Ques. Suppose we consider a large number of containers each containing initially 10000 atoms of a radioactive material with a half life of 1 year. After 1 year,
(a) all the containers will have 5000 atoms of the material
(b) all the containers will contain the same number of atoms of the material but that number will only be approximately 5000
(c) the containers will in general have different numbers of the atoms of the material but their average will be close to 5000
(d) none of the containers can have more than 5000 atoms
NUCLEI
MCQ (Single correct option)
Thinking process :
T2=λln2,λ→ decay constant
Ans. (c)
Radioactivity is a process due to which a radioactive material spontaneously decays. In half-life (t = 1 year) of the material on the average half the number of atoms will decay.
Therefore, the containers will in general have different number of atoms of the material, but their average will be approx 5000.
NUCLEI
MCQ (Single correct option)
Ques. The gravitational force between a H-atom and another particle of mass m will be given by Newton’s law:
F=Gr2M.m, where r is in km and
(a) M=mproton+melectron
(b) M=mproton+melectron−c2B (B = 13.6 eV)
(c) M is not relate to the mass of the hydrogen atom
(d) M=mproton+melectron−c2∣V∣ (|V| = magnitude of the potential energy of electron in the H-atom)
NUCLEI
MCQ (Single correct option)
Ans. (b)
Given, F=r2GMm
M = effective mass of hydrogen atom
mass of electron + mass of proton −CB2
where B is BE of hydrogen atom =13.6eV.
NUCLEI
MCQ (Single correct option)
Ques. 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 α and β-radioactivity but not for γ-radioactivity
(c) change for α-radioactivity but not for others
(d) change for β-radioactivity but not for others
NUCLEI
MCQ (Single correct option)
Ans. (b)
α−β particle carries one unit of negative charge, an α-particle carries 2 units of positive charge and γ (particle) carries no charge, therefore electronic energy levels of the atom charges for α and β decay, but not for γ-decay.
NUCLEI
MCQ (Single correct option)
Ques.Mx and My denote the atomic masses of the parent and the daughter nuclei respectively in radioactive decay. The Q-value for a β−decay is Q1 and that for a β+decay is Q2. If me denotes the mass of an electron, then which of the following statements is correct?
(a) Q1=(Mx−My)c2 and Q2=[Mx−My−2me]c2
(b) Q1=(Mx−My)c2 and Q2=(Mx−My)c2
(c) Q1=(Mx−My−2me)c2 and Q2=(Mx−My+2ce)c2
(d) Q1=(Mx−My+2me)c2 and Q2=(Mx−my+2me)c2
NUCLEI
MCQ (Single correct option)
Ans. (a)
Let the nucleus is zXA.β+decay is represented as
zXA→z−1YA++1e0+v+Q2
∴Q2=[m(zXA)−mn(z−1YA)−me]c2
=[m(zXA)+zme−mn(z−1YA)−(z−1)me−2me]c2
=[m(zXA)−m(z−1YA)−2me]c2
Sol. to be continued …
NUCLEI
MCQ (Single correct option)
=(Mx−My−2me)c2
β−decay is represented as
∴zXA→z−1yA++1e0+v+Q2
α1=[m(zXA)−mn(z−1YA)−me]c2
=[m(zXA)+zme−mn(z+1YA)−(z−1)me]c2
=[m(zXA)−m(z−1YA)]c2
=(Mx−My)c2
NUCLEI
MCQ (Single correct option)
Ques. Tritium is an isotope of hydrogen whose nucleus triton contains 2 neutrons and 1 proton. Free neutrons decay into p+eˉ+nˉ. If one of the neutrons in Triton decays, it would transform into He3 nucleus. This does not happen. This is because
(a) Triton energy is than that of a He3 nucleus
(b) The electron created in the beta decay process cannot remain in the nucleus
(c) both the neutons in triton have to decay simultaneously resulting in a nucleus with 3 protons, which is not a He3 nucleus.
(d) free neutrons decay due to external perturbations which is absent in triton nucleus
NUCLEI
MCQ (Single correct option)
Thinking process :
Isotopes of an element are having same atomic numbers and different mass numbers.
Ans. (a)
Tritium (1H3) contains 1 proton and 2 neutrons. A neutron decays as n→P+eˉ+vˉ, the nucleus may have 2 protons and one neutron, i.e., tritium will transform into 2He3 (2 protons and 1 neutron).
Triton energy is less than that of 2He3 nucleus, i.e., transformation is not allowed energetically.
NUCLEI
MCQ (Single correct option)
Ques. Heavy stable nuclei have more neutrons than protons. This is because of the fact that
(a) neutrons are heavier than protons
(b) electrostatic force between protons are repulsive
(c) neutrons decay into protons through beta decay
(d) nuclear forces between neutrons are weaker than that between protons
NUCLEI
MCQ (Single correct option)
Ans. (b)
Stable heavy nuclei have more neutrons than protons. This is because electrostatic force between protons is repulsive, which may reduce stability.
NUCLEI
MCQ (Single correct option)
Ques. In a nuclear reactor, moderators slow down the neutrons which come out in a fission process. The moderator used have light nuclei. Heavy nuclei will not serve the purpose, because
(a) they will break up
(b) elastic collision of neutrons with heavy nuclei will not slow them down
(c) the net weight of the reactor would be unbearably high
(d) substances with heavy nuclei do not occur in liquid or gaseous state at room temperature
NUCLEI
MCQ (Single correct option)
Thinking process :
When there is an elastic collision between two bodies of same mass their velocities are exchanged.
Ans. (b)
According to the question, the moderator used have light nuclei (like proton). When protons undergo perfectly elastic collision with the neutron emitted their velocities are exchanged, i.e., neutrons come to rest and protons move with the velocity of neutrons. Heavy nuclei will not serve the purpose because elastic collisions of neutrons with heavy nuclei will not slow them down.
NUCLEI
MCQ (More than one correct option)
Ques. Fusion processes, like combining two deuterons to form a He nucleus are impossible at ordinary temperatures and pressure.The reasons for this can be traced to the fact
(a) nuclear forces have short range
(b) nuclei are positively charged
(c) the original nuclei must be completely ionized before fusion can take place
(d) the original nuclei must first break up before combining with each other
NUCLEI
MCQ (More than one correct option)
Ans. (a, b)
Fusion processes are impossible at ordinary temperatures and pressures. The reason is nuclei are positively charged and nuclear forces are short range strongest forces.
NUCLEI
MCQ (More than one correct option)
Ques. Samples of two radioactive nuclides A and B are taken λA and λB are the disintegration constants of A and B respectively. In which of the following cases, the two samples can simultaneously have the same decay rate at any time?
(a) Initial rate of decay of A is twice the initial rate of decay of B and λA=λB
(b) Initial rate of decay of A is twice the initial rate of decay of B and λA>λB
(c) Initial rate of decay of B is twice the initial rate of decay of A and λA>λB
(d) Initial rate of decay of B is same as the rate of decay of A at t=2h and λB<λA
NUCLEI
MCQ (More than one correct option)
Ans. (b, d)
The two samples of the two radioactive nuclides A and B can simultaneously have the same decay rate at any time if initial rate of decay of A is twice the initial rate of decay of B and λA>λB. Also, when initial rate of decay of B is same as rate of decay of A at t=2 hand λB<λA.
NUCLEI
MCQ (More than one correct option)
Ques. The variation of decay rate of two radioactive samples A and B with time is shown in figure.
Which of the following statements are true?
Ques. to be continued …
NUCLEI
MCQ (More than one correct option)
Options:
(a) Decay constant of A is greater than that of B, hence A always decays faster than B
(b) Decay constant of B is greater than that of A but its decay rate is always smaller than that of A
(c) Decay constant of A is greater than that of B but it does not always decay faster than B
(d) Decay constant of B is smaller than that of A but still its decay rate becomes equal to that of A at a later instant
NUCLEI
MCQ (More than one correct option)
Ans. (c, d)
From the given figure, it is clear that slope of curve A is greater than that of curve B. So rate of decay is faster for A than that of B.
Sol. to be continued …
NUCLEI
MCQ (More than one correct option)
We know that (dtdN)∝,λ, at any instant of time hence we can say that λA>λB.
At point P shown in the diagram the two curve intersect.
Hence at point P, rate of decay for both A and B is the same.
NUCLEI
Very short answer type questions
Ques.He23 and He13 nuclei have the same mass number. Do they have the same binding energy?
NUCLEI
Very short answer type questions
Ans.
Nuclei He23 and He13 have the same mass number. He23 has two proton and one neutron. He13 has one proton and two neutron.
The repulsive force between protons is missing in 1He3 so the binding energy of 1He3 is greater than that of 2He3.
NUCLEI
Very short answer type questions
Ques. Draw a graph showing the variation of decay rate with number of active nuclei.
NUCLEI
Very short answer type questions
Ans.
We know that, rate of decay =dt−dN=λN
where decay constant (λ) is constant for a given radioactive material. Therefore, graph between N and dtdN is a straight line as shown in the diagram.
NUCLEI
Very short answer type questions
Ques. Which sample A or B shown in figure has shorter mean-life?
NUCLEI
Very short answer type questions
Ans.
From the given figure, we can say that
at t=0,(dtdN)A=(dtdN)B
⇒(N0)A=(N0)B
Considering any instant t by drawing a line perpendicular to time axis, we find that (dtdN)A>(dtdN)B
⇒λANA>λBNB
Sol. to be continued …
NUCLEI
Very short answer type questions
∵NA>NB (rate of decay of B is slower)
∴λB>λA
⇒τA>τB
[∵ Average life τ=λ1]
NUCLEI
Very short answer type questions
Ques. Which one of the following cannot emit radiation and why?
Excited nucleus, excited electron.
NUCLEI
Very short answer type questions
Ans.
Excited electron cannot emit radiation because energy of electronic energy levels is in the range of eV and not MeV (mega electron volt ). γ-radiations have energy of the order of MeV.
NUCLEI
Very short answer type questions
Ques. In pair annihilation, an electron and a positron destroy each other to produce gamma radiations. How is the momentum conserved?
NUCLEI
Very short answer type questions
Ans.
In pair annihilation, an electron and a positron destroy each other to produce 2γ photons which move in opposite directions to conserve linear momentum.
The annihilation is shown below 0e−1+0e+1→2γ ray photons.
NUCLEI
Short answer type questions
Ques. Why do stable nuclei never have more protons than neutrons?
NUCLEI
Short answer type questions
Ans.
Because protons are positively charged and repel one another electrically. This repulsion becomes so great in nuclei with more than 10 protons or so, that an excess of neutrons which produce only attractive forces, is required for stability.
NUCLEI
Short answer type questions
Ques. Consider a radioactive nucleus A which decays to a stable nucleus C through the following sequence:
A→B→C
Here B is an intermediate nuclei which is also radioactive. Considering that there are N0 atoms of A initially, plot the graph showing the variation of number of atoms of A and B versus time.
NUCLEI
Short answer type questions
Thinking process : Based or decay law of unstable radioactive nuclei.
Ans. Consider the situation shown in the graph.
Sol. to be continued …
NUCLEI
Short answer type questions
At t=0,NA=N0 (maximum) while NB=0.
As time increases, NA decreases exponentially and the number of atoms of B increases.
They becomes (NB) maximum and finally drop to zero exponentially by radioactive decay law.
NUCLEI
Short answer type questions
Ques. A piece of wood from the ruins of an ancient building was found to have a 14C activity of 12 disintegrations per minute per gram of its carbon content. The 14C activity of the living wood is 16 disintegrations per minute per gram. How long ago did the tree, from which the wooden sample came, die? Given half-life of 14C is 5760 years.
NUCLEI
Short answer type questions
Thinking process :
Carbon dating is a technique that uses the decay of carbon −14(14C) to estimate the age of organic materials, such as wood and leather.
Ans.
Given, R = 12 dis/min per g,R0=16 dis/min per g,T1/2=5760year Let t be the span of the tree
According to radioactive decay law,
R=R0e−λt or R0R=e−λt or eλt=RR0
Sol. to be continued …
NUCLEI
Short answer type questions
Taking log on both the sides
λtlogee=logeRR0
⇒λt=log101216×2.303
t=λ2.303(log4−log3)
=0.69312.303(0.6020−4.771)×5760
[∵λ=T1/20.6931]
= 2391.20 year
NUCLEI
Short answer type questions
Ques. Are the nucleons fundamental particles, or do they consist of still smaller parts? One way to find out is to probe a nucleon just as Rutherford probed an atom. What should be the kinetic energy of an electron for it to be able to probe a nucleon? Assume the diameter of a nucleon to be approximately 10−15 m.
NUCLEI
Short answer type questions
Thinking process :
We have to use de-Broglie formula (λ=h/p) to find momentum of the particle.
Ans.
Each particle (neutron and proton) present inside the nucleus is called a nucleon.
Let λ be the wavelength λ=10−15m
To detect separate parts inside a nucleon, the electron must have wavelength less than 10−15m.
Sol. to be continued …
NUCLEI
Short answer type questions
We know that
Energy
λ=ph and KE=PE
=λhc
From Eq. (i) and Eq. (ii),
kinetic energy of electron =PE=λhc−10−15×1.6×10−196.6×10−34×3×108eV
KE=109eV
NUCLEI
Short answer type questions
Ques. A nuclide 1 is said to be the mirror isobar of nuclide 2 if Z1=N2 and Z2=N1.(a) What nuclide is a mirror isobar of 1123Na ? (b) Which nuclide out of the two mirror isobars have greater binding energy and why?
NUCLEI
Short answer type questions
Thinking process :
Based on the mirror isobar concept and binding energy concept.
Ans.
(a) According to question, a nuclide 1 is said to be mirror isobar of nuclide 2 , if Z1=N2 and Z2=N1.
Now in 11Na23,Z1=11,N1=23−11=12
∴ Mirror isobar of 11Na23 is 12,Mg23, for which Z2=12=N1 and N2=23−12=11=Z1
(b) As 1223Mg contains even number of protons (12) against 1123Na which has odd number of protons (11), therefore 1123Mg has greater binding energy than 11Na23.
NUCLEI
Long answer type questions
Ques. Sometimes a radioactive nucleus decays into a nucleus which itself is radioactive. An example is:
Assume that we start with 1000 38S nuclei at time t=0. The number of 38Cl is of count zero at t=0 and will again be zero at t=∞. At what value of t, would the number of counts be a maximum?
NUCLEI
Long answer type questions
Thinking process :
To solve this problem concept of chain of two decays will be used. For the process,
A→B→C, the law of decay is dtdNB=λBNB+λANA.
Ans.
Consider the chain of two decays
38S2.48h38Cl0.62h38Ar
Sol. to be continued …
NUCLEI
Long answer type questions
At time t, Let 38S have N1(t) active nuclei and 38Cl have N2(t) active nuclei.
Note Do not apply directly the formula of radioactive. Apply formulae related to chain decay.
NUCLEI
Long answer type questions
Ques. Deuteron is a bound state of a neutron and a proton with a binding energy B=2.2MeV. A γ-ray of energy E is aimed at a deuteron nucleus to try to break it into a (neutron + proton) such that the n and p move in the direction of the incident γ-ray. If E=B, show that this cannot happen. Hence, calculate how much bigger than B must be E be for such a process to happen.
NUCLEI
Long answer type questions
Thinking process :
Apply conservation of energy as well as conservation of momentum.
Ans.
Given binding energy
B=2.2MeV
From the energy conservation law,
Sol. to be continued …
NUCLEI
Long answer type questions
E−B=Kn+Kp=2mpn2+2mpp2 ……… (i)
From conservation of momentum,
pn+pp=CE ……… (ii)
As
E=B, Eq. (i) pn2+pp2=0
It only happen if pn=pp=0
Sol. to be continued …
NUCLEI
Long answer type questions
So, the Eq. (ii) cannot satisfied and the process cannot take place.
Let E=B+X, where X«B for the process to take place.
Put value of pn from Eq. (ii) in Eq. (i), we get
X=2m(cE−pp)+2mpp2
or2pp2−c2Epp+c2E2−2mX=0
Using the formula of quadratic equation, we get
Sol. to be continued …
NUCLEI
Long answer type questions
pp=4c2E±c24E2−8(c2E2−2mX)
For the real value pp, the discriminant is positive
c24E2=8(c2E2−2mX)
16mX=c24E2
X=4mc2E2≈4mc2B2
[∵X<<B⇒E≅B]
NUCLEI
Long answer type questions
Ques. The deuteron is bound by nuclear forces just as H-atom is made up of p and e bound by electrostatic forces. If we consider the force between neutron and proton in deuteron as given in the form a coulomb potential but with an effective charge e′:
F=4πε01re‘2
estimate the value of (e′/e) given that the binding energy of a deuteron is 2.2 MeV.
NUCLEI
Long answer type questions
Ans.
The binding energy is H-atom
E=πε02h2me4=13.6eV…(i)
If proton and neutron had charge e′ each and were governed by the same electrostatic force, then in the above equation we would need to replace electronic mass m by the reduced mass m′ of proton-neutron and the electronic charge e by e′.
m′=M+NM×N=2M
Sol. to be continued …
NUCLEI
Long answer type questions
=21836m=918m
Here, M represents mass of a neutron/proton
∴ Binding energy = 8ε02h2918m(e′)4=2.2MeV ……… (ii)
Dividing Eqs. (ii) and (i), we get
918(ee′)4=13.6eV2.2MeV=13.62.2×106
(ee′)4=13.6×9182.2×106=176.21
ee′=(176.21)1/4=3.64
NUCLEI
Long answer type questions
Ques. Before the neutrino hypothesis, the beta decay process was throught to be the transition.
n→p+e
If this was true, show that if the neutron was at rest, the proton and electron would emerge with fixed energies and calculate them. Experimentally, the electron energy was found to have a large range.
NUCLEI
Long answer type questions
Ans.
Before β-decay, neutron is at rest. Hence, En=mnc2,pn=0
pn=pp+pe
Orpp+pe=0⇒∣pp∣=∣pe∣=p
Also,
Ep=(mp2c4+pp2c2)21,
Sol. to be continued …
NUCLEI
Long answer type questions
Ee=(me2c4+pp2c2)21
=(me2c4+pe2c2)21
From conservation of energy,
(mp2c4+p2c2)21+=(me2c4+p2c2)21=mnc2
mpc2≈936MeV,mnc2≈938MeV,mec2=0.51MeV
Since, the energy difference between n and p is small, pc will be small, $p c«
Sol. to be continued …
NUCLEI
Long answer type questions
⇒mpc2+2mp2c4p2c2=mnc2−pc
To first order pc≃mnc2−mpc2=938MeV−936MeV=2MeV
This gives the momentum of proton or neutron. Then,
Ep=(mp2c4+p2c2)21=9362+22
≃936MeV
Ee=(me2c4+p2c2)21=(0.51)2+22
≃2.06MeV
NUCLEI
Long answer type questions
Ques. The activity R of an unknown radioactive nuclide is measured at hourly intervals. The result found are tabulated as follows:
t(h)
0
1
2
3
4
R(MBq)
100
35.36
12.51
4.42
1.56
(i) Plot the graph of R versus t and calculate half-life from the graph.
(ii) Plot the graph of In (R0R) versus t and obtain the value of half-life from the graph.
NUCLEI
Long answer type questions
Thinking process :
Based on Decay law and half-life.
Ans.
In the table given below, we have listed values of R(MBq) and lnR0R.
t(h)
0
1
2
3
4
R(MBq)
100
35.36
12.51
4.42
1.56
R0R
-
-1.04
-2.08
-3.11
-4.16
Sol. to be continued …
NUCLEI
Long answer type questions
(i) When we plot the graph of R versust, we obtain an exponential curve as shown.
From the graph we can say that activity R reduces to 50 in t=OB≈40min So, t1/2≈40min.
(ii) The adjacent figure shows the graph of ln(R/R0) versus t.
Sol. to be continued …
NUCLEI
Long answer type questions
Slope of this graph =−λ
from the graph,
Sol. to be continued …
NUCLEI
Long answer type questions
Half-life
λ=−1−4.16−3.11⇒1.05h−1
T1/2=λ0.693=1.050.693=0.66h
=39.6min≈40min
NUCLEI
Long answer type questions
Ques. Nuclei with magic number of proton Z=2,8,20,28,50,52 and magic number of neutrons N=2,8,20,28,50,82 and 126 are found to be very stable.
(i) Verify this by caculating the proton. separation energy Sp for 120Sn(Z=50) and 121Sb(Z=51).
The proton separation energy for a nuclide is the minimum energy required to separate the least tightly bound proton from a nucleus of that nuclide. It is given by
Sp=(MZ−1,N+MH−MZ,N)c2.
Given, 119In = 118.9058 u, 120Sn = 199.902199 u, 121Sb = 120.903824 u, 1H = 1.0078252 u.
(ii) What does the existence of magic number indicate?
NUCLEI
Long answer type questions
Ans.
(i) The proton separation energy is given by
pSn=(M119.70+MH−M120,70)c2
=(118.9058+1.0078252−119.902199)c2
=0.0114362c2
SpSp=(M120,70+MH−M121,70)c2
Sol. to be continued …
NUCLEI
Long answer type questions
=(119.902199+1.0078252−120.903822)c2
=0.0059912c2
Similarly
Since, SpSn>SpSb, Sn nucleus is more stable than Sb nucleus.
(ii) The existence of magic numbers indicates that the shell structure of nucleus similar to the shell structure of an atom. This also explains the peaks in binding energy/nucleon curve.