously emit a proton. Given
$^{238}_{92}U=238.05079$u,$^{237}_{91}Pu=237.05121$u, $^{1}_{1}H=1.00783$u Question 3
Calculate the energy liberated in KWH when 100 g of $^7_3Li$ is converted into $^4_2He$ by proton ($^1_1H$) bombardment.
Given
m( $^7_3Li$ ) = 7.0183 amu
m( $^4_2He$)=4.0040 amu
m($^1_1H$)=1.0081 amu
1 amu= 931 MeV
1 MeV= $1.6 \times 10^{-13}$ J
1 KWH= $3.6 \times 10^6$ J Solution
The complete nuclear reaction is
$^{3}_{7}Li + ^1_1H \rightarrow ^{2}_{4}He + ^{2}_{4}He$
Initial Mass= 7.0183 + 1.0081=8.0264 amu
Final mass=2 * 4.0040 amu=8.0080 amu
Energy liberated = 8.0264 - 8.0080=.0184 amu
Now 1 Amu is equivalent to 931 MeV
So energy liberated = .0184 * 931 MeV= 17.1 MeV
Now Number of Lithium atoms in 7 gm are $6.023 \times 10^{23}$
So Atoms in 100g of Lithium will be given by
=$\frac { 6.023 \times 10^{23}}{7} \times 100 = 8.6 \times 10^{24}$
So energy Produced from 100 g Lithium is
$=17.1 \times 8.6 \times 10^{24}=1.47 \times 10^{26}$ MeV
$=1.47 \times 10^{26} \times 1.6 \times 10^{-13}= 2.35 \times 10^{13}$ J
$=\frac {2.35 \times 10^{13}}{3.6 \times 10^6} = 6.5 \times 10^6$ KWH
Question 4
The half lie of $^{238}_{92}U$ against a alpha decay is $4.5 \times 10^9$ Years. Calculate the activity of 1 g of sample of $^{238}_{92}U$? Solution
$T_{1/2} =4.5 \times 10^9$ Years = $ 4.5 \times 10^9 \times 3.156 \times 10^7$ sec
m=1g , M=238
Now
Number of atoms in 1 G Uranium is given by
$N= \frac {m}{M} \times \text{Avogadra's Number}$
$= \frac {1 \times 6.023 \times 10^{23}}{238} $ atoms
Actvity of the Sample is given by
$R =\lambda N = \frac {.693}{T_{1/2}} \times N$
Substituting the values
$= 1.235 \times 10^4$ Bq
Question 5
A radioactive isotope X has a half life of 3 seconds. At t=0 , a given sample of this isotope X contains 8000 atoms.Calculate
(i)Decay constant
(ii) Time when the atoms remaining undecayed is 1000 Solution
i. Decay constant is given by
$\lambda = \frac {.693}{T_{1/2}}= \frac {.693}{3}= .231$ /sec
ii. Now as per laws of Radioactivity
$N=N_0 e^{-\lambda t}$
Now $ \frac {N}{N_0} = \frac {1000}{8000} = \frac {1}{8}$
Therefore
$ \frac {1}{8}=e^{-\lambda t}$
or
$t= \frac {1}{\lambda} ln 8 = 9$s
