HC Verma solutions for Concepts of Physics Vol. 2 [English] Class 11 and 12 chapter 24 - The Nucleus [Latest edition]

Consider two pairs of neutrons. In each pair, the separation between the neutrons is the same. Can the force between the neutrons have different magnitudes for the two pairs?

A molecule of hydrogen contains two protons and two electrons. The nuclear force between these two protons is always neglected while discussing the behaviour of a hydrogen molecule. Why?

Suppose we have 12 protons and 12 neutrons. We can assemble them to form either a ²⁴Mg nucleus or two ¹²C nuclei. In which of the two cases more energy will be liberated?

What is the difference between cathode rays and beta rays? When the two are travelling in space, can you make out which is the cathode ray and which is the beta ray?

If the nucleons of a nucleus are separated from each other, the total mass is increased. Where does this mass come from?

In beta decay, an electron (or a positron) is emitted by a nucleus. Does the remaining atom get oppositely charged?

When a boron nucleus `(""_5^10"B")` is bombarded by a neutron, a α-particle is emitted. Which nucleus will be formed as a result?

In a typical fission reaction, the nucleus is split into two middle-weight nuclei of unequal masses. Which of the two (heavier or lighter) has greater kinetic energy? Which one has greater liner momentum? 

If three helium nuclei combine to form a carbon nucleus, energy is liberated. Why can't helium nuclei combine on their own and minimise the energy?

The mass of a neutral carbon atom in ground state is

The mass number of a nucleus is equal to

As compared to ¹²C atom, ¹⁴C atom has

The mass number of a nucleus is

The graph of ln `("R"/"R"_0)` versus ln A (R = radius of a nucleus and A = its mass number) is ______.

Let F_pp, F_pn and F_nn denote the magnitudes of the net force by a proton on a proton, by a proton on a neutron and by a neutron on a neutron respectively.  neglect gravitational force. When the separation is 1 fm.

Let F_pp, F_pn and F_nn denote the magnitudes of the net force by a proton on a proton, by a proton on a neutron and by a neutron on a neutron respectively.  neglect gravitational force. When the separation is 1 fm.

Two protons are kept at a separation of 10 nm. Let F_n and F_e be the nuclear force and the electromagnetic force between them.

As the mass number A increases, the binding energy per nucleon in a nucleus

Which of the following is a wrong description of binding energy of a nucleus?

In one average-life,

In a radioactive decay, neither the atomic number nor the mass number changes. Which of the following particles is emitted in the decay?

During a negative beta decay,

A freshly prepared radioactive source of half-life 2 h emits radiation of intensity which is 64 times the permissible safe level. The minimum time after which it would be possible to work safely with this source is

The decay constant of a radioactive sample is λ. The half-life and the average-life of the sample are respectively

An α-particle is bombarded on ¹⁴N. As a result, a ¹⁷O nucleus is formed and a particle is emitted. This particle is a

Ten grams of ⁵⁷Co kept in an open container beta-decays with a half-life of 270 days. The weight of the material inside the container after 540 days will be very nearly

Free ²³⁸U nuclei kept in a train emit alpha particles. When the train is stationary and a uranium nucleus decays, a passenger measures that the separation between the alpha particle and the recoiling nucleus becomes x in time t after the decay. If a decay takes place when the train is moving at a uniform speed v, the distance between the alpha particle and the recoiling nucleus at a time t after the decay, as measured by the passenger will be

During a nuclear fission reaction,

As the mass number A increases, which of the following quantities related to a nucleus do not change?

The heavier nuclei tend to have larger N/Z ratio because
(a) a neutron is heavier than a proton
(b) a neutron is an unstable particle
(c) a neutron does not exert electric repulsion
(d) Coulomb forces have longer range compared to the nuclear forces.

A free neutron decays to a proton but a free proton does not decay to a neutron. This is because

Consider a sample of a pure beta-active material.

In which of the following decays the element does not change?

In which of the following decays the atomic number decreases?

(a) α-decay
(b) β⁺-decay
(c) β⁻-decay
(d) γ-decay

Magnetic field does not cause deflection in

Which of the following are electromagnetic waves?

Two lithium nuclei in a lithium vapour at room temperature do not combine to form a carbon nucleus because

For nuclei with A > 100,
(a) the binding energy of the nucleus decreases on an average as A increases
(b) the binding energy per nucleon decreases on an average as A increases
(c) if the nucleus breaks into two roughly equal parts, energy is released
(d) if two nuclei fuse to form a bigger nucleus, energy is released.

Assume that the mass of a nucleus is approximately given by M = Am_p where A is the mass number. Estimate the density of matter in kgm⁻³ inside a nucleus. What is the specific gravity of nuclear matter?

A neutron star has a density equal to that of the nuclear matter. Assuming the star to be spherical, find the radius of a neutron star whose mass is 4.0 × 10³⁰ kg (twice the mass of the sun).

How much energy is released in the following reaction : ⁷Li + p → α + α.
Atomic mass of ⁷Li = 7.0160 u and that of ⁴He = 4.0026 u.

(Use Mass of proton m_p = 1.007276 u, Mass of `""_1^1"H"` atom = 1.007825 u, Mass of neutron m_n = 1.008665 u, Mass of electron = 0.0005486 u ≈ 511 keV/c²,1 u = 931 MeV/c².)

Find the binding energy per nucleon of `""_79^197"Au"` if its atomic mass is 196.96 u.

(Use Mass of proton m_p = 1.007276 u, Mass of `""_1^1"H"` atom = 1.007825 u, Mass of neutron m_n = 1.008665 u, Mass of electron = 0.0005486 u ≈ 511 keV/c²,1 u = 931 MeV/c².)

(a) Calculate the energy released if ²³⁸U emits an α-particle. (b) Calculate the energy to be supplied to ²³⁸U it two protons and two neutrons are to be emitted one by one. The atomic masses of ²³⁸U, ²³⁴Th and ⁴He are 238.0508 u, 234.04363 u and 4.00260 u respectively.

(Use Mass of proton m_p = 1.007276 u, Mass of `""_1^1"H"` atom = 1.007825 u, Mass of neutron m_n = 1.008665 u, Mass of electron = 0.0005486 u ≈ 511 keV/c²,1 u = 931 MeV/c².)

Find the energy liberated in the reaction
²²³Ra → ²⁰⁹Pb + ¹⁴C.
The atomic masses needed are as follows.
²²³Ra         ²⁰⁹Pb        ¹⁴C
22..018 u  208.981 u  14.003 u

Show that the minimum energy needed to separate a proton from a nucleus with Zprotons and N neutrons is `ΔE = (M_(Z-1,N) + M_B - M_(Z,N))c^2` 

where M_Z,N = mass of an atom with Z protons and N neutrons in the nucleus and M_B = mass of a hydrogen atom. This energy is known as proton-separation energy.

Calculate the minimum energy needed to separate a neutron from a nucleus with Zprotons and N neutrons it terms of the masses M_Z.N' M_Z,N−1 and the mass of the neutron.

³²P beta-decays to ³²S. Find the sum of the energy of the antineutrino and the kinetic energy of the β-particle. Neglect the recoil of the daughter nucleus. Atomic mass of ³²P = 31.974 u and that of ³²S = 31.972 u.

A free neutron beta-decays to a proton with a half-life of 14 minutes. (a) What is the decay constant? (b) Find the energy liberated in the process.

(Use Mass of proton m_p = 1.007276 u, Mass of `""_1^1"H"` atom = 1.007825 u, Mass of neutron m_n = 1.008665 u, Mass of electron = 0.0005486 u ≈ 511 keV/c²,1 u = 931 MeV/c².)

Complete the following decay schemes.

(a) `"" _88^226Ra → alpha+`

(b) `""_8^19O → _9^19F+`

(c) `""_13^25Al → ""_12^25Mg+`

In the decay ⁶⁴Cu → ⁶⁴Ni + e⁺ + v, the maximum kinetic energy carried by the positron is found to be 0.650 MeV.
(a) What is the energy of the neutrino which was emitted together with a positron of kinetic energy 0.150 MeV?
(b) What is the momentum of this neutrino in kg m s⁻¹?
Use the formula applicable to a photon.

Potassium-40 can decay in three modes. It can decay by β⁻-emission, B*-emission of electron capture. (a) Write the equations showing the end products. (b) Find the Q-values in each of the three cases. Atomic masses of `""_18^40Ar` , `""_19^40K` and `""_20^40Ca` are 39.9624 u, 39.9640 u and 39.9626 u respectively.

(Use Mass of proton m_p = 1.007276 u, Mass of `""_1^1"H"` atom = 1.007825 u, Mass of neutron m_n = 1.008665 u, Mass of electron = 0.0005486 u ≈ 511 keV/c²,1 u = 931 MeV/c².)

Lithium (Z = 3) has two stable isotopes ⁶Li and ⁷Li. When neutrons are bombarded on lithium sample, electrons and α-particles are ejected. Write down the nuclear process taking place.

The masses of ¹¹C and ¹¹B are respectively 11.0114 u and 11.0093 u. Find the maximum energy a positron can have in the β*-decay of ¹¹C to ¹¹B.

(Use Mass of proton m_p = 1.007276 u, Mass of `""_1^1"H"` atom = 1.007825 u, Mass of neutron m_n = 1.008665 u, Mass of electron = 0.0005486 u ≈ 511 keV/c²,1 u = 931 MeV/c².)

²⁸Th emits an alpha particle to reduce to ²²⁴Ra. Calculate the kinetic energy of the alpha particle emitted in the following decay:

`""^228"Th" → ""^224"Ra"^(∗) + alpha`

`""^224"Ra"^(∗) → ""^224"Ra" + γ (217 "keV")`.

Atomic mass of ²²⁸Th is 228.028726 u, that of ²²⁴Ra is 224.020196 u and that of  `""_2^4H` is 4.00260 u.

(Use Mass of proton m_p = 1.007276 u, Mass of `""_1^1"H"` atom = 1.007825 u, Mass of neutron m_n = 1.008665 u, Mass of electron = 0.0005486 u ≈ 511 keV/c²,1 u = 931 MeV/c².)

Calculate the maximum kinetic energy of the beta particle emitted in the following decay scheme:
¹²N → ¹²C* + e⁺ + v
¹²C* → ¹²C + γ (4.43MeV).
The atomic mass of ¹²N is 12.018613 u.

(Use Mass of proton m_p = 1.007276 u, Mass of `""_1^1"H"` atom = 1.007825 u, Mass of neutron m_n = 1.008665 u, Mass of electron = 0.0005486 u ≈ 511 keV/c²,1 u = 931 MeV/c².)

The decay constant of `""_80^197`Hg (electron capture to `""_79^197`Au) is 1.8 × 10⁻⁴ S⁻¹. (a) What is the half-life? (b) What is the average-life? (c) How much time will it take to convert 25% of this isotope of mercury into gold?

The half-life of ¹⁹⁹Au is 2.7 days. (a) Find the activity of a sample containing 1.00 µg of ¹⁹⁸Au. (b) What will be the activity after 7 days? Take the atomic weight of ¹⁹⁸Au to be 198 g mol⁻¹.

Radioactive ¹³¹I has a half-life of 8.0 days. A sample containing ¹³¹I has activity 20 µCi at t = 0. (a) What is its activity at t = 4 days? (b) What is its decay constant at t = 4.0 days?

The decay constant of ²³⁸U is 4.9 × 10⁻¹⁸ S⁻¹. (a) What is the average-life of ²³⁸U? (b) What is the half-life of ²³⁸U? (c) By what factor does the activity of a ²³⁸U sample decrease in 9 × 10⁹ years?

A certain sample of a radioactive material decays at the rate of 500 per second at a certain time. The count rate falls to 200 per second after 50 minutes. (a) What is the decay constant of the sample? (b) What is its half-life?

The count rate from a radioactive sample falls from 4.0 × 10⁶ per second to 1.0 × 10⁶per second in 20 hours. What will be the count rate 100 hours after the beginning?

The half-life of ²²⁶Ra is 1602 y. Calculate the activity of 0.1 g of RaCl₂ in which all the radium is in the form of ²²⁶Ra. Taken atomic weight of Ra to be 226 g mol⁻¹ and that of Cl to be 35.5 g mol⁻¹.

The half-life of a radioisotope is 10 h. Find the total number of disintegration in the tenth hour measured from a time when the activity was 1 Ci.

The selling rate of a radioactive isotope is decided by its activity. What will be the second-hand rate of a one month old ³²P(t_1/2 = 14.3 days) source if it was originally purchased for 800 rupees?

⁵⁷Co decays to ⁵⁷Fe by β⁺- emission. The resulting ⁵⁷Fe is in its excited state and comes to the ground state by emitting γ-rays. The half-life of β⁺- decay is 270 days and that of the γ-emissions is 10⁻⁸ s. A sample of ⁵⁷Co gives 5.0 × 10⁹ gamma rays per second. How much time will elapse before the emission rate of gamma rays drops to 2.5 × 10⁹per second?

Carbon (Z = 6) with mass number 11 decays to boron (Z = 5). (a) Is it a β⁺-decay or a β⁻decay? (b) The half-life of the decay scheme is 20.3 minutes. How much time will elapse before a mixture of 90% carbon-11 and 10% boron-11 (by the number of atoms) converts itself into a mixture of 10% carbon-11 and 90% boron-11?

4 × 10²³ tritium atoms are contained in a vessel. The half-life of decay tritium nuclei is 12.3 y. Find (a) the activity of the sample, (b) the number of decay in the next 10 hours (c) the number of decays in the next 6.15 y.

A point source emitting alpha particles is placed at a distance of 1 m from a counter which records any alpha particle falling on its 1 cm² window. If the source contains 6.0 × 10¹⁶ active nuclei and the counter records a rate of 50000 counts/second, find the decay constant. Assume that the source emits alpha particles uniformly in all directions and the alpha particles fall nearly normally on the window.

²³⁸U decays to ²⁰⁶Pb with a half-life of 4.47 × 10⁹ y. This happens in a number of steps. Can you justify a single half for this chain of processes? A sample of rock is found to contain 2.00 mg of ²³⁸U and 0.600 mg of ²⁰⁶Pb. Assuming that all the lead has come from uranium, find the life of the rock.

When charcoal is prepared from a living tree, it shows a disintegration rate of 15.3 disintegrations of ¹⁴C per gram per minute. A sample from an ancient piece of charcoal shows ¹⁴C activity to be 12.3 disintegrations per gram per minute. How old is this sample? Half-life of ¹⁴C is 5730 y.

Natural water contains a small amount of tritium (`""_1^3H`). This isotope beta-decays with a half-life of 12.5 years. A mountaineer while climbing towards a difficult peak finds debris of some earlier unsuccessful attempt. Among other things he finds a sealed bottled of whisky. On returning, he analyses the whisky and finds that it contains only 1.5 per cent of the `""_1^3H` radioactivity as compared to a recently purchased bottle marked '8 years old'. Estimate the time of that unsuccessful attempt.

The count rate of nuclear radiation coming from a radiation coming from a radioactive sample containing ¹²⁸I varies with time as follows.

(a) Plot In (R₀/R) against t. (b) From the slope of the best straight line through the points, find the decay constant λ. (c) Calculate the half-life t_1/2.

The half-life of ⁴⁰K is 1.30 × 10⁹ y. A sample of 1.00 g of pure KCI gives 160 counts s⁻¹. Calculate the relative abundance of ⁴⁰K (fraction of ⁴⁰K present) in natural potassium.

`""_80^197`Hg decay to `""_79^197`Au through electron capture with a decay constant of 0.257 per day. (a) What other particle or particles are emitted in the decay? (b) Assume that the electron is captured from the K shell. Use Moseley's law √v = a(Z − b) with a = 4.95 × 10⁷s^−1/2 and b = 1 to find the wavelength of the K_α X-ray emitted following the electron capture.

A radioactive isotope is being produced at a constant rate dN/dt = R in an experiment. The isotope has a half-life t_1/2. Show that after a time t >> t_1/2 the number of active nuclei will become constant. Find the value of this constant.

Consider the situation of the previous problem. Suppose the production of the radioactive isotope starts at t = 0. Find the number of active nuclei at time t.

In an agricultural experiment, a solution containing 1 mole of a radioactive material (t_1/2= 14.3 days) was injected into the roots of a plant. The plant was allowed 70 hours to settle down and then activity was measured in its fruit. If the activity measured was 1 µCi, what per cent of activity is transmitted from the root to the fruit in steady state?

A vessel of volume 125 cm³ contains tritium (³H, t_1/2 = 12.3 y) at 500 kPa and 300 K. Calculate the activity of the gas.

`""_83^212"Bi"` can disintegrate either by emitting an α-particle of by emitting a β⁻-particle. (a) Write the two equations showing the products of the decays. (b) The probabilities of disintegration α-and β-decays are in the ratio 7/13. The overall half-life of ²¹²Bi is one hour. If 1 g of pure ²¹²Bi is taken at 12.00 noon, what will be the composition of this sample at 1 P.m. the same day?

A sample contains a mixture of ¹⁰⁸Ag and ¹¹⁰Ag isotopes each having an activity of 8.0 × 10⁸ disintegration per second. ¹¹⁰Ag is known to have larger half-life than ¹⁰⁸Ag. The activity A is measured as a function of time and the following data are obtained.
 

(a) Plot ln (A/A₀) versus time. (b) See that for large values of time, the plot is nearly linear. Deduce the half-life of ¹¹⁰Ag from this portion of the plot. (c) Use the half-life of ¹¹⁰Ag to calculate the activity corresponding to ¹⁰⁸Ag in the first 50 s. (d) Plot In (A/A₀) versus time for ¹⁰⁸Ag for the first 50 s. (e) Find the half-life of ¹⁰⁸Ag.

A human body excretes (removes by waste discharge, sweating, etc.) certain materials by a law similar to radioactivity. If technetium is injected in some form in a human body, the body excretes half the amount in 24 hours. A patient is given an injection containing ⁹⁹Tc. This isotope is radioactive with a half-life of 6 hours. The activity from the body just after the injection is 6 μCi. How much time will elapse before the activity falls to 3 μCi?

A charged capacitor of capacitance C is discharged through a resistance R. A radioactive sample decays with an average-life τ. Find the value of R for which the ratio of the electrostatic field energy stored in the capacitor to the activity of the radioactive sample remains constant in time.

Radioactive isotopes are produced in a nuclear physics experiment at a constant rate dN/dt = R. An inductor of inductance 100 mH, a resistor of resistance 100 Ω and a battery are connected to form a series circuit. The circuit is switched on at the instant the production of radioactive isotope starts. It is found that i/N remains constant in time where i is the current in the circuit at time t and N is the number of active nuclei at time t. Find the half-life of the isotope.

Calculate the energy released by 1g of natural uranium assuming 200 MeV is released in each fission event and that the fissionable isotope ²³⁵U has an abundance of 0.7% by weight in natural uranium.

A uranium reactor develops thermal energy at a rate of 300 MW. Calculate the amount of ²³⁵U being consumed every second. Average released per fission is 200 MeV.

A town has a population of 1 million. The average electric power needed per person is 300 W. A reactor is to be designed to supply power to this town. The efficiency with which thermal power is converted into electric power is aimed at 25%. (a) Assuming 200 MeV to thermal energy to come from each fission event on an average, find the number of events that should take place every day. (b) Assuming the fission to take place largely through ²³⁵U, at what rate will the amount of ²³⁵U decrease? Express your answer in kg per day. (c) Assuming that uranium enriched to 3% in ²³⁵U will be used, how much uranium is needed per month (30 days)?

Calculate the Q-values of the following fusion reactions :-

(a) `""_1^2H + ""_1^2H → ""_1^3H + ""_1^1H`

(b) `""_1^2H + ""_1^2H → ""_2^3H + n`

(c) `""_1^2H + ""_1^3H → _2^4H + n`.

Atomic masses are `m(""_1^2H) = 2.014102 "u", m(""_1^3H) = 3.016049 "u", m(""_2^3He) = 3.016029 "u", m(""_2^4He) = 4.002603 "u".`

(Use Mass of proton m_p = 1.007276 u, Mass of `""_1^1"H"` atom = 1.007825 u, Mass of neutron m_n = 1.008665 u, Mass of electron = 0.0005486 u ≈ 511 keV/c²,1 u = 931 MeV/c².)

Consider the fusion in helium plasma. Find the temperature at which the average thermal energy 1.5 kT equals the Coulomb potential energy at 2 fm.

Calculate the Q-value of the fusion reaction ⁴He + ⁴He = ⁸Be. Is such a fusion energetically favourable? Atomic mass of ⁸Be is 8.0053 u and that of ⁴He is 4.0026 u.

(Use Mass of proton m_p = 1.007276 u, Mass of `""_1^1"H"` atom = 1.007825 u, Mass of neutron m_n = 1.008665 u, Mass of electron = 0.0005486 u ≈ 511 keV/c²,1 u = 931 MeV/c².)

Calculate the energy that can be obtained from 1 kg of water through the fusion reaction ²H + ²H → ³H + p. Assume that 1.5 × 10⁻²% of natural water is heavy water D₂O (by number of molecules) and all the deuterium is used for fusion.

(Use Mass of proton m_p = 1.007276 u, Mass of `""_1^1"H"` atom = 1.007825 u, Mass of neutron m_n = 1.008665 u, Mass of electron = 0.0005486 u ≈ 511 keV/c²,1 u = 931 MeV/c².)