IB Physics How to Solve: IB Physics HL – Quantum & Nuclear (Photoelectric, Bohr, De Broglie, Binding Energy, Decay Law)

How to Solve: IB Physics HL – Quantum & Nuclear (Photoelectric, Bohr, De Broglie, Binding Energy, Decay Law)

Complete Guide

Introduction

Mastering quantum and nuclear physics unlocks 10–15% of your IB Physics HL Paper 3—and real-world tech like solar panels, cancer treatment, and nuclear power. One photoelectric effect question alone can be worth 6 marks—miss it, and you’re fighting for a 6.

WHAT YOU NEED TO KNOW FIRST

1. Energy conservation – Energy can’t be created or destroyed, only transferred.
2. Wave-particle duality – Light behaves as both a wave (interference) and a particle (photon).
3. Basic atomic structure – Protons, neutrons, electrons, and the nucleus.

KEY TERMS & FORMULAS

1. Photoelectric Effect

Key Terms: - Threshold frequency (f₀) – Minimum frequency of light needed to eject electrons. - Work function (Φ) – Minimum energy required to remove an electron from a metal. - Stopping potential (Vₛ) – Voltage needed to stop the fastest ejected electrons.

Formulas:
1. Einstein’s photoelectric equation [ E_{\text{max}} = hf - \Phi ] - ( E_{\text{max}} ) = Maximum kinetic energy of ejected electrons (J or eV) - ( h ) = Planck’s constant (MEMORISE THIS: ( 6.63 \times 10^{-34} \, \text{Js} )) - ( f ) = Frequency of incident light (Hz) - ( \Phi ) = Work function of the metal (J or eV)

1. Stopping potential [ eV_s = E_{\text{max}} ]
2. ( e ) = Charge of an electron (MEMORISE THIS: ( 1.6 \times 10^{-19} \, \text{C} ))

3. ( V_s ) = Stopping potential (V)

4. Threshold frequency [ \Phi = hf_0 ]

5. ( f_0 ) = Threshold frequency (Hz)

2. Bohr Model of the Hydrogen Atom

Key Terms: - Energy levels (n) – Discrete orbits where electrons exist. - Ground state (n=1) – Lowest energy level. - Ionization energy – Energy needed to remove an electron from the atom.

Formulas:
1. Energy of an electron in the nth level [ E_n = -\frac{13.6}{n^2} \, \text{eV} ] - ( E_n ) = Energy of the electron in level ( n ) (eV) - ( n ) = Principal quantum number (1, 2, 3, ...) (MEMORISE THIS: ( 13.6 \, \text{eV} ) is the ionization energy for hydrogen.)

1. Wavelength of emitted/absorbed photon [ \Delta E = hf = \frac{hc}{\lambda} ]
2. ( \Delta E ) = Energy difference between levels (J or eV)
3. ( \lambda ) = Wavelength of photon (m)
4. ( c ) = Speed of light (MEMORISE THIS: ( 3.0 \times 10^8 \, \text{m/s} ))

3. De Broglie Wavelength

Key Terms: - Matter waves – Particles exhibit wave-like properties. - Wavelength (λ) – Associated with a moving particle.

Formula: [ \lambda = \frac{h}{p} = \frac{h}{mv} ] - ( \lambda ) = De Broglie wavelength (m) - ( p ) = Momentum of the particle (kg·m/s) - ( m ) = Mass of the particle (kg) - ( v ) = Velocity of the particle (m/s)

4. Nuclear Binding Energy & Mass Defect

Key Terms: - Mass defect (Δm) – Difference between mass of nucleus and sum of its nucleons. - Binding energy (E_b) – Energy needed to split a nucleus into its nucleons. - Binding energy per nucleon – Average energy per nucleon (higher = more stable nucleus).

Formulas:
1. Mass defect [ \Delta m = Zm_p + Nm_n - M_{\text{nucleus}} ] - ( Z ) = Number of protons - ( m_p ) = Mass of a proton (MEMORISE THIS: ( 1.673 \times 10^{-27} \, \text{kg} )) - ( N ) = Number of neutrons - ( m_n ) = Mass of a neutron (MEMORISE THIS: ( 1.675 \times 10^{-27} \, \text{kg} )) - ( M_{\text{nucleus}} ) = Mass of the nucleus (kg)

1. Binding energy (Einstein’s equation) [ E_b = \Delta m \cdot c^2 ]
2. ( E_b ) = Binding energy (J)
3. ( \Delta m ) = Mass defect (kg)

4. ( c ) = Speed of light (MEMORISE THIS: ( 3.0 \times 10^8 \, \text{m/s} ))

5. Binding energy per nucleon [ \text{Binding energy per nucleon} = \frac{E_b}{A} ]

6. ( A ) = Mass number (total nucleons)

5. Radioactive Decay Law

Key Terms: - Half-life (t₁/₂) – Time for half the radioactive nuclei to decay. - Decay constant (λ) – Probability of decay per unit time. - Activity (A) – Number of decays per second (Bq).

Formulas:
1. Decay law (exponential decay) [ N = N_0 e^{-\lambda t} ] - ( N ) = Number of undecayed nuclei at time ( t ) - ( N_0 ) = Initial number of nuclei - ( \lambda ) = Decay constant (s⁻¹) - ( t ) = Time (s)

1. Half-life and decay constant [ t_{1/2} = \frac{\ln 2}{\lambda} ]
2. ( t_{1/2} ) = Half-life (s)

3. ( \ln 2 ) ≈ 0.693 (MEMORISE THIS)

4. Activity [ A = \lambda N ]

5. ( A ) = Activity (Bq)

STEP-BY-STEP METHOD

1. Photoelectric Effect Problems

Step 1: Identify given values (frequency ( f ), work function ( \Phi ), stopping potential ( V_s )). Step 2: If frequency is given, use ( E_{\text{max}} = hf - \Phi ). Step 3: If stopping potential is given, use ( eV_s = E_{\text{max}} ). Step 4: Solve for the unknown (e.g., ( E_{\text{max}} ), ( \Phi ), ( f )). Step 5: Check units (J vs. eV—convert if needed).

2. Bohr Model Problems

Step 1: Identify initial and final energy levels (( n_i ) and ( n_f )). Step 2: Calculate energy difference ( \Delta E = E_f - E_i ). Step 3: Use ( \Delta E = \frac{hc}{\lambda} ) to find wavelength. Step 4: If wavelength is given, rearrange to find ( \Delta E ). Step 5: Check if the transition is emission (( n_i > n_f )) or absorption (( n_i < n_f )).

3. De Broglie Wavelength Problems

Step 1: Identify mass (( m )) and velocity (( v )) of the particle. Step 2: Calculate momentum ( p = mv ). Step 3: Use ( \lambda = \frac{h}{p} ). Step 4: Check units (kg for mass, m/s for velocity).

4. Binding Energy Problems

Step 1: Calculate mass defect ( \Delta m = Zm_p + Nm_n - M_{\text{nucleus}} ). Step 2: Convert mass defect to kg (if in u, use ( 1 \, \text{u} = 1.66 \times 10^{-27} \, \text{kg} )). Step 3: Use ( E_b = \Delta m \cdot c^2 ). Step 4: For binding energy per nucleon, divide by ( A ).

5. Radioactive Decay Problems

Step 1: Identify given values (( N_0 ), ( N ), ( t ), ( t_{1/2} ), ( \lambda )). Step 2: If half-life is given, find ( \lambda = \frac{\ln 2}{t_{1/2}} ). Step 3: Use ( N = N_0 e^{-\lambda t} ) to find remaining nuclei. Step 4: For activity, use ( A = \lambda N ).

WORKED EXAMPLES

Example 1 – Photoelectric Effect (Basic)

Question: Light of frequency ( 8.0 \times 10^{14} \, \text{Hz} ) shines on a metal with a work function of ( 3.0 \, \text{eV} ). What is the maximum kinetic energy of the ejected electrons?

Solution:
1. Convert work function to joules: ( \Phi = 3.0 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} = 4.8 \times 10^{-19} \, \text{J} ).
2. Use ( E_{\text{max}} = hf - \Phi ): ( E_{\text{max}} = (6.63 \times 10^{-34} \, \text{Js})(8.0 \times 10^{14} \, \text{Hz}) - 4.8 \times 10^{-19} \, \text{J} ).
3. Calculate: ( E_{\text{max}} = 5.304 \times 10^{-19} \, \text{J} - 4.8 \times 10^{-19} \, \text{J} = 5.04 \times 10^{-20} \, \text{J} ).

What we did and why: We used Einstein’s photoelectric equation to find the leftover energy after overcoming the work function. Always convert eV to joules when using ( h ).

Example 2 – Bohr Model (Medium)

Question: An electron transitions from ( n=3 ) to ( n=2 ) in a hydrogen atom. What is the wavelength of the emitted photon?

Solution:
1. Calculate energy levels: ( E_3 = -\frac{13.6}{3^2} = -1.51 \, \text{eV} ), ( E_2 = -\frac{13.6}{2^2} = -3.4 \, \text{eV} ).
2. Find energy difference: ( \Delta E = E_2 - E_3 = -3.4 \, \text{eV} - (-1.51 \, \text{eV}) = -1.89 \, \text{eV} ). (Negative sign indicates emission—ignore for magnitude.)
3. Convert to joules: ( \Delta E = 1.89 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} = 3.024 \times 10^{-19} \, \text{J} ).
4. Use ( \lambda = \frac{hc}{\Delta E} ): ( \lambda = \frac{(6.63 \times 10^{-34} \, \text{Js})(3.0 \times 10^8 \, \text{m/s})}{3.024 \times 10^{-19} \, \text{J}} = 6.58 \times 10^{-7} \, \text{m} ).

What we did and why: We calculated the energy difference between levels and used it to find the photon wavelength. Always check if the transition is emission or absorption.

Example 3 – Binding Energy (Exam-Style)

Question: The mass of a ( ^4_2\text{He} ) nucleus is ( 6.644 \times 10^{-27} \, \text{kg} ). Calculate its binding energy per nucleon.

Solution:
1. Calculate mass defect: ( \Delta m = (2 \times 1.673 \times 10^{-27} \, \text{kg}) + (2 \times 1.675 \times 10^{-27} \, \text{kg}) - 6.644 \times 10^{-27} \, \text{kg} ). ( \Delta m = 6.696 \times 10^{-27} \, \text{kg} - 6.644 \times 10^{-27} \, \text{kg} = 5.2 \times 10^{-29} \, \text{kg} ).
2. Calculate binding energy: ( E_b = \Delta m \cdot c^2 = (5.2 \times 10^{-29} \, \text{kg})(3.0 \times 10^8 \, \text{m/s})^2 = 4.68 \times 10^{-12} \, \text{J} ).
3. Convert to MeV: ( 4.68 \times 10^{-12} \, \text{J} \div 1.6 \times 10^{-13} \, \text{J/MeV} = 29.25 \, \text{MeV} ).
4. Binding energy per nucleon: ( \frac{29.25 \, \text{MeV}}{4} = 7.31 \, \text{MeV/nucleon} ).

What we did and why: We found the mass defect, converted it to energy, and divided by the number of nucleons. Always convert to MeV for nuclear energy.

COMMON MISTAKES

1. Mistake: Forgetting to convert eV to joules. Why it happens: Using ( h ) in Js with energy in eV. Correct approach: Multiply eV by ( 1.6 \times 10^{-19} ) to get joules.

2. Mistake: Mixing up emission and absorption in Bohr model. Why it happens: Not checking if ( n_i > n_f ) (emission) or ( n_i < n_f ) (absorption). Correct approach: Emission = energy lost (negative ( \Delta E )), absorption = energy gained (positive ( \Delta E )).

3. Mistake: Using atomic mass instead of nuclear mass. Why it happens: Forgetting to subtract electron masses. Correct approach: For binding energy, use nuclear mass (or subtract ( Z \times m_e ) from atomic mass).

4. Mistake: Confusing half-life and decay constant. Why it happens: Using ( t_{1/2} ) directly in ( N = N_0 e^{-\lambda t} ). Correct approach: First find ( \lambda = \frac{\ln 2}{t_{1/2}} ).

5. Mistake: Ignoring units in binding energy. Why it happens: Using mass in u without converting to kg. Correct approach: Convert u to kg (( 1 \, \text{u} = 1.66 \times 10^{-27} \, \text{kg} )).

EXAM TRAPS

1. Trap: Giving frequency in kHz or MHz instead of Hz. How to spot it: Units like "kHz" or "MHz" in the question. How to avoid it: Convert to Hz (multiply by ( 10^3 ) or ( 10^6 )).

2. Trap: Asking for "energy of the photon" but giving stopping potential. How to spot it: Question mentions ( V_s ) but asks for ( E ). How to avoid it: Use ( eV_s = E_{\text{max}} ), then ( E_{\text{photon}} = E_{\text{max}} + \Phi ).

3. Trap: Using atomic mass instead of nuclear mass in binding energy. How to spot it: Question gives "atomic mass" but asks for binding energy. How to avoid it: Subtract ( Z \times m_e ) from atomic mass to get nuclear mass.

1-MINUTE RECAP

You: "Okay, quantum and nuclear physics—let’s crush this in 60 seconds. First, photoelectric effect: ( E_{\text{max}} = hf - \Phi ). If they give stopping potential, ( eV_s = E_{\text{max}} ). Bohr model: ( E_n = -13.6/n^2 ), then ( \Delta E = hc/\lambda ). De Broglie: ( \lambda = h/mv ). Binding energy: mass defect ( \times c^2 ), then divide by ( A ) for per nucleon. Decay law: ( N = N_0 e^{-\lambda t} ), and ( \lambda = \ln 2 / t_{1/2} ). Common traps? Units—always convert eV to joules, kHz to Hz, u to kg. Half-life vs. decay constant—don’t mix them up. And binding energy? Use nuclear mass, not atomic mass. You’ve got this—go ace that exam!