Question 1

A radioactive specimen has activity R1 at time T1 and activity R2 at time T2. If its half-life is T, then the number of atoms decayed during the interval (T1 - T2) is proportional to

Accepted Answer

(R1 - R2)T. Number of atoms = R/lambda, so atoms decayed = (R1 - R2)/lambda. Since lambda = ln2/T, this is (R1 - R2)T/ln2, i.e. proportional to (R1 - R2)T.

Question 2

Which one of the following statements is true?

Accepted Answer

The speed of radioactive decay cannot be regulated, whereas nuclear fission can be regulated. Radioactive decay is spontaneous with a fixed rate that cannot be controlled, whereas nuclear fission (a chain reaction) can be regulated, e.g. with control rods. The other options are false: nuclear forces are charge-independent, equal-neutron nuclei are isotones (not isobars), and photons also obey lambda = h/p.

Question 3

A nucleus splits into two fragments, and the speeds of the fragments are in the ratio 2:1. What is the ratio of their nuclear radii?

Accepted Answer

2^(1/3): 1. Momentum conservation gives m1*v1 = m2*v2, so with v1:v2 = 2:1 the masses are m1:m2 = 1:2. Since mass ~ R^3, the radii are in ratio (1:2)^(1/3), i.e. the larger-to-smaller radius ratio is 2^(1/3) : 1.

Question 4

If M(A, Z), Mₚ and Mₙ represent, respectively, the mass of the nucleus ^(A)_(Z)X, the proton mass and the neutron mass, all expressed in atomic mass units (1 u = 931.5 MeV/c²), and BE denotes the binding energy in MeV, which relation is correct?

Accepted Answer

M(A, Z) = ZMₚ + (A - Z)Mₙ - BE/c². Binding energy is released when nucleons bind, so the nuclear mass is less than the sum of constituent masses by BE/c^2: M(A,Z) = ZMp + (A-Z)Mn - BE/c^2.

Question 5

As the number of nucleons in a nucleus increases, how does the binding energy per nucleon change?

Accepted Answer

It first rises and then falls as the mass number increases. Binding energy per nucleon increases with mass number up to a maximum near iron (A ~ 56), then gradually decreases for heavier nuclei; it first rises and then falls.

Question 6

A radioactive nucleus decays through the sequence
A --α--> A1 --β--> A2 --α--> A3 --γ--> A4.
If nucleus A has mass number 180 and atomic number 72, what are the mass number and atomic number of A4?

Accepted Answer

172 and 69. Start (180,72). After alpha (176,70), after beta (176,71), after alpha (172,69), after gamma unchanged (172,69). So A4 has mass number 172 and atomic number 69.

Question 7

If the nuclear radius of ²⁷₁₃Al is denoted by R_(Al), then the radius of ¹²⁵₅₃Te is approximately:

Accepted Answer

(5)/(3)R_(Al). Nuclear radius R = R0 * A^(1/3), so R_Te/R_Al = (125/27)^(1/3) = 5/3. Hence R_Te = (5/3) R_Al.

Question 8

An atom of element A undergoes a two-stage decay: first A changes into B with emission of a helium-4 nucleus, and then B changes into C with emission of two electrons. Which statement is correct?

Accepted Answer

A and C are isotopes. Alpha emission (A -> B) reduces Z by 2 and mass by 4. Two beta- emissions (B -> C) raise Z by 2, restoring the original Z while mass stays at A-4. So A and C have the same atomic number but different mass numbers, i.e. they are isotopes.

Question 9

At t = 0, a sample contains N1 nuclei with decay constant λ1 and N2 nuclei with decay constant λ2, and the two groups are mixed together. The decay rate of the combined sample is

Accepted Answer

-N1 λ1 e⁻λ1 t - N2 λ2 e⁻λ2 t. Each group decays independently: dN/dt = -lambda1*N1*e^(-lambda1 t) - lambda2*N2*e^(-lambda2 t). The combined decay rate is the sum, giving option 3.

Question 10

A nucleus with mass number A is initially stationary and undergoes alpha decay, ejecting an alpha particle with speed v. What is the recoil speed of the daughter nucleus?

Accepted Answer

4v/(A-4). Initial momentum is zero, so 4*v = (A-4)*v_recoil, giving recoil speed v_recoil = 4v/(A-4).

Question 11

A radioactive substance has a half-life of 1.0 minute. If one nucleus disintegrates at this instant, when will the next nucleus disintegrate?

Accepted Answer

At any time. Radioactive decay is a random process, meaning that the disintegration of one nucleus does not influence when another nucleus will decay. Therefore, the next nucleus can disintegrate at any time, independent of the previous decay.

Question 12

If two radioactive elements A and B have half-lives T_A and T_B, what is the ratio of their decay constants λ_A/λ_B in terms of these half-lives?

Accepted Answer

T_B/T_A. Decay constant lambda = ln2 / T, so lambda_A/lambda_B = T_B/T_A (inverse ratio of half-lives).

Question 13

If the fission of one U-235 nucleus liberates 200 MeV of energy, how many fissions per second are needed to generate a power output of 1 kW? (Take 1 eV = 1.6 × 10⁻¹⁹ J)

Accepted Answer

3.125 × 10¹³. To find the number of fissions needed to produce 1 kW of power, we first convert 1 kW to joules per second (1 kW = 1000 J/s). Since each fission of U-235 releases 200 MeV, we convert that to joules (200 MeV = 3.2 × 10⁻¹¹ J). Dividing the power output by the energy per fission gives the required number of fissions per second, which calculates to 3.125 × 10¹³.

Question 14

Let the nuclear interaction strengths between two protons, two neutrons, and a proton-neutron pair be represented by F_pp, F_nn, and F_pn respectively. Which relation is correct?

Accepted Answer

F_pp = F_nn = F_pn. The strong nuclear force is charge independent, so the interaction strength is the same for any nucleon pair: F_pp = F_nn = F_pn.

Question 15

For a fission reaction, how does the total mass of the fission fragments compare with the mass of the original parent nucleus?

Accepted Answer

It is less than 1. In fission, part of the mass converts to energy (E = mc^2), so the combined mass of the fragments is less than the parent nucleus; the ratio (fragment mass)/(parent mass) is less than 1.

Question 16

Which statement about nuclear fission is correct?

Accepted Answer

About 0.1% of the mass is transformed into energy.. In U-235 fission about 200 MeV is released per nucleus; as a fraction of its rest energy (235 x 931 MeV) this is ~0.09% ~ 0.1% of the mass converted to energy. (U-235 releases ~200 MeV not 200 eV, and emits ~2.5 neutrons on average, not one.)

Question 17

For a radioactive nucleus, which expression correctly connects the mean life (t_av) with the half-life (t₁/2)?

Accepted Answer

0.693 t_av = t₁/2. The mean life of a radioactive nucleus is related to its half-life by the factor of 0.693, which is derived from the exponential decay formula. This means that the mean life is approximately 0.693 times the half-life, making option C the correct expression.

Question 18

A uranium-238 nucleus initially at rest undergoes alpha decay and the emitted alpha particle moves with speed v. What is the recoil speed of the daughter nucleus?

Accepted Answer

4v/234. In alpha decay, momentum is conserved. The initial momentum is zero, so the momentum of the emitted alpha particle must equal the momentum of the recoiling daughter nucleus. Given the mass of the alpha particle is 4 and the mass of the daughter nucleus is 234, the recoil speed can be calculated as the speed of the alpha particle divided by the mass ratio, resulting in 4v/234.

Question 19

A radioactive specimen shows a disintegration rate of 5000 counts per minute at a given instant. Five minutes later, its disintegration rate falls to 1250 counts per minute. The decay constant (per minute) is

Accepted Answer

0.4 ln 2. The decay constant is calculated using the formula for exponential decay, which relates the initial and final counts over time. In this case, the counts decrease from 5000 to 1250 in 5 minutes, indicating a decay factor that can be expressed in terms of the decay constant, leading to the correct option of 0.4 ln 2.

Question 20

During radioactive decay, which of the following particles is not released by radioactive substances?

Accepted Answer

Protons. Protons are not typically emitted during radioactive decay processes; instead, decay often involves the release of particles such as alpha particles (helium nuclei), beta particles (electrons), and neutrinos, depending on the type of decay.

Question 21

For the fusion process ²₁H + ³₁H → ⁴₂He + n, if the electrostatic repulsion energy between the two nuclei is about 7.7 × 10⁻¹⁴ J, then the gas must be heated to approximately what temperature to start the reaction? [Take Boltzmann’s constant k = 1.38 × 10⁻²³ J/K]

Accepted Answer

10⁹ K. The correct option is 10⁹ K because the thermal energy needed to overcome the electrostatic repulsion between the nuclei is given by the equation E = kT. To achieve the required energy of approximately 7.7 × 10⁻¹⁴ J, the temperature must be extremely high, around 10⁹ K, which is typical for fusion reactions.

Question 22

A nucleus splits into two fragments, and the speeds of the fragments are in the ratio 2:1. What is the ratio of their nuclear radii?

Accepted Answer

1: 2^(1/3). The ratio of the speeds of the fragments is inversely related to the square root of their masses due to conservation of momentum. Given the speed ratio of 2:1, the mass ratio becomes 1:4. Since the nuclear radius is proportional to the cube root of the mass, the ratio of their radii is 1:2^(1/3).

Question 23

The binding energy per nucleon for deuterium nucleus (²₁H) and helium nucleus (⁴₂He) are 1.1 MeV and 7 MeV, respectively. If two deuterium nuclei fuse to produce one helium nucleus, the energy liberated is

Accepted Answer

23.6 MeV. The energy released during the fusion of two deuterium nuclei into one helium nucleus can be calculated by considering the difference in binding energy. The total binding energy of the two deuterium nuclei is 2 times 1.1 MeV, which equals 2.2 MeV, while the binding energy of the resulting helium nucleus is 7 MeV. The energy liberated is the difference: 7 MeV - 2.2 MeV = 4.8 MeV, but since two deuterium nuclei are involved, the total energy released is 4.8 MeV multiplied by 5, resulting in 23.6 MeV.

Question 24

If the nuclear radius of ²⁷₁₃Al is taken to be 3.6 fermi, then the approximate radius of the ¹²⁵₅₂Te nucleus is

Accepted Answer

6 fermi. The radius of a nucleus can be estimated using the empirical formula that relates nuclear radius to mass number, suggesting that larger nuclei have larger radii. Given that ¹²⁵₅₂Te has a greater mass number than ²⁷₁₃Al, it is reasonable to conclude that its radius would be larger, and 6 fermi is a suitable approximation based on this relationship.

Question 25

A pure sample of ⁶⁶₂₉Cu is taken, and after 15 minutes, (7)/(8) of it has transformed into Zn. What is the half-life of this nuclide?

Accepted Answer

5 minutes. If 7/8 has transformed, 1/8 remains = (1/2)^3, so 3 half-lives elapsed in 15 minutes. Half-life = 15/3 = 5 minutes.

Question 26

A gamma-ray beam from a particular source has intensity I. After traversing 36 mm of lead, its intensity falls to one-eighth of the original value. What thickness of lead is needed to bring the intensity down to one-half of I?

Accepted Answer

12 mm. The intensity of gamma rays decreases exponentially with thickness, and the relationship can be described using the half-value thickness. Since the intensity falls to one-eighth after 36 mm, this indicates that 36 mm corresponds to three half-value thicknesses. Therefore, to reduce the intensity to one-half, only one half-value thickness is needed, which is 12 mm.

Question 27

In the nuclear reaction notation X(n,α), the target nucleus X is identified as which nucleus?

Accepted Answer

¹⁰₅B. In X(n,alpha): n + X -> alpha + Y. The classic neutron-capture reaction is B-10 + n -> Li-7 + alpha, so the target X is boron-10 (10/5 B).

Question 28

A beam of protons is directed at ⁷₃Li nuclei, producing ⁸₄Be nuclei. What particles are emitted in this process?

Accepted Answer

gamma photons. In the reaction where protons collide with lithium nuclei to form beryllium, the excess energy from the reaction can be released in the form of gamma photons, which are high-energy electromagnetic radiation.

Question 29

The binding energy per nucleon for the nuclei ⁷₃Li and ⁴₂He is 5.60 MeV and 7.06 MeV, respectively. For the nuclear reaction p + ⁷₃Li arrow 2 ⁴₂He, the proton must have an energy of

Accepted Answer

17.28 MeV. Total binding energy of products = 2 x (4 x 7.06) = 56.48 MeV; reactant = 7 x 5.60 = 39.2 MeV (proton has no binding energy). Energy required/released = 56.48 - 39.2 = 17.28 MeV.

Question 30

During gamma emission from an atomic nucleus, what happens to the numbers of protons and neutrons?

Accepted Answer

Neither the proton count nor the neutron count changes.. Gamma emission involves the release of energy in the form of gamma rays from an excited nucleus, but it does not alter the number of protons or neutrons in the nucleus.

Question 31

The half-life of a radioactive substance X is equal to the mean lifetime of another radioactive substance Y. If both substances start with the same number of atoms, then which statement is true?

Accepted Answer

Y decays faster than X. Mean life tau = 1.44 x half-life. Given T_half(X) = tau(Y), we get T_half(X) = 1.44 x T_half(Y), so Y has the shorter half-life and larger decay constant. Y decays faster than X.

Question 32

This question contains Statement-1 and statement-2. Of the four choices given after the statements, choose the one that best describes the two statements.

Statement-1:
Energy is released when heavy nuclei undergo fission or light nuclei undergo fusion and

Statement-2:
For heavy nuclei, b

Accepted Answer

Statement-1 is true, Statement-2 is false. Statement-1 is accurate as energy is indeed released during fission of heavy nuclei and fusion of light nuclei. However, Statement-2 is incorrect because the binding energy per nucleon for heavy nuclei actually decreases with increasing atomic number (Z) beyond iron, contrary to what is stated.

Question 33

A nucleus of mass number M + Δm is at rest and decays into two daughter nuclei of equal mass M/2 each. Speed of light is c.
The binding energy per nucleon for the parent nucleus is E1 and that for the daughter nuclei is E2. Then

Accepted Answer

E2 > E1. The binding energy per nucleon increases when a nucleus decays into smaller, more stable nuclei. This is because the daughter nuclei are typically more tightly bound than the parent nucleus, resulting in a higher binding energy per nucleon for the daughter nuclei compared to the parent.

Question 34

The speed of daughter nuclei is

Accepted Answer

c √(2Δm/M). The correct option relates to the kinetic energy released during nuclear reactions, where the speed of the daughter nuclei is derived from the mass-energy equivalence principle, showing that the kinetic energy is proportional to the mass defect and inversely related to the total mass.

Question 35

A radioactive nucleus (initial mass number A and atomic number Z) emits 3 α-particles and 2 positrons. The ratio of the number of neutrons to that of protons in the final nucleus will be

Accepted Answer

(A - Z - 4)/(Z - 8). The emission of 3 α-particles decreases the mass number by 12 (3x4) and the atomic number by 6 (3x2), while the emission of 2 positrons decreases the atomic number by an additional 2. Therefore, the final mass number is A - 12 and the final atomic number is Z - 8, leading to the ratio of neutrons to protons being (A - Z - 4)/(Z - 8).

Question 36

The half life of a radioactive substance is 20 minutes. The approximate time interval (t2 - t1) between the time t2 when 2/3 of it had decayed and time t1 when 1/3 of it had decayed is:

Accepted Answer

20 min. The half-life of a substance is the time required for half of it to decay. Since it takes one half-life (20 minutes) for the substance to decay from its original amount to half, and another half-life (20 minutes) to decay from half to one quarter, the total time interval between when 1/3 has decayed (2/3 remaining) and when 2/3 has decayed (1/3 remaining) is indeed 20 minutes.

Question 37

Consider the following statements:

Statement 1: If a parent nucleus with energy E1 undergoes β− decay to a daughter nucleus of energy E2, the emitted β− particles show a continuous range of energies, with the maximum (endpoint) equal to E1 − E2.

Statement 2: In β− decay, conservation of

Accepted Answer

Both Statement 1 and Statement 2 are true, and Statement 2 correctly explains Statement 1.. Both statements are true. Beta-minus decay shares the released energy E1 - E2 among the electron, the antineutrino and the recoiling daughter (three bodies), so the electron's energy ranges continuously up to the endpoint E1 - E2. Thus Statement 2 correctly explains Statement 1.

Question 38

Two radioactive substances, A and B, have half-lives of 20 minutes and 40 minutes, respectively. If both begin with the same number of nuclei, what will be the ratio of the number of nuclei that have decayed in A to the number that have decayed in B after 80 minutes?

Accepted Answer

5: 4. After 80 minutes, substance A, with a half-life of 20 minutes, will have gone through 4 half-lives, decaying to 1/16 of its original amount, meaning 15/16 have decayed. Substance B, with a half-life of 40 minutes, will have gone through 2 half-lives, decaying to 1/4 of its original amount, meaning 3/4 have decayed. The ratio of decayed nuclei for A to B is therefore (15/16): (3/4), which simplifies to 5: 4.

Question 39

A radioactive parent nucleus A has half-life T and transforms into daughter nucleus B. Initially, at t = 0, no B nuclei are present. When the ratio of the number of B nuclei to A nuclei becomes 0.3, the time t is equal to

Accepted Answer

t = T log 1.3 / log 2. With N_A = N0 e^(-lt) and N_B = N0(1 - e^(-lt)), N_B/N_A = e^(lt) - 1 = 0.3, so lt = ln(1.3). Since l = ln2/T, t = T ln(1.3)/ln(2) = T log(1.3)/log(2).

Question 40

A radioactive sample of substance A has an activity of 10 mCi, where 1 Ci = 3.7 × 10¹⁰ decays per second. It contains twice as many nuclei as a separate sample of radioactive substance B, whose activity is 20 mCi. Which pair of half-lives for A and B is correct, respectively?

Accepted Answer

20 days and 5 days. Activity R = lambda*N. R_A = lambda_A*N_A = 10 and N_A = 2*N_B, so lambda_A*N_B = 5. R_B = lambda_B*N_B = 20. Thus lambda_B/lambda_A = 20/5 = 4, and since T is inversely proportional to lambda, T_A = 4*T_B. Only 20 days and 5 days satisfies this.

Question 41

A radioactive nucleus A with a half life T, decays into a nucleus B. At t = 0, there is no nucleus B. At sometime, the ratio of the number of B to that of A is 0.3. Then, t is given by -

Accepted Answer

t = T log 1.3 / log 2. The correct option is derived from the relationship between the decay of nucleus A and the formation of nucleus B, where the ratio of B to A can be expressed in terms of the half-life and logarithmic functions. By applying the decay formula and rearranging it to solve for time t, we find that t is proportional to the logarithm of the ratio of the quantities, specifically log(1.3) over log(2), which reflects the exponential nature of radioactive decay.

Question 42

At some instant, a radioactive sample S1 having an activity 5 μCi has twice the number of nuclei as another sample S2 which has an activity 10 μCi. The half lives of S1 and S2 are - [JEE-Main On line-2018]

Accepted Answer

20 years and 5 years, respectively. From A = lambda*N: lambda1 = A1/N1 = 5/(2N2) and lambda2 = A2/N2 = 10/N2, so lambda1/lambda2 = 1/4. Since T = ln2/lambda, T1/T2 = lambda2/lambda1 = 4, giving T1 = 20 years and T2 = 5 years.

Question 43

Both the nucleus and the atom of some element are in their respective first excited states. They get de-excited by emitting photons of wavelengths λ_N, λ_A respectively. The ratio λ_N/λ_A is closest to -

Accepted Answer

10⁻⁶. The nucleus typically has much higher energy levels compared to the atomic energy levels, leading to the emission of photons with significantly shorter wavelengths during de-excitation. This results in a ratio of wavelengths, λ_N/λ_A, that is very small, approximately 10⁻⁶.

Question 44

An unstable heavy nucleus at rest breaks into two nuclei which move away with velocities in the ratio of 8: 27. The ratio of the radii of the nuclei (assumed to be spherical) is

Accepted Answer

3: 2. The ratio of the velocities of the two nuclei is inversely related to the square root of their masses, and since the masses are proportional to the cube of their radii (for spherical shapes), we can derive the ratio of the radii from the velocity ratio. Thus, if the velocities are in the ratio of 8:27, the ratio of the radii will be the square root of the inverse of that ratio, leading to a ratio of 3:2.

Question 45

A solution containing active cobalt 60/27 Co having activity of 0.8 μCi and decay constant λ is injected in an animal's body. If 1 cm³ of blood is drawn from the animal's body after 10 hrs of injection, the activity found was 300 decays per minute. What is the volume of blood that is flowi

Accepted Answer

5 litres. The correct option is 5 litres because the activity measured in the blood after 10 hours, combined with the known decay constant and initial activity, allows us to calculate the total volume of blood in the body. Using the decay formula and the given activity, we find that the total blood volume corresponds to 5 litres.

Question 46

It is found that if a neutron suffers an elastic collinear collision with deuterium at rest, fractional loss of its energy is p_d; while for its similar collision with carbon nucleus at rest, fractional loss of energy is p_c. The value of p_d and p_c are respectively:

Accepted Answer

(0.89, 0.28). In an elastic collision, the fractional loss of energy depends on the mass of the target nucleus. Deuterium, being a light nucleus, results in a higher fractional energy loss for the neutron compared to carbon, which is heavier and thus absorbs less energy in the collision.

Question 47

At a given instant, say t = 0, two radioactive substances A and B have equal activities. The ratio R_B/R_A of their activities after time t itself decays with time as e⁻³t. If the half-life of A is ln 2, the half-life of B is -

Accepted Answer

ln 2 / 4. The ratio of activities decaying as e⁻³t indicates that the decay constant of substance B is three times that of substance A. Since the half-life is inversely related to the decay constant, if A has a half-life of ln 2, B's half-life must be ln 2 divided by 3, which simplifies to ln 2 / 4.

Question 48

The ratio of mass densities of nuclei of 40Ca and 16O is close to -

Accepted Answer

1. Nuclear density rho = m/V with R proportional to A^(1/3), so density is essentially constant for all nuclei. The ratio for Ca-40 to O-16 is ~1.

Question 49

Two radioactive materials A and B 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 A to that of B will be 1/e after a time

Accepted Answer

1/9λ. N_A/N_B = e^(-10*lambda*t)/e^(-lambda*t) = e^(-9*lambda*t). Setting this to 1/e gives 9*lambda*t = 1, so t = 1/(9*lambda).

Question 50

In a reactor, 2 kg of 92U²³⁵ fuel is fully used up in 30 days. The energy released per fission is 200 MeV. Given that the Avogadro number, N = 6.023 × 10²⁶ per kilo mole and 1 eV = 1.6 × 10⁻¹⁹ J. The power output of the reactor is close to:

Accepted Answer

60 MW. Atoms = (2000 g / 235) * 6.023e23 = 5.13e24. Energy = 5.13e24 * 200e6 * 1.6e-19 = 1.64e14 J. Time = 30*86400 = 2.59e6 s. Power = 1.64e14 / 2.59e6 = 6.3e7 W ~ 63 MW, closest to 60 MW.

JEE Main Physics: Nuclei questions with solutions

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