JEE Advanced Physics Questions & Answers

Q 1 :

A heavy nucleus Q of half-life 20 minutes undergoes alpha-decay with probability of 60% and beta-decay with probability of 40%. Initially, the number of Q nuclei is 1000. The number of alpha-decays of Q in the first one hour is [2021]

50
75
350
525

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(4)

Out of 1000 nuclei of $Q$ , 60% may go $α$ -decay

i.e., 600 nuclei may have $α$ -decay

Decay constant

$t = 1 hour = 60 minutes$

From, $N = N_{0} e^{- λ i} = 600 \times e^{\frac{\ln 2}{20} \times 60}$

$∴$ $N = 75$

i.e., 75 nuclei are left after one hour.

So, number of nuclei decayed $= 600 - 75 = 525$

Q 2 :

In a radioactive sample, ${}_{19}^{40}K$ nuclei either decay into stable ${}_{20}^{40}{Ca}$ nuclei with decay constant $4.5 \times 10^{- 10}$ per year or into stable ${}_{18}^{40}{Ar}$ nuclei with decay constant $0.5 \times 10^{- 10}$ per year. Given that in this sample all the stable ${}_{20}^{40}{Ca}$ and ${}_{18}^{40}{Ar}$ nuclei are produced by the ${}_{19}^{40}K$ nuclei only. In time $t \times 10^{9}$ years, if the ratio of the sum of stable ${}_{20}^{40}{Ca}$ and ${}_{18}^{40}{Ar}$ nuclei to the radioactive ${}_{19}^{40}K$ nuclei is 99, the value of $t$ will be [Given: $l n 10 = 2.3$ ] [2019]

9.2
4.6
1.15
2.3

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(1)

Here $- \frac{d N}{d t} = λ_{1} N + λ_{2} N$

In integrating on both sides,

$t = \frac{2.303}{λ_{1} + λ_{2}} \log_{10} (\frac{N_{0}}{N})$ $\Rightarrow t = \frac{2.303}{5 \times 10^{- 10}} \log_{10} (\frac{100}{1})$

$[∵ λ_{1} + λ_{2} = 4.5 \times 10^{- 10} + 0.5 \times 10^{- 10} and \frac{N_{0}}{N} = 100]$

$∴$ $t = 9.2 \times 10^{9} year$

Q 3 :

A radioactive sample $S_{1}$ having an activity $5 μ Ci$ has twice the number of nuclei as another sample $S_{2}$ which has an activity of $10 μ Ci$ . The half-lives of $S_{1}$ and $S_{2}$ can be [2008]

20 years and 5 years, respectively
20 years and 10 years, respectively
10 years each
5 years each

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(1)

Let $λ_{1}$ and $λ_{2}$ be the decay constants and $N_{1}$ and $N_{2}$ be the number of active nuclei present for the two samples $S_{1}$ and $S_{2}$ respectively.

i.e., $λ_{1} N_{1} = 5 μ Ci$ and $λ_{2} N_{2} = 10 μ Ci$ $\Rightarrow λ_{2} N_{2} = 2 λ_{1} N_{1}$

Also, $N_{1} = 2 N_{2}$

$∴$ $λ_{2} N_{2} = 2 λ_{1} (2 N_{2})$ or $λ_{2} = 4 λ_{1}$ $\Rightarrow \frac{λ_{2}}{λ_{1}} = 4$

Also, $T_{1 / 2} = \frac{0.693}{λ}$

$∴$ ${(T_{1 / 2})}_{1} = 4 {(T_{1 / 2})}_{2}$

Q 4 :

${}_{87}^{221}{Ra}$ is a radioactive substance having a half-life of 4 days. Find the probability that a nucleus undergoes decay after two half-lives. [2006]

$1$
$\frac{1}{2}$
$\frac{3}{4}$
$\frac{1}{4}$

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(3)

For a nucleus to disintegrate in two half lives, $\frac{1}{2} + \frac{1}{4} = \frac{3}{4}$ i.e., The probability is $\frac{3}{4}$ as 75% of the nuclei will disintegrate in this time.

Q 5 :

A 280 days old radioactive substance shows an activity of 6000 dps. 140 days later, its activity becomes 3000 dps. What was its initial activity? [2004]

20000 dps
24000 dps
12000 dps
6000 dps

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(2)

In two half lives, the activity will remain $\frac{1}{4}$ of its initial activity.

$∴$ $Initial activity = 4 \times 6000 = 24000 dps$

Q 6 :

A nucleus with mass number 220, initially at rest, emits an $α$ -particle. If the Q-value of the reaction is 5.5 MeV, calculate the kinetic energy of the $α$ -particle. [2003]

4.4 MeV
5.4 MeV
5.6 MeV
6.5 MeV

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(2)

By conservation of momentum, $p_{1} = p_{2}$

$\sqrt{2 K_{1} m_{1}} = \sqrt{2 K_{2} m_{2}}$ $[∵ P = \sqrt{2 K m}]$

$\Rightarrow \sqrt{2 K_{1} (216)} = \sqrt{2 K_{2} (4)}$

$∴$ $K_{2} = 54 K_{1}$ $\dots (i)$

And given, $K_{1} + K_{2} = 5.5 MeV$ $\dots (i i)$

Solving equations (i) and (ii), we get $K_{α} = K_{2} = 5.4 MeV$

Q 7 :

Which of the following processes represents a $γ$ -decay? [2002]

${}^{A}X_{Z} + γ ⟶ {}^{A}{X_{Z - 1} + a + b}$
${}^{A}X_{Z} + {}^{1}n_{0} ⟶ {}^{A - 3}X_{Z - 2} + c$
${}^{A}X_{Z} ⟶ {}^{A}X_{Z} + f$
${}^{A}X_{Z} + {}^{0}e_{- 1} ⟶ {}^{A}{X_{Z - 1} + g}$

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(3)

In $γ$ -decay, the atomic number (Z) and mass number (A) do not change.

Q 8 :

The half-life of ${}^{215}{At}$ is $100 μ s$ . The time taken for the radioactivity of a sample of ${}^{215}{At}$ to decay to ${\frac{1}{16}}^{th}$ of its initial value is [2002]

$400 μ s$
$6.3 μ s$
$40 μ s$
$300 μ s$

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(1)

$A = A_{0} {(\frac{1}{2})}^{n};$ $n = number of half lives.$

$\frac{A_{0}}{16} = A_{0} {(\frac{1}{2})}^{n}$ $∴$ ${(\frac{1}{2})}^{4} = {(\frac{1}{2})}^{n}$

$∴$ $n = 4$

$Therefore, time taken to decay to$ $\frac{1}{16^{th}}$ $of its initial value.$

$∴$ $t = n \times t_{1 / 2} = (4 \times 100) μ s = 400 μ s$

Q 9 :

A radioactive sample consists of two distinct species having equal number of atoms initially. The mean lifetime of one species is $τ$ and that of the other is $5 τ$ . The decay products in both cases are stable. A plot is made of the total number of radioactive nuclei as a function of time. Which of the following figures best represent the form of this plot? [2001]

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(4)

$N_{1} = N_{0} e^{- λ_{1} t} = N_{0} e^{- t / τ}$ $\dots (i)$

Mean life time $τ = \frac{1}{λ_{1}}$

Similarly, $N_{2} = N_{0} e^{- λ_{2} t} = N_{0} e^{- t / 5 τ}$ $\dots (i i)$

as $5 τ = \frac{1}{λ_{2}}$

Adding equations (i) and (ii), we get

$N = N_{1} + N_{2} = N_{0} (e^{- t / τ} + e^{- t / 5 τ})$

The total number of radioactive nuclei $(N)$ as a function of time only decreases exponentially. Hence graph (4) correctly depicts this behavior.

Q 10 :

The electron emitted in beta radiation originates from [2001]

inner orbits of atoms
free electrons existing in nuclei
decay of a neutron in a nucleus
photon escaping from the nucleus

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(3)

In a nucleus a neutron converts into a proton as follows

$n \to p^{+} + e^{- 1}$

Therefore, decay of neutron is responsible for $β$ -radiation origin.

Q 11 :

Two radioactive materials $X_{1}$ and $X_{2}$ have decay constants $10 λ$ and $λ$ , respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of $X_{1}$ to that of $X_{2}$ will be $\frac{1}{e}$ after a time [2000]

$\frac{1}{10 λ}$
$\frac{1}{11 λ}$
$\frac{11}{10 λ}$
$\frac{1}{9 λ}$

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(4)

$Number of nuclei of X_{1},$ $N_{1} = N_{0} e^{- 10 λ t}$ $and number of nuclei of X_{2}$ $N_{2} = N_{0} e^{- λ t} .$

$∴$ $\frac{N_{1}}{N_{2}} = \frac{e^{- 10 λ t}}{e^{- λ t}} = \frac{1}{e^{9 λ t}}$

$Given,$ $\frac{N_{1}}{N_{2}} = \frac{1}{e};$

$∴$ $\frac{1}{e^{9 λ t}} = \frac{1}{e}$ $(∵ \frac{N_{1}}{N_{2}} = \frac{1}{e} is given)$

$\Rightarrow 9 λ t = 1$ $∴$ $t = (\frac{1}{9 λ})$

$i.e., after time$ $t = \frac{1}{9 λ},$ $the ratio of the number of nuclei of$ $X_{1}$ $to that of$ $X_{2}$ $will be$ $\frac{1}{e} .$

Q 12 :

In a radioactive decay process, the activity is defined as $A = - \frac{d N}{d t},$ where $N (t)$ is the number of radioactive nuclei at time $t$ . Two radioactive sources, $S_{1}$ and $S_{2}$ , have the same activity at time $t = 0$ . At a later time, the activities of $S_{1}$ and $S_{2}$ are $A_{1}$ and $A_{2}$ , respectively. When $S_{1}$ and $S_{2}$ have just completed their $3^{rd}$ and $7^{th}$ half-lives, respectively, the ratio $\frac{A_{1}}{A_{2}}$ is _________. [2023]

(16)

When radioactive sources just completed their $3^{rd}$ and $7^{th}$ half-lives, then the ratio

$\frac{A_{1}}{A_{2}} = \frac{A_{0} e^{- 3 \ln 2}}{A_{0} e^{- 7 \ln 2}} = \frac{2^{- 3}}{2^{- 7}} = \frac{1}{2^{- 4}} = 2^{4} = 16$

Q 13 :

${}^{131}I$ is an isotope of Iodine that B decays to an isotope of Xenon with a half-life of 8 days. A small amount of a serum labelled with ${}^{131}I$ is injected into the blood of a person. The activity of the amount of ${}^{131}I$ injected was $2.4 \times 10^{5} Bq$ . It is known that the injected serum will get distributed uniformly in the bloodstream in less than half an hour. After 11.5 hours, 2.5 ml of blood is drawn from the person's body, and gives an activity of 115 Bq. The total volume of blood in the person's body, in litres, is approximately (you may use $e^{x} \approx 1 + x$ for $| x |$ $< < 1$ and $\ln 2 \approx 0.7$ ). [2017]

(5)

$According to question,$

$I^{131} \underset{T_{1 / 2} = 8 days}{\to} X e^{131} + β$

$A_{0} = 2.4 \times 10^{5} Bq = λ N_{0}$

$Let the volume be V$

$Given:$ $At$ $t = 0$ , $A_{0} = λ N_{0} = 2.4 \times 10^{5} Bq$

$t = 11.5 hrs,$ $A = λ N$

$After$ $t = 11.5 h$ , $2.5$ $ml of blood is drawn from the person's body and gives an activity of$ $115$ $Bq.$

$∴$ $115 = λ (\frac{N}{V} \times 2.5);$ $115 = \frac{λ}{V} \times 2.5 \times N_{0} e^{- λ t}$

$\Rightarrow 115 = \frac{λ N_{0}}{V} \times 2.5 \times e^{- \frac{\ln 2}{8 days} (11.5 hrs)}$

$\Rightarrow V = \frac{2.4 \times 10^{5}}{115} \times 2.5 (1 - \frac{1}{24})$ $[from approximation e^{x} \approx 1 + x]$

$\Rightarrow V = \frac{2.4 \times 10^{5}}{115} \times 2.5 \times \frac{23}{24} = 5 \times 10^{3} ml = 5 litres$

Q 14 :

For a radioactive material, its activity $A$ and rate of change of its activity $R$ are defined as $A = - \frac{d N}{d t}$ and $R = - \frac{d A}{d t},$ where $N (t)$ is the number of nuclei at time $t$ . Two radioactive sources $P$ (mean life $τ$ ) and $Q$ (mean life $2 τ$ ) have the same activity at $t = 0$ . Their rates of change of activities at $t = 2 τ$ are $R_{P}$ and $R_{Q}$ , respectively. If $\frac{R_{P}}{R_{Q}} = \frac{n}{e},$ then the value of $n$ is [2015]

(2)

$Activity,$

$R = - \frac{d A}{d t} = - \frac{d}{d t} [- \frac{d N}{d t}] = \frac{d^{2} N}{d t^{2}} = \frac{d^{2}}{d t^{2}} (N_{0} e^{- λ t})$

$∴$ $R = N_{0} λ^{2} e^{- λ t} = (N_{0} λ) λ e^{- λ t} = A_{0} λ e^{- λ t}$ $[∵ A_{0} = N_{0} λ]$

$∴$ $\frac{R_{P}}{R_{Q}} = \frac{λ_{P} e^{- λ_{P} t}}{λ_{Q} e^{- λ_{Q} t}} = \frac{λ_{P}}{λ_{Q}} \times \frac{e^{λ_{Q} t}}{e^{λ_{P} t}} = \frac{2 τ}{τ} \times \frac{e^{\frac{2 τ}{2 τ}}}{e^{\frac{2 τ}{τ}}} = \frac{2}{e} = \frac{n}{e}$

$∴$ $n = 2$

Q 15 :

A freshly prepared sample of a radioisotope of half-life 1386 s has activity $10^{3}$ disintegrations per second. Given that $l n 2 = 0.693,$ the fraction of the initial number of nuclei (expressed in nearest integer percentage) that will decay in the first 80 s after preparation of the sample is [2013]

(4)

$For a radioactive decay$

$N = N_{0} (1 - e^{- λ t})$

$∴$ $\frac{N}{N_{0}} = e^{- λ t}$ $∴$ $1 - \frac{N}{N_{0}} = 1 - e^{- λ t}$

$∴$ $\frac{N_{0} - N}{N_{0}} = 1 - e^{- \frac{0.693}{t_{1 / 2}} t} = 1 - e^{- 0.04} = 1 - (1 - 0.04)$ $[∵ e^{- x} = 1 - x, x ≪ 1]$

$% decayed \approx 0.04 \times 100 = 4 %$

Q 16 :

The activity of a freshly prepared radioactive sample is $10^{10}$ disintegrations per second, whose mean life is $10^{9} s$ . The mass of an atom of this radioisotope is $10^{- 25} kg$ . The mass (in mg) of the radioactive sample is [2011]

(1)

$We know that,$ $| \frac{d N}{d t} |$ $= λ N = \frac{1}{T_{mean}} N$ $[∵ λ = \frac{1}{T_{mean}}]$

$∴$ $10^{10} = \frac{1}{10^{9}} \times N$

$∴$ $N = 10^{19}$

$i.e.,$ $10^{19}$ $radioactive atoms are present in the freshly prepared sample.$

$∴$ $Mass of the sample = N \times mass of one atom$

$= 10^{19} \times 10^{- 25} kg = 10^{- 6} kg = 1 mg$

Q 17 :

To determine the half-life of a radioactive element, a student plots a graph of $l n$ $| \frac{dN (t)}{dt} |$ versus $t$ . Here, $| \frac{d N (t)}{d t} |$ is the rate of radioactive decay at time $t$ . If the number of radioactive nuclei of this element decreases by a factor of $p$ after 4.16 years, the value of $p$ is [2010]

(8)

$We know that$ $N = N_{0} e^{- λ t}$

$∴$ $\frac{d N}{d t} = N_{0} e^{- λ t} (- λ) = - λ N_{0} e^{- λ t}$

$Taking log on both sides,$

$\log_{e}$ $| \frac{d N}{d t} |$ $= \log_{e} (λ N_{0}) - λ t$

$Comparing it with the graph line,$

$Decay constant,$ $λ = \frac{1}{2} {yr}^{- 1}$ $[\frac{A C}{B C} = \frac{1}{2}]$

$∴$ $T_{1 / 2} = \frac{0.693}{λ} = 0.693 \times 2 = 1.386 years$

$n (t_{1 / 2}) = 4.16$

$∴$ $n = \frac{4.16}{1.386} \approx 3$

$∴$ $N = N_{0} {(\frac{1}{2})}^{3}$

$∴$ $\frac{1}{P} = \frac{1}{8}$

$∴$ $P = 8$

Q 18 :

The minimum kinetic energy needed by an alpha particle to cause the nuclear reaction ${}_{7}^{16}N + {}_{2}^{4}{He} \to {}_{1}^{1}H + {}_{8}^{19}O$ in a laboratory frame is $n$ (in MeV). Assume that ${}_{7}^{16}N$ is at rest in the laboratory frame. The masses of ${}_{7}^{16}N$ , ${}_{2}^{4}{He}$ , ${}_{1}^{1}H$ and ${}_{8}^{19}O$ can be taken to be $16.006 u$ , $4.003 u$ , $1.008 u$ and $19.003 u$ , respectively, where $1 u = 930 MeV c^{- 2} .$ The value of $n$ is _______. [2022]

(2.33)

$We have$

$Q = Δ m c^{2}$

$= (m_{N} + m_{He} - m_{H} - m_{O}) \times 930 MeV$

$= (16.006 + 4.003 - 1.008 - 19.003) \times 930 MeV$

$= - 1.86 MeV$ $= 1.86 MeV energy absorbed$

This $Q$ is equal to the maximum loss in kinetic energy of the $α$ -particle.

Considering collision as inelastic, we get maximum loss in

$K . E . = \frac{1}{2} (\frac{4 m \times 16 m}{4 m + 16 m}) v^{2}$

$\Rightarrow Q = (\frac{1}{2} \times 4 m \times v^{2}) \times \frac{16 m}{20 m}$

$\Rightarrow Q = {(K . E .)}_{\min} \times \frac{4}{5}$

$\Rightarrow {(K . E .)}_{\min} = \frac{5}{4} Q = (\frac{5}{4} \times 1.86) MeV = 2.325 MeV$

$∴$ $n = 2.33$

Q 19 :

In a radioactive decay chain, ${}_{90}^{232}{Th}$ nucleus decays to ${}_{82}^{212}{Pb}$ nucleus. Let $N_{α}$ and $N_{β}$ be the number of $α$ and $β^{-}$ particles, respectively, emitted in this decay process. Which of the following statement(s) is(are) true? [2018]

$N_{α} = 5$
$N_{α} = 6$
$N_{β} = 2$
$N_{β} = 4$

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4

Select one or more options

(1, 3)

$No. of α -particles emitted,$ $N_{α} = \frac{232 - 212}{4} = \frac{20}{4} = 5$

$∴$ ${}_{90}^{232}{Th} \to {}_{82}^{212}{Pb} + 5 {}_{2}^{4}{He} + 2 {}_{- 1}^{0}β$

Considering the laws of conservation of mass number $(A)$ and atomic number $(Z)$ , the number of $β$ -particles emitted is $N_{β} = 2$

Q 20 :

List-I shows different radioactive decay processes and List-II provides possible emitted particles. Match each entry in List-I with an appropriate entry from List-II, and choose the correct option. [2023]

List-I List-II

(P) ${}_{92}^{238}U \to {}_{91}^{234}{Pa}$ (1) one $α$ particle and one $β^{+}$ particle

(Q) ${}_{82}^{214}{Pb} \to {}_{82}^{210}{Pb}$ (2) three $β^{-}$ particles and one $α$ particle

(R) ${}_{81}^{210}{Tl} \to {}_{82}^{206}{Pb}$ (3) two $β^{-}$ particles and one $α$ particle

(S) ${}_{91}^{228}{Pa} \to {}_{88}^{224}{Ra}$ (4) one $α$ particle and one $β^{-}$ particle

(5) one $α$ particle and two $β^{+}$ particles

P → 4, Q → 3, R → 2, S → 1
P → 4, Q → 1, R → 2, S → 5
P → 5, Q → 3, R → 1, S → 4
P → 5, Q → 1, R → 3, S → 2

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(1)

In $α$ -decay, mass number (A) decreases by 4 units and atomic number (Z) decreases by 2 units.

In $β^{-}$ decay, A does not change but Z increases by 1 unit.

In $β^{+}$ decay, A does not change but Z decreases by 1 unit.

(P) ${}_{92}U^{238} \to {}_{91}{Pa}^{234}$

$N_{1} = \frac{238 - 234}{4} = 1 \to 1 α$

$N_{2} - N_{3} = (92 - 91) - \frac{4}{2} = - 1 \to 1 β^{-}$

$∴$ $1 α and 1 β^{-} emission.$

(Q) ${}_{82}{Pb}^{214} \to {}_{82}{Pb}^{210}$

$N_{1} = \frac{214 - 210}{4} = 1 \to 1 α$

$N_{2} - N_{3} = (82 - 82) - \frac{4}{2} = - 2 \to 2 β^{-}$

$∴$ $1 α and 2 β^{-} emission.$

(R) ${}_{81}{Tl}^{210} \to {}_{82}{Pb}^{206}$

$N_{1} = \frac{210 - 206}{4} = 1 \to 1 α$

$N_{2} - N_{3} = (81 - 82) - \frac{4}{2} = - 3 \to 3 β^{-}$

$∴$ $1 α and 3 β^{-} emission.$

(S) ${}_{91}{Pa}^{228} \to {}_{88}{Ra}^{224}$

$N_{1} = \frac{228 - 224}{4} = 1 \to 1 α$

$N_{2} - N_{3} = (91 - 88) - \frac{4}{2} = 1 \to 1 β^{+}$

$∴$ $1 α and 1 β^{+} emission.$

Q 21 :

Match the nuclear processes given in Column I with the appropriate option(s) in Column II. [2015]

Column I Column II

(A) Nuclear fusion (p) Absorption of thermal neutrons by ${}_{92}^{235}U$

(B) Fission in a nuclear reactor (q) ${}_{27}^{60}{Co}$ nucleus

(C) $β$ -decay (r) Energy production in stars via hydrogen conversion to helium

(D) $γ$ -ray emission (s) Heavy water

(t) Neutrino emission

A → p, q, r, t; B → p, s; C → p, q, r, t; D → r, t
A → p, q, r, t; B → p, q, r, t; C → p, s; D → r, t
A → r, t; B → p, s; C → p, q, r, t; D → p, q, r, t.
A → p, q, r, t; B → p, s; C → p, q, r, t; D → r, t

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(3)

For A → r, t; B → p, s; C → p, q, r, t; D → p, q, r, t.

Q 22 :

Match List I of the nuclear processes with List II containing parent nucleus and one of the end products of each process and then select the correct answer using the codes given below the lists: [2013]

List I List II

P. Alpha decay 1. ${}_{8}^{15}O \to {}_{7}^{15}O + \dots$

Q. $β^{+}$ decay 2. ${}_{92}^{238}U \to {}_{90}^{234}{Th} + \dots$

R. Fission 3. ${}_{83}^{185}{Bi} \to {}_{82}^{184}{Pb} + \dots$

S. Proton emission 4. ${}_{94}^{239}{Pu} \to {}_{57}^{140}{La} + \dots$

Codes:

P - 4, Q - 2, R - 1, S - 3
P - 1, Q - 3, R - 2, S - 4
P - 2, Q - 1, R - 4, S - 3
P - 4, Q - 3, R - 2, S - 1

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(3)

$In β^{+} -decay, atomic number (Z) decreases by 1 and mass number (A) remains unchanged.$

${}_{8}^{15}O \to {}_{7}^{15}N + \underset{β^{+} particle}{{}_{+ 1}^{0}β}$

$In α -decay, mass number (A) decreases by 4 units and atomic number (Z) by 2 units.$

${}_{92}^{238}U \to {}_{90}^{234}{Th} + \underset{α -particle}{{}_{2}^{4}{He}}$

$In proton ({}_{1}^{1}H) emission, both (A) and (Z) decrease by 1 .$

${}_{83}^{185}{Bi} \to {}_{82}^{184}{Pb} + \underset{proton}{{}_{1}^{1}H}$

$In fission process, heavier nucleus breaks into two fragments.$

${}_{94}^{239}{Pu} \to {}_{57}^{140}{La} + {}_{37}^{99}X$

Q 23 :

Given below are certain matching type questions, where two columns (each having 4 items) are given. Immediately after the columns the matching grid is given, where each item of Column I has to be matched with the items of Column II, by encircling the correct match(es). Note that an item of Column I can match with more than one item of Column II. All the items of Column II must be matched. Match the following: [2006]

Column I Column II

(A) Nuclear fusion (p) Converts some matter into energy

(B) Nuclear fission (q) Generally possible for nuclei with low atomic number

(C) $β$ -decay (r) Generally possible for nuclei with higher atomic number

(D) Exothermic nuclear reaction (s) Essentially proceeds by weak nuclear forces

A → p, q, r; B → p, r; C → p, s; D → p, q
A → p, q; B → p, r; C → p, s; D → p, q, r
A → p, q, r; B → p, s; C → p, r; D → p, q
A → p, r; B → p, q, r; C → p, s; D → p, q

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(2)

A → p, q; B → p, r; C → p, s; D → p, q, r

In a nuclear fusion reaction, two or more lighter nuclei combine to give a comparatively heavier nucleus and some matter is converted into energy.

In a nuclear fission reaction, a heavy nucleus breaks into two or more lighter nuclei and some matter is converted into energy.

$β$ -decay essentially proceeds by weak nuclear forces and converts some matter into energy.

Exothermic nuclear reaction is possible for both nuclei with low and high atomic number and releases energy.

Q 24 :

The mass of a nucleus ${}_{Z}^{A}X$ is less than the sum of the masses of $(A - Z)$ number of neutrons and Z number of protons in the nucleus. The energy equivalent to the corresponding mass difference is known as the binding energy of the nucleus. A heavy nucleus of mass M can break into two light nuclei of masses $m_{1}$ and $m_{2}$ only if $(m_{1} + m_{2}) < M$ . Also two light nuclei of masses $m_{3}$ and $m_{4}$ can undergo complete fusion and form a heavy nucleus of mass $M'$ only if $(m_{3} + m_{4}) > M'$ . The masses of some neutral atoms are given in the table below:

Q. The kinetic energy (in keV) of the alpha particle, when the nucleus ${}_{84}^{210}{Po}$ at rest undergoes alpha decay, is [2013]

5319
5422
5707
5818

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(1)

${}_{84}^{210}{Po} ⟶ {}_{82}^{206}{Pb} + {}_{2}^{4}{He}$

$Mass defect$

$Δ m = [209.982876 - (205.974455 + 4.002603)] = 0.005818 μ$

$= 5.422 MeV = 5422 keV$

$∴$ $Q = (Δ m) \times 932 = (0.005818) \times 932$

$K_{α} = (\frac{A - 4}{A}) Q = (\frac{210 - 4}{210}) \times 5422$

$∴$ $K_{α} = 5319 keV$

Q 25 :

The mass of a nucleus ${}_{Z}^{A}X$ is less than the sum of the masses of $(A - Z)$ number of neutrons and $Z$ number of protons in the nucleus. The energy equivalent to the corresponding mass difference is known as the binding energy of the nucleus. A heavy nucleus of mass $M$ can break into two light nuclei of masses $m_{1}$ and $m_{2}$ only if $(m_{1} + m_{2}) < M$ . Also two light nuclei of masses $m_{3}$ and $m_{4}$ can undergo complete fusion and form a heavy nucleus of mass $M'$ only if $(m_{3} + m_{4}) > M'$ . The masses of some neutral atoms are given in the table below:

Q. The correct statement is [2013]

The nucleus ${}_{3}^{6}{Li}$ can emit an alpha particle
The nucleus ${}_{84}^{210}{Po}$ can emit a proton
Deuteron and alpha particle can undergo complete fusion
The nuclei ${}_{30}^{70}{Zn}$ and ${}_{34}^{82}{Se}$ can undergo complete fusion

1

2

3

4

(3)

In case of (3) only, mass defect $(Δ m)$ is positive. In all other cases (1), (2) and (4), $Δ m$ is negative.

Hence, deuteron and $α$ -particle can undergo complete fusion.

Topic Question Set

Q 1 :

A heavy nucleus Q of half-life 20 minutes undergoes alpha-decay with probability of 60% and beta-decay with probability of 40%. Initially, the number of Q nuclei is 1000. The number of alpha-decays of Q in the first one hour is [2021]

Q 3 :

A radioactive sample $S_{1}$ having an activity $5 μ Ci$ has twice the number of nuclei as another sample $S_{2}$ which has an activity of $10 μ Ci$ . The half-lives of $S_{1}$ and $S_{2}$ can be [2008]

Q 4 :

${}_{87}^{221}{Ra}$ is a radioactive substance having a half-life of 4 days. Find the probability that a nucleus undergoes decay after two half-lives. [2006]

Q 5 :

A 280 days old radioactive substance shows an activity of 6000 dps. 140 days later, its activity becomes 3000 dps. What was its initial activity? [2004]

Q 6 :

A nucleus with mass number 220, initially at rest, emits an $α$ -particle. If the Q-value of the reaction is 5.5 MeV, calculate the kinetic energy of the $α$ -particle. [2003]

Q 7 :

Which of the following processes represents a $γ$ -decay? [2002]

Q 8 :

The half-life of ${}^{215}{At}$ is $100 μ s$ . The time taken for the radioactivity of a sample of ${}^{215}{At}$ to decay to ${\frac{1}{16}}^{th}$ of its initial value is [2002]

Q 10 :

The electron emitted in beta radiation originates from [2001]

Q 11 :

Two radioactive materials $X_{1}$ and $X_{2}$ have decay constants $10 λ$ and $λ$ , respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of $X_{1}$ to that of $X_{2}$ will be $\frac{1}{e}$ after a time [2000]

Q 16 :

The activity of a freshly prepared radioactive sample is $10^{10}$ disintegrations per second, whose mean life is $10^{9} s$ . The mass of an atom of this radioisotope is $10^{- 25} kg$ . The mass (in mg) of the radioactive sample is [2011]

Q 19 :

In a radioactive decay chain, ${}_{90}^{232}{Th}$ nucleus decays to ${}_{82}^{212}{Pb}$ nucleus. Let $N_{α}$ and $N_{β}$ be the number of $α$ and $β^{-}$ particles, respectively, emitted in this decay process. Which of the following statement(s) is(are) true? [2018]

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	List-I		List-II
(P)	${}_{92}^{238}U \to {}_{91}^{234}{Pa}$	(1)	one $α$ particle and one $β^{+}$ particle
(Q)	${}_{82}^{214}{Pb} \to {}_{82}^{210}{Pb}$	(2)	three $β^{-}$ particles and one $α$ particle
(R)	${}_{81}^{210}{Tl} \to {}_{82}^{206}{Pb}$	(3)	two $β^{-}$ particles and one $α$ particle
(S)	${}_{91}^{228}{Pa} \to {}_{88}^{224}{Ra}$	(4)	one $α$ particle and one $β^{-}$ particle
		(5)	one $α$ particle and two $β^{+}$ particles

	Column I		Column II
(A)	Nuclear fusion	(p)	Absorption of thermal neutrons by ${}_{92}^{235}U$
(B)	Fission in a nuclear reactor	(q)	${}_{27}^{60}{Co}$ nucleus
(C)	$β$ -decay	(r)	Energy production in stars via hydrogen conversion to helium
(D)	$γ$ -ray emission	(s)	Heavy water
		(t)	Neutrino emission

	List I		List II
P.	Alpha decay	1.	${}_{8}^{15}O \to {}_{7}^{15}O + \dots$
Q.	$β^{+}$ decay	2.	${}_{92}^{238}U \to {}_{90}^{234}{Th} + \dots$
R.	Fission	3.	${}_{83}^{185}{Bi} \to {}_{82}^{184}{Pb} + \dots$
S.	Proton emission	4.	${}_{94}^{239}{Pu} \to {}_{57}^{140}{La} + \dots$

Topic Question Set

Q 1 : A heavy nucleus Q of half-life 20 minutes undergoes alpha-decay with probability of 60% and beta-decay with probability of 40%. Initially, the number of Q nuclei is 1000. The number of alpha-decays of Q in the first one hour is [2021]

Q 3 : A radioactive sample S1 having an activity 5μCi has twice the number of nuclei as another sample S2 which has an activity of 10μCi. The half-lives of S1 and S2 can be [2008]

Q 4 : Ra87221 is a radioactive substance having a half-life of 4 days. Find the probability that a nucleus undergoes decay after two half-lives. [2006]

Q 5 : A 280 days old radioactive substance shows an activity of 6000 dps. 140 days later, its activity becomes 3000 dps. What was its initial activity? [2004]

Q 6 : A nucleus with mass number 220, initially at rest, emits an α-particle. If the Q-value of the reaction is 5.5 MeV, calculate the kinetic energy of the α-particle. [2003]

Q 7 : Which of the following processes represents a γ-decay? [2002]

Q 8 : The half-life of At215 is 100μs. The time taken for the radioactivity of a sample of At215 to decay to 116th of its initial value is [2002]

Q 10 : The electron emitted in beta radiation originates from [2001]

Q 11 : Two radioactive materials X1 and X2 have decay constants 10λ and λ, respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of X1 to that of X2 will be 1e after a time [2000]

Q 16 : The activity of a freshly prepared radioactive sample is 1010 disintegrations per second, whose mean life is 109 s. The mass of an atom of this radioisotope is 10-25 kg. The mass (in mg) of the radioactive sample is [2011]

Q 17 : To determine the half-life of a radioactive element, a student plots a graph of ln|dN(t)dt| versus t. Here, |dN(t)dt| is the rate of radioactive decay at time t. If the number of radioactive nuclei of this element decreases by a factor of p after 4.16 years, the value of p is [2010]

Q 19 : In a radioactive decay chain, Th90232 nucleus decays to Pb82212 nucleus. Let Nα and Nβ be the number of α and β- particles, respectively, emitted in this decay process. Which of the following statement(s) is(are) true? [2018]

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Q 1 :

A heavy nucleus Q of half-life 20 minutes undergoes alpha-decay with probability of 60% and beta-decay with probability of 40%. Initially, the number of Q nuclei is 1000. The number of alpha-decays of Q in the first one hour is [2021]

Q 3 :

A radioactive sample $S_{1}$ having an activity $5 μ Ci$ has twice the number of nuclei as another sample $S_{2}$ which has an activity of $10 μ Ci$ . The half-lives of $S_{1}$ and $S_{2}$ can be [2008]

Q 4 :

${}_{87}^{221}{Ra}$ is a radioactive substance having a half-life of 4 days. Find the probability that a nucleus undergoes decay after two half-lives. [2006]

Q 5 :

A 280 days old radioactive substance shows an activity of 6000 dps. 140 days later, its activity becomes 3000 dps. What was its initial activity? [2004]

Q 6 :

A nucleus with mass number 220, initially at rest, emits an $α$ -particle. If the Q-value of the reaction is 5.5 MeV, calculate the kinetic energy of the $α$ -particle. [2003]

Q 7 :

Which of the following processes represents a $γ$ -decay? [2002]

Q 8 :

The half-life of ${}^{215}{At}$ is $100 μ s$ . The time taken for the radioactivity of a sample of ${}^{215}{At}$ to decay to ${\frac{1}{16}}^{th}$ of its initial value is [2002]

Q 10 :

The electron emitted in beta radiation originates from [2001]

Q 11 :

Two radioactive materials $X_{1}$ and $X_{2}$ have decay constants $10 λ$ and $λ$ , respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of $X_{1}$ to that of $X_{2}$ will be $\frac{1}{e}$ after a time [2000]

Q 16 :

The activity of a freshly prepared radioactive sample is $10^{10}$ disintegrations per second, whose mean life is $10^{9} s$ . The mass of an atom of this radioisotope is $10^{- 25} kg$ . The mass (in mg) of the radioactive sample is [2011]

Q 19 :

In a radioactive decay chain, ${}_{90}^{232}{Th}$ nucleus decays to ${}_{82}^{212}{Pb}$ nucleus. Let $N_{α}$ and $N_{β}$ be the number of $α$ and $β^{-}$ particles, respectively, emitted in this decay process. Which of the following statement(s) is(are) true? [2018]