Nuclear and Quantum Physics — Free IB Physics HL Practice Questions

1FoundationMCQQuantum theory and uncertainty principle1 markPaper 1~2 min

An electron has a kinetic energy of

100 \text{ eV}

. What is its de Broglie wavelength?

(m_e = 9.11 \times 10^{-31} \text{ kg},\ h = 6.63 \times 10^{-34} \text{ J s},\ 1 \text{ eV} = 1.60 \times 10^{-19} \text{ J})

Solutions

2MasteryMCQPhotoelectric effect1 markPaper 1~2 min

A zinc plate, initially negatively charged, is connected to a gold-leaf electroscope. The plate is illuminated with ultraviolet radiation whose frequency exceeds the threshold frequency for zinc. Which observation is correct?

Solutions

3FoundationMCQQuantum theory and uncertainty principle1 markPaper 1~2 min

An electron is trapped in a one-dimensional infinite potential well of length

L = 0.20

nm. The ground state energy is

E_1 = 9.4

eV. What is the energy of the

n = 4

state?

Solutions

4FoundationMCQQuantum theory and uncertainty principle1 markPaper 1~2 min

A quantum dot confines an electron in a one-dimensional box of length

L

. The ground state energy is given by

E_1 = \dfrac{h^2}{8mL^2}

. When

L

is doubled, the ground state energy changes by a factor of

Solutions

5FoundationMCQTypes of radiation: Alpha, beta, gamma1 markPaper 1~2 min

A Geiger-Müller tube connected to a counter is used to investigate three radioactive sources. Source X is completely stopped by a sheet of paper. Source Y penetrates 3 mm of aluminium but is stopped by 10 cm of lead. Source Z is deflected toward the negative plate when passed between two oppositely charged parallel plates. Which row correctly identifies the nature of the radiation from Source Y?

Solutions

6FoundationMCQTypes of radiation: Alpha, beta, gamma1 markPaper 1~2 min

A radioactive source emits alpha particles, beta particles, and gamma rays. Which row correctly ranks these three types of radiation in order of decreasing ionising power?

Solutions

7FoundationMCQTypes of radiation: Alpha, beta, gamma1 markPaper 1~2 min

A researcher compares the penetrating power of four types of nuclear radiation by passing each through increasing thicknesses of lead shielding. Which type of radiation requires the greatest thickness of lead to reduce its intensity to half its original value?

Solutions

8FoundationMCQTypes of radiation: Alpha, beta, gamma1 markPaper 1~2 min

Gamma radiation is produced by a radioactive source in a vacuum. A student claims that gamma radiation travels faster than visible light because gamma photons carry more energy than visible light photons. Which of the following correctly evaluates this claim?

Solutions

9FoundationMCQNuclear fission process1 markPaper 1~2 min

A uranium-235 nucleus absorbs a thermal neutron and undergoes fission, producing barium-141, krypton-92, and three neutrons. Using the atomic masses

m(^{235}\text{U}) = 235.044\text{ u}

,

m(^{141}\text{Ba}) = 140.914\text{ u}

,

m(^{92}\text{Kr}) = 91.926\text{ u}

, and

m_n = 1.009\text{ u}

, what is the energy released in this reaction?

Solutions

10FoundationMCQNuclear fission process1 markPaper 1~2 min

In a nuclear fission reactor, control rods absorb neutrons. What is the effect of inserting the control rods deeper into the reactor core?

Solutions

11FoundationMCQNuclear fission process1 markPaper 1~2 min

A uranium-235 nucleus absorbs a slow neutron and undergoes nuclear fission. Which statement correctly describes what happens to the nucleus?

Solutions

12FoundationMCQNuclear fission process1 markPaper 1~2 min

A uranium-235 nucleus absorbs a neutron and undergoes fission, releasing three neutrons. The total mass of the reactants is

236.053 \text{ u}

and the total mass of the products is

235.844 \text{ u}

. What is the energy released in this fission event?

Solutions

13FoundationMCQNuclear fusion and energy production in stars1 markPaper 1~2 min

In the core of a main-sequence star like the Sun, the primary nuclear fusion reaction converts hydrogen nuclei into helium, releasing energy. Which of the following correctly identifies both the fuel and the approximate temperature required for this process to occur?

Solutions

14FoundationMCQNuclear fusion and energy production in stars1 markPaper 1~2 min

In the core of a main-sequence star, nuclear fusion occurs between hydrogen nuclei. Which conditions are necessary for the electrostatic repulsion between nuclei to be overcome so that fusion can take place?

Solutions

15FoundationMCQNuclear fusion and energy production in stars1 markPaper 1~2 min

A nuclear fusion reaction releases energy because the total rest mass of the products is less than the total rest mass of the reactants. Which equation correctly relates the energy released,

E

, to the mass deficit,

\Delta m

, in this reaction?

Solutions

16FoundationMCQNuclear fusion and energy production in stars1 markPaper 1~2 min

In a star, four hydrogen nuclei (protons) fuse to form one helium-4 nucleus. The mass of a proton is

1.6726 \times 10^{-27}\ \text{kg}

and the mass of a helium-4 nucleus is

6.6447 \times 10^{-27}\ \text{kg}

. Which statement correctly describes the total nuclear mass before and after this fusion reaction?

Solutions

17FoundationMCQIsotopes and atomic mass1 markPaper 1~2 min

A nucleus contains 6 protons and 8 neutrons. Which statement correctly identifies the mass number of this nucleus?

Solutions

18MasteryMCQIsotopes and atomic mass1 markPaper 1~2 min

Naturally occurring boron consists of two stable isotopes:

^{10}

B with atomic mass

10.01\text{ u}

and

^{11}

B with atomic mass

11.01\text{ u}

. The relative atomic mass of naturally occurring boron is

10.81\text{ u}

. What is the percentage abundance of

^{10}

B?

Solutions

19FoundationMCQElectron configuration and energy levels1 markPaper 1~2 min

An electron in a hydrogen atom is in the

n = 2

energy level. A photon of wavelength

\lambda = 486\text{ nm}

is incident on the atom. The first four energy levels of hydrogen are

E_1 = -13.6\text{ eV}

,

E_2 = -3.4\text{ eV}

,

E_3 = -1.5\text{ eV}

, and

E_4 = -0.85\text{ eV}

. Which statement correctly describes whether this photon is absorbed?

Solutions

20FoundationMCQIsotopes and atomic mass1 markPaper 1~2 min

Iodine-131 has a mass number of 131 and contains 78 neutrons in its nucleus. Which quantity correctly identifies the atomic number of this isotope?

Solutions

21FoundationMCQIsotopes and atomic mass1 markPaper 1~2 min

A mass spectrometer detects two isotopes of lithium in a sample. The atomic mass of lithium-6 is

6.015 \text{ u}

and the atomic mass of lithium-7 is

7.016 \text{ u}

. The relative atomic mass of the sample is

6.94 \text{ u}

. What is the approximate natural abundance of lithium-7 in this sample?

Solutions

22MasteryMCQIsotopes and atomic mass1 markPaper 1~2 min

A mass spectrometer analyses a magnesium sample and identifies three isotopes with the following data. Isotope — Relative abundance — Atomic mass

^{24}

Mg — 78.99% — 23.99 u

^{25}

Mg — 10.00% — 24.99 u

^{26}

Mg — 11.01% — 25.98 u What is the relative atomic mass of this magnesium sample?

Solutions

23MasterySAQ-SQuantum theory and uncertainty principle5 marksPaper 2~8 min

A particle is confined in a one-dimensional infinite potential well of width

L = 0.20 \text{ nm}

. The particle is in the first excited state (

n = 2

), with wavefunction:

\psi(x) = \sqrt{\frac{2}{L}} \sin\!\left(\frac{2\pi x}{L}\right), \quad 0 \leq x \leq L

Useful integral:

\displaystyle\int \sin^2(ax)\, dx = \frac{x}{2} - \frac{\sin(2ax)}{4a} + C

(a)

State what is meant by a quantum state of a particle in this system. [1 mark]

(b)

Calculate the probability of finding the particle in the region

0 \leq x \leq \dfrac{L}{4}

. [3 marks]

(c)

The wavefunction has a node at

x = \dfrac{L}{2}

. Deduce whether the probability of finding the particle in the region

\dfrac{L}{4} \leq x \leq \dfrac{L}{2}

is greater than, equal to, or less than the probability found in (b). Justify your answer without further integration. [1 mark]

Solutions

24MasterySAQ-SQuantum theory and uncertainty principle8 marksPaper 2~12 min

In a quantum optics laboratory, a photon is emitted from a laser source and passes through a 50:50 beam splitter. The photon can be reflected (path A) or transmitted (path B) with equal probability. - Planck constant

h = 6.63 \times 10^{-34}

J s

(a)

State what is meant by the collapse of the wavefunction in this context. [1 mark]

(b)

The position of the photon along path A is measured with an uncertainty of

0.10\ \mu\text{m}

. Calculate the minimum uncertainty in the momentum of the photon along path A. [3 marks]

(c)

A student claims: "If the detectors along both paths are made less sensitive so they only detect the photon 50% of the time, the interference pattern will be partially restored." Evaluate this claim with reference to which-path information and wave–particle duality. [4 marks]

Solutions

25ChallengeSAQ-LQuantum theory and uncertainty principle9 marksPaper 2~14 min

Data

h = 6.63 \times 10^{-34} \text{ J s}, \quad \hbar = 1.05 \times 10^{-34} \text{ J s}, \quad m_e = 9.11 \times 10^{-31} \text{ kg}, \quad 1 \text{ eV} = 1.60 \times 10^{-19} \text{ J}

A scanning tunnelling microscope (STM) uses a sharp conducting tip to image surfaces at the atomic scale. The tip is held a distance

d = 0.50 \text{ nm}

from a conducting surface. Electrons with kinetic energy

E = 5.0 \text{ eV}

must tunnel through a vacuum barrier of height

V_0 = 9.5 \text{ eV}

. The following data are

(a)

Calculate the de Broglie wavelength of an electron with kinetic energy

5.0 \text{ eV}

. [2 marks]

(b)

The tunnelling probability through a rectangular barrier is approximated by:

T \approx e^{-2\alpha d}, \quad \text{where} \quad \alpha = \sqrt{\frac{2m_e(V_0 - E)}{\hbar^2}}

Calculate the tunnelling probability

T

for

d = 0.50 \text{ nm}

. [3 marks]

(c)

Explain, with reference to the Heisenberg uncertainty principle, why quantum tunnelling can occur even though the electron's energy is less than the barrier height

V_0

. [2 marks]

(d)

The tip is retracted so that

d = 1.0 \text{ nm}

. Calculate the new tunnelling probability and determine the ratio

T(1.0 \text{ nm}) / T(0.50 \text{ nm})

. Hence comment on the sensitivity of the STM to changes in tip–surface distance. [2 marks]

Solutions

26MasterySAQ-SQuantum theory and uncertainty principle14 marksPaper 2~21 min

Data

1\ \text{eV} = 1.60 \times 10^{-19}\ \text{J}

;

h = 6.63 \times 10^{-34}\ \text{J s}

;

m_e = 9.11 \times 10^{-31}\ \text{kg}

;

c = 3.00 \times 10^8\ \text{m s}^{-1}

A medical linear accelerator (linac) produces a beam of electrons for cancer treatment. The electrons are accelerated through a potential difference of 6.0 MV, giving them a kinetic energy of 6.0 MeV. The beam passes through a narrow slit of width

1.0\ \mu\text{m}

to collimate it.

(a)

State the Heisenberg uncertainty principle as it applies to position and momentum. [1 mark]

(b)

Calculate the minimum uncertainty in the transverse momentum of an electron after passing through the slit. [2 marks]

(c)

The rest-mass energy of an electron is

0.511\ \text{MeV}

. (i) Show that the total energy of a 6.0 MeV electron is approximately

6.5\ \text{MeV}

, and hence determine its relativistic momentum. [2]
(ii) Explain, with reference to your answers to (b) and (c)(i), why the uncertainty in transverse momentum is significant for the precision of the beam in medical treatment. [2] Total: [7 marks]

Solutions

27ChallengeSAQ-LRadioactive decay and applications7 marksPaper 2~11 min

Data

- Half-life of

^{60}\text{Co}

:

T_{1/2} = 5.27\,\text{years}

- Activity on 1 January 2020:

A_0 = 1.8 \times 10^{12}\,\text{Bq}

- Minimum usable activity:

A_{\min} = 5.0 \times 10^{11}\,\text{Bq}

A nuclear reactor monitoring system uses a cobalt-60 source to test the integrity of steel reactor vessels. Cobalt-60 (

^{60}_{27}\text{Co}

) decays via beta-minus emission to nickel-60 (

^{60}_{28}\text{Ni}

). A technician measures the activity of a sealed source to be

1.8 \times 10^{12}\,\text{Bq}

on 1 January 2020.

(a)

Calculate the decay constant

\lambda

for cobalt-60, in units of

\text{year}^{-1}

. [2 marks]

(b)

Calculate the activity of the source on 1 January 2030. [2 marks]

(c)

The source must maintain activity of at least

5.0 \times 10^{11}\,\text{Bq}

to remain usable. Determine the latest year in which the source is still usable, and evaluate whether it will remain usable on 1 January 2040. [3 marks]

Solutions

28ChallengeSAQ-LHalf-life and decay constant7 marksPaper 2~11 min

A medical physics laboratory tests a sealed iodine-123 source for thyroid imaging. The source has an initial activity of

4.8 \times 10^{8}\,\text{Bq}

. Iodine-123 has a half-life of

13.2\,\text{hours}

. The source is considered safe for patient handling when its activity falls below

1.5 \times 10^{7}\,\text{Bq}

. Relevant equations:

A = A_0 e^{-\lambda t}

,

\lambda = \dfrac{\ln 2}{T_{1/2}}

,

\ln 2 \approx 0.693

(a)

State the decay constant

\lambda

for iodine-123, in

\text{h}^{-1}

. [1 mark]

(b)

Calculate the time, in hours, after which the source can be safely handled. [3 marks]

(c)

Deduce the number of half-lives that must elapse before safe handling, and hence verify your answer to (b). [2 marks]

(d)

Evaluate whether the exponential decay model with a constant half-life is appropriate for predicting the safe handling time of this source. [1 mark]

Solutions

29ChallengeSAQ-LHalf-life and decay constant7 marksPaper 2~11 min

A nuclear reactor used for research produces technetium-99m, a medical tracer. The decay constant of technetium-99m is

3.22 \times 10^{-5}\,\text{s}^{-1}

. A sample is prepared for a hospital and must contain at least

2.0 \times 10^{12}

undecayed nuclei at the time of injection. Preparation and transport take

6.0

hours from the reactor to the hospital.

(a)

Calculate the half-life of technetium-99m in hours. [2 marks]

(b)

Calculate the minimum number of technetium-99m nuclei that must be present in the sample at the reactor. [3 marks]

(c)

The sample is stored at

-20\,^\circ\text{C}

during transport. A technician claims the decay constant will change because lower temperatures reduce atomic activity. Deduce, with reference to the nature of radioactive decay, whether this claim is correct. [2 marks]

Solutions

30ChallengeSAQ-LRadioactive decay and applications9 marksPaper 2~14 min

Data

-

1\,\text{year} = 3.16 \times 10^{7}\,\text{s}

- Decay constant of cobalt-60:

\lambda = 4.17 \times 10^{-9}\,\text{s}^{-1}

-

1\,\text{MeV} = 1.60 \times 10^{-13}\,\text{J}

A medical physicist is calibrating a cobalt-60 (

^{60}_{27}\text{Co}

) source for use in cancer radiotherapy. The source has an initial activity of

1.2 \times 10^{15}\,\text{Bq}

at the time of manufacture. Cobalt-60 decays by beta-minus emission to an excited state of nickel-60 (

^{60}_{28}\text{Ni}^*

), which then promptly emits two gamma photons of energies

1.17\,\text{MeV}

and

1.33\,\text{MeV}

to reach the ground state. The half-life of cobalt-60 is

5.27

years.

(a)

State the relationship between activity

A

, decay constant

\lambda

, and number of undecayed nuclei

N

. [1 mark]

(b)

Calculate the number of cobalt-60 atoms present in the source at the time of manufacture. [2 marks]

(c)

Calculate the total energy, in joules, released as gamma radiation per decay of a cobalt-60 nucleus. [2 marks]

(d)

Explain why the activity of the cobalt-60 source decreases over time. [2] (e) Calculate the activity of the source after

10.0

years. [2 marks]

Solutions

31ChallengeSAQ-LRadioactive decay and applications9 marksPaper 2~14 min

Data

-

1\,\text{hour} = 3600\,\text{s}

- Energy of emitted gamma photon:

140\,\text{keV}

-

1\,\text{eV} = 1.60 \times 10^{-19}\,\text{J}

Formulae:

A = \lambda N

,

\quad N = N_0 e^{-\lambda t}

,

\quad A = A_0 e^{-\lambda t}

,

\quad \lambda = \dfrac{\ln 2}{T_{1/2}}

A nuclear power plant uses uranium-235 fuel. Neutron-induced fission produces technetium-99m (

^{99\text{m}}_{43}\text{Tc}

), a metastable isomer that decays by gamma emission to technetium-99 (

^{99}_{43}\text{Tc}

) with a half-life of

T_{1/2} = 6.01\,\text{hours}

. The technetium-99m is extracted and used as a medical tracer. A sample is prepared for a hospital with an initial activity of

A_0 = 8.0 \times 10^{8}\,\text{Bq}

.

(a)

Calculate the decay constant

\lambda

of technetium-99m in

\text{s}^{-1}

. [2 marks]

(b)

Determine the initial number of technetium-99m nuclei in the sample. [2 marks]

(c)

The bone scan procedure requires the activity to be at least

2.0 \times 10^{8}\,\text{Bq}

. Calculate the maximum time, in hours, after preparation that the sample can still be used. [3 marks]

(d)

Evaluate the suitability of technetium-99m for medical imaging, considering its half-life and the type of radiation emitted. [2 marks]

Solutions

32ChallengeSAQ-LHalf-life and decay constant7 marksPaper 2~11 min

A medical physics laboratory is testing Technetium-99m (

^{99m}\text{Tc}

) for use in cardiac imaging. The decay of

^{99m}\text{Tc}

follows first-order kinetics.

A_0 = 8.0 \times 10^8\,\text{Bq} \qquad A = 2.0 \times 10^8\,\text{Bq after } t = 12.0\,\text{hours}

(a)

Calculate the decay constant

\lambda

of

^{99m}\text{Tc}

. [2 marks]

(b)

Determine the half-life

T_{1/2}

of

^{99m}\text{Tc}

, giving your answer in hours. [2 marks]

(c)

A cardiac imaging procedure requires an isotope whose half-life is long enough to complete a 4.0-hour scan, yet short enough so that the patient's residual activity falls below

1.0 \times 10^7\,\text{Bq}

within 48 hours of administration. Evaluate whether

^{99m}\text{Tc}

is suitable for this procedure. [3 marks]

Solutions

33FoundationSAQ-LNuclear fission process7 marksPaper 2~11 min

Data

1\,\text{u} = 931.5\,\text{MeV}\,c^{-2}

A uranium-235 nucleus undergoes asymmetric fission, producing strontium-94 and xenon-139 fragments along with neutrons. The complete reaction is:

^{235}_{92}\text{U} + ^{1}_{0}n \rightarrow ^{94}_{38}\text{Sr} + ^{139}_{54}\text{Xe} + 3\,^{1}_{0}n

Atomic masses:

^{235}\text{U} = 235.0439\,\text{u}

,

\;^{94}\text{Sr} = 93.9154\,\text{u}

,

\;^{139}\text{Xe} = 138.9188\,\text{u}

, neutron

= 1.0087\,\text{u}

.

(a)

Determine the mass defect, in u, for this reaction. [1 mark]

(b)

Calculate the total energy released, in MeV, by this fission reaction. [2 marks]

(c)

Explain why the fission fragments have high ionizing power as they travel through matter. [2 marks]

(d)

The total energy released in part (b) is approximately 179 MeV. In a reactor, the measured kinetic energy of the fission fragments and prompt neutrons accounts for about 168 MeV. Evaluate whether the assumption that all released energy becomes kinetic energy of the products is valid, identifying the physical processes responsible for the remaining energy. [2 marks]

Solutions

34FoundationSAQ-SNuclear fission process7 marksPaper 2~11 min

A fission reactor uses enriched uranium fuel rods. The graph below shows the number of neutrons produced per fission as a function of the energy of the incident neutron for uranium-235.

(a)

Describe how the number of neutrons produced per fission varies with the energy of the incident neutron, as shown in the graph. [2 marks]

(b)

For a reactor to be critical, exactly one neutron from each fission must cause a further fission. The average number of neutrons produced per fission is 2.4. Calculate the fraction of neutrons produced per fission that must successfully cause fission to maintain a critical chain reaction. [2 marks]

(c)

Control rods absorb neutrons. A reactor operating at criticality has its control rods partially inserted further, increasing the fraction of neutrons absorbed. Deduce and explain the effect this has on the reactor power output. [3 marks]

Solutions

35ChallengeSAQ-LNuclear fission process9 marksPaper 2~14 min

Data

- Mass of

{}^{239}_{94}\text{Pu}

atom

= 239.0522\,\text{u}

- Mass of

{}^{133}_{51}\text{Sb}

atom

= 132.9152\,\text{u}

- Mass of

{}^{104}_{43}\text{Tc}

atom

= 103.9115\,\text{u}

- Mass of neutron

= 1.00867\,\text{u}

-

1\,\text{u} = 931.5\,\text{MeV}\,c^{-2}

In a research reactor, a sample of plutonium-239 undergoes fission after absorbing a slow neutron. One possible fission reaction is:

{}^{239}_{94}\text{Pu} + {}^{1}_{0}\text{n} \rightarrow {}^{133}_{51}\text{Sb} + {}^{104}_{43}\text{Tc} + 3\,{}^{1}_{0}\text{n}

(a)

Calculate the energy released in this fission event. Give your answer in MeV. [3 marks]

(b)

Determine the neutron-to-proton ratio

N/Z

for each fission fragment immediately after fission. State, with a reason, which fragment is further from the valley of stability. [3 marks]

(c)

The fission fragments undergo beta-minus decay. Explain why the fragments are unstable immediately after fission and how beta-minus decay moves them towards stability. [3 marks]

Solutions

36FoundationSAQ-SChain reaction and critical mass5 marksPaper 2~8 min

The shows the mass of uranium-235 in a reactor core as fuel is gradually added. The critical mass for this reactor geometry is 48 kg.

(a)

State what is meant by critical mass and explain why it depends on the geometry of the reactor core. [2 marks]

(b)

Using the graph, determine the additional mass of uranium-235 that must be added when the core contains 42 kg in order to reach criticality. [1 mark]

(c)

The core is loaded to a total mass of 55 kg of uranium-235. Deduce whether the core is subcritical, critical, or supercritical, and explain the consequence for the neutron population over time. [2 marks]

Solutions

37FoundationSAQ-SChain reaction and critical mass5 marksPaper 2~8 min

The shows how neutron flux varies with distance from the centre of a cylindrical reactor core. The neutron flux at the centre is

\Phi_0 = 2.4 \times 10^{14} \ \text{neutrons m}^{-2}\text{s}^{-1}

and falls to zero at the critical radius

r_c = 0.50 \ \text{m}

.

(a)

Explain why a minimum core size is required for a fission chain reaction to be self-sustaining. [2 marks]

(b)

The actual core has radius

r = 0.45 \ \text{m}

. The neutron flux at the core boundary is

\Phi_b = 3.8 \times 10^{13} \ \text{neutrons m}^{-2}\text{s}^{-1}

. Calculate the ratio

\dfrac{\Phi_b}{\Phi_0}

for the actual core and use this, together with the surface-to-volume ratio, to justify whether this reactor is subcritical or supercritical. [3 marks]

Solutions

38FoundationSAQ-SChain reaction and critical mass8 marksPaper 2~12 min

A nuclear reactor uses uranium-235 fuel rods. Neutrons produced by fission have an average speed of

2.2 \times 10^4 \text{ m s}^{-1}

and a mean free path of

0.12 \text{ m}

in the fuel. A moderator slows these neutrons before they interact further with the fuel.

(a)

(i) State what is meant by a moderator in a nuclear reactor. [1]
(ii) Explain why slowing neutrons increases the probability of fission in uranium-235. [2 marks]

(b)

Calculate the average time between successive collisions for a neutron travelling through the fuel before moderation. [2 marks]

(c)

Determine the kinetic energy of a neutron travelling at

2.2 \times 10^4 \text{ m s}^{-1}

. Hence deduce whether this neutron would be classified as a thermal neutron, given that thermal neutrons have kinetic energies of approximately

0.025 \text{ eV}

.

m_n = 1.675 \times 10^{-27} \text{ kg}, \quad 1 \text{ eV} = 1.6 \times 10^{-19} \text{ J}

[3 marks]

Solutions

39MasterySAQ-SLife cycle of stars6 marksPaper 2~9 min

A main sequence star has a mass of

2.0 \times 10^{30}\,\text{kg}

and a luminosity of

3.8 \times 10^{26}\,\text{W}

. During the main sequence phase, hydrogen is fused into helium at a rate of

6.0 \times 10^{11}\,\text{kg s}^{-1}

. Speed of light:

c = 3.0 \times 10^{8}\,\text{m s}^{-1}

(a)

State the two opposing forces that maintain hydrostatic equilibrium during the main sequence phase of a star. [1 mark]

(b)

Calculate the fraction of the mass consumed per second that is converted into energy. [3 marks]

(c)

Only approximately 10 percent of the star's total mass is available as hydrogen fuel in the core. Deduce the main sequence lifetime of this star in years. [2 marks]

Solutions

40MasterySAQ-SLife cycle of stars7 marksPaper 2~11 min

A red giant star has the following properties: Quantity — Value Radius —

5.0 \times 10^{10}\,\text{m}

Luminosity —

1.0 \times 10^{28}\,\text{W}

Stefan–Boltzmann constant

\sigma

—

5.67 \times 10^{-8}\,\text{W\,m}^{-2}\,\text{K}^{-4}

(a)

State what happens to the core of a main-sequence star when its hydrogen fuel is exhausted. [1 mark]

(b)

Calculate the surface temperature of this red giant. [3 marks]

(c)

The peak wavelength of radiation emitted by the Sun is

5.0 \times 10^{-7}\,\text{m}

. Using your answer to (b) and Wien's displacement law (

\lambda_{\max} T = 2.90 \times 10^{-3}\,\text{m\,K}

), deduce whether this red giant emits peak radiation at a longer or shorter wavelength than the Sun, and calculate the ratio

\dfrac{\lambda_{\max,\,\text{giant}}}{\lambda_{\max,\,\odot}}

. [3 marks]

Solutions

41ChallengeSAQ-LNuclear fusion and energy production in stars10 marksPaper 2~15 min

Data

- Mass of one proton:

m_p = 1.6726 \times 10^{-27}\,\text{kg}

- Mass of a helium-

4

nucleus:

m_{\text{He}} = 6.6447 \times 10^{-27}\,\text{kg}

- Mass of one positron:

m_{e^+} = 9.109 \times 10^{-31}\,\text{kg}

- Mass of one electron neutrino:

\approx 0\,\text{kg}

- Speed of light:

c = 3.00 \times 10^{8}\,\text{m\,s}^{-1}

- Sun's total power output:

L_\odot = 3.85 \times 10^{26}\,\text{W}

The proton–proton chain the core of a star like the Sun converts four protons into a helium-

4

nucleus. The overall reaction is:

4\,^1_1\text{H} \rightarrow\, ^4_2\text{He} + 2e^+ + 2\nu_e + \gamma

(a)

Calculate the mass defect

\Delta m

for a single fusion event in which four protons fuse to produce one helium-

4

nucleus and two positrons. [2 marks]

(b)

Calculate the energy released per fusion event in joules. [1 mark]

(c)

Calculate the number of fusion events occurring in the Sun per second. [2 marks]

(d)

Explain why the positrons produced in the reaction do not represent a long-term mass loss from the star. [2] (e) The neutrinos produced in the proton–proton chain carry away approximately

2\,\text{percent}

of the energy released per fusion event. Evaluate the significance of neutrino emission for the energy balance of the Sun, referring to both the immediate energy budget and the long-term stability of the star. [3 marks]

Solutions

42MasterySAQ-SLife cycle of stars5 marksPaper 2~8 min

A star has an initial mass of

8.0 \times 10^{30}\,\text{kg}

. It converts hydrogen to helium;

0.71\%

of the mass consumed is released as energy. The star's luminosity (power output) is

3.8 \times 10^{26}\,\text{W}

. Only

10\%

of the star's mass is available for fusion in the core.

(a)

State the condition that determines when a star leaves the main sequence. [1 mark]

(b)

Calculate the rate at which the star must consume hydrogen mass to sustain its luminosity of

3.8 \times 10^{26}\,\text{W}

. [2 marks]

(c)

Calculate the time, in years, that the star spends on the main sequence. [2 marks]

Solutions

43MasterySAQ-SLife cycle of stars5 marksPaper 2~8 min

A star cluster contains stars of various masses. - Star A: mass

M_A = 3.0 \times 10^{30}\,\text{kg}

, main sequence lifetime

\tau_A = 5.0 \times 10^9\,\text{years}

- Star B: mass

M_B = 1.2 \times 10^{31}\,\text{kg}

For main sequence stars:

L \propto M^3

(a)

State how the main sequence lifetime of a star depends on its mass. [1 mark]

(b)

Explain, with reference to

L \propto M^3

, why more massive stars have shorter main sequence lifetimes. [2 marks]

(c)

Calculate the main sequence lifetime of star B. [2 marks]

Solutions

44MasterySAQ-SStellar nucleosynthesis8 marksPaper 2~12 min

A massive star of approximately 20 solar masses undergoes nucleosynthesis, fusing elements up to iron-56 in its core. The graph of binding energy per nucleon versus nucleon number peaks at iron-56, with a binding energy per nucleon of

8.8\,\text{MeV}

.

1\,\text{MeV} = 1.60 \times 10^{-13}\,\text{J}

(a)

(i) Describe the trend in binding energy per nucleon for nuclei from hydrogen (

A = 1

) to iron-56 (

A = 56

). [1]
(ii) Explain why this trend means that fusion of elements up to iron releases energy in a stellar core. [2 marks]

(b)

Calculate the total binding energy of an iron-56 nucleus. Give your answer in joules. [2 marks]

(c)

A supernova explosion produces elements heavier than iron. Evaluate why the extreme conditions of a supernova are necessary for this process, whereas the conditions in a stable stellar core are not sufficient. [3 marks]

Solutions

45MasterySAQ-SAtomic models (Bohr, quantum model)5 marksPaper 2~8 min

A student investigates the photoelectric effect using a sodium metal surface. The work function of sodium is

\phi = 1.80\,\text{eV}

. She illuminates the surface with light from a hydrogen discharge lamp emitting photons from the

n = 3

to

n = 2

transition.

E_n = -\frac{13.6}{n^2}\,\text{eV}, \quad K_{\text{max}} = hf - \phi, \quad 1\,\text{eV} = 1.60 \times 10^{-19}\,\text{J}, \quad h = 6.63 \times 10^{-34}\,\text{J\,s}

(a)

State what is meant by the ground state in the Bohr model of the atom. [1 mark]

(b)

Calculate the maximum kinetic energy, in eV, of photoelectrons ejected from the sodium surface when illuminated by the

n = 3

to

n = 2

photons. [2 marks]

(c)

The student replaces the lamp with one emitting photons from the

n = 4

to

n = 3

transition in hydrogen. Deduce, with a calculation, whether the photoelectric effect occurs for this transition, and explain why the result differs from that in (b). [2 marks]

Solutions

46MasterySAQ-SAtomic models (Bohr, quantum model)7 marksPaper 2~11 min

Data

h = 6.63 \times 10^{-34}\,\text{J\,s}

,

c = 3.00 \times 10^{8}\,\text{m\,s}^{-1}

,

1\,\text{eV} = 1.60 \times 10^{-19}\,\text{J}

E_n = -\frac{13.6}{n^2}\,\text{eV}

A physicist uses a particle accelerator to excite hydrogen gas. She measures the wavelengths of emitted photons and identifies a line at

\lambda = 102.6\,\text{nm}

in the ultraviolet region. This line belongs to the Lyman series (transitions ending at

n = 1

).

(a)

State one difference between the Bohr model and the quantum mechanical model regarding electron location. [1 mark]

(b)

Calculate the energy, in eV, of the photon of wavelength

102.6\,\text{nm}

. [2 marks]

(c)

Show that the upper energy level of this transition is

n = 3

. [2 marks]

(d)

The Lyman series has a series limit a minimum wavelength. Deduce the wavelength of this series limit. [2 marks]

Solutions

47ChallengeSAQ-LAtomic models (Bohr, quantum model)7 marksPaper 2~11 min

Data

- Planck constant:

h = 6.63 \times 10^{-34}\,\text{J\,s}

- Speed of light:

c = 3.00 \times 10^{8}\,\text{m\,s}^{-1}

-

1\,\text{eV} = 1.60 \times 10^{-19}\,\text{J}

A medical imaging technique uses a gas of excited singly ionized helium (

\text{He}^+

) atoms that emit photons when electrons transition between energy levels. The energy levels of

\text{He}^+

are given by:

E_n = -\frac{Z^2 \times 13.6\,\text{eV}}{n^2}, \quad Z = 2

(a)

Calculate the wavelength of the photon emitted when an electron in

\text{He}^+

transitions from

n = 3

to

n = 2

. [3 marks]

(b)

State what is meant by an atomic orbital in the quantum model, and explain how this differs from an electron orbit in the Bohr model. [2 marks]

(c)

The Bohr model accurately predicts the emission spectrum of

\text{He}^+

but fails for neutral helium. Evaluate whether this difference in success is a fundamental limitation of the Bohr model or merely a computational one. [2 marks]

Solutions

48MasterySAQ-SAtomic models (Bohr, quantum model)5 marksPaper 2~8 min

Data

h = 6.63 \times 10^{-34}\,\text{J\,s}

,

c = 3.00 \times 10^{8}\,\text{m\,s}^{-1}

,

1\,\text{eV} = 1.60 \times 10^{-19}\,\text{J}

E_n = -\frac{13.6}{n^2}\,\text{eV}

A hydrogen atom in an excited state emits a photon of wavelength

1875\,\text{nm}

when the electron falls to the

n = 3

level (Paschen series).

(a)

State one feature of the Bohr model that explains why hydrogen emits only specific wavelengths of light. [1 mark]

(b)

Show that the energy of the

n = 3

level is

-1.51\,\text{eV}

. [1 mark]

(c)

Calculate the energy, in eV, of the photon emitted at

1875\,\text{nm}

. [2 marks]

(d)

Determine the quantum number

n

of the initial energy level for this transition. [1 mark]

Solutions

49ChallengeLAQWave-particle duality12 marksPaper 3~18 min

Data

-

\hbar = 1.055 \times 10^{-34}\,\text{J\,s}

-

m_e = 9.11 \times 10^{-31}\,\text{kg}

-

1\,\text{eV} = 1.60 \times 10^{-19}\,\text{J}

-

T \propto e^{-2\alpha d}

A metal tip in a scanning tunnelling microscope (STM) is held at distance

d

from a conducting surface. Electrons in the metal have total energy

E

less than the barrier height

V_0

at the tip–surface gap.

(a)

Analyse the phenomenon of quantum tunnelling as evidence for the wave nature of particles, and evaluate how the STM exploits this phenomenon to achieve atomic-scale resolution. In your answer: - explain why classical mechanics predicts zero transmission probability for a particle with

E < V_0

- explain how the quantum wavefunction gives a non-zero transmission probability through a barrier of finite width

d

- describe the role of the de Broglie wavelength and the exponential decay constant

\alpha = \sqrt{2m(V_0 - E)}/\hbar

inside the barrier - evaluate how the exponential dependence of tunnelling current on tip–surface separation enables atomic-scale vertical resolution [6 marks]

(b)

The tip is held at

d = 0.50\,\text{nm}

from the surface. The barrier height above the electron energy is

U = V_0 - E = 1.0\,\text{eV}

. (i) Calculate the decay constant

\alpha

, where

\alpha = \sqrt{2m_e U}/\hbar

. [2 marks]
(ii) The tip is retracted to

d = 0.60\,\text{nm}

. Using

T \propto e^{-2\alpha d}

, calculate the ratio

T_2/T_1

of the tunnelling probability at

d = 0.60\,\text{nm}

to that

d = 0.50\,\text{nm}

. [2 marks]
(iii) Evaluate what the result of (b)(ii) implies about the ability of the STM to resolve individual atoms separated by

0.10\,\text{nm}

vertically. [2 marks]

Solutions

50ChallengeLAQQuantum theory and uncertainty principle10 marksPaper 3~15 min

A team of experimental physicists investigates the behaviour of electrons in a quantum dot device. The quantum dot is modelled as a one-dimensional infinite potential well of width

L = 5.00\,\text{nm}

. An electron is prepared in a superposition of the ground state (

n = 1

) and the first excited state (

n = 2

) with equal probability amplitudes. The following data are available:

\psi_n(x) = \sqrt{\frac{2}{L}}\sin\!\left(\frac{n\pi x}{L}\right), \quad E_n = \frac{n^2 h^2}{8m_e L^2}, \quad \Delta x\,\Delta p \geq \frac{h}{4\pi}

h = 6.63 \times 10^{-34}\,\text{J\,s}, \quad m_e = 9.11 \times 10^{-31}\,\text{kg}

(a)

State the normalised wavefunction

\Psi(x,0)

of the electron at

t = 0

as a linear combination of stationary-state wavefunctions. [1 mark]

(b)

Calculate the expectation value of the energy

\langle E \rangle

for this superposition state. [3 marks]

(c)

Determine the minimum uncertainty in the electron's momentum

\Delta p_{\min}

when the position uncertainty is taken as the well width

L

. Explain what this result implies about the validity of a classical description of the electron. [3 marks]

(d)

An energy measurement is performed and the result

E_2

is obtained. Calculate the probability that a subsequent position measurement locates the electron in the left half of the well,

0 \leq x \leq L/2

. [2] (e) The energy measurement in (d) yields a precise value of

E_2

. Evaluate whether this precise energy knowledge is consistent with the Heisenberg uncertainty principle, and deduce what this implies for the electron's momentum uncertainty after the measurement. [1 mark]

Solutions

Nuclear and Quantum Physics — Free Physics HL Practice Questions