Physics Modern and Semiconductor - How to Solve: Nuclei (Radioactive Decay, Half-Life, Nuclear Binding Energy, Fission/Fusion) – NEET UG Guide

How to Solve: Nuclei (Radioactive Decay, Half-Life, Nuclear Binding Energy, Fission/Fusion) – NEET UG Guide

Introduction Mastering nuclei unlocks 10-12 marks in NEET UG—enough to push you from a 600 to a 650+ score. It’s the difference between a medical seat and a waitlist. Plus, it explains nuclear power, cancer treatment, and how stars shine—real-world impact you’ll see in both physics and chemistry.

WHAT YOU NEED TO KNOW FIRST

1. Exponential functions – Understand how quantities decay over time (e.g., ( N = N_0 e^{-\lambda t} )).
2. Atomic structure – Know protons, neutrons, mass number (A), and atomic number (Z).
3. Energy-mass equivalence – ( E = mc^2 ) (given on exam sheet, but you must know how to apply it).

KEY TERMS & FORMULAS

1. Radioactive Decay

• Decay constant (λ) – Probability of decay per unit time (s⁻¹).
• Activity (A) – Number of decays per second (Bq or Ci). Formula: ( A = \lambda N )
• ( A ) = Activity (Bq)
• ( \lambda ) = Decay constant (s⁻¹)

• ( N ) = Number of undecayed nuclei MEMORISE THIS

• Decay law: ( N = N_0 e^{-\lambda t} )

• ( N ) = Remaining nuclei after time ( t )
• ( N_0 ) = Initial nuclei
• ( \lambda ) = Decay constant
• ( t ) = Time elapsed MEMORISE THIS

2. Half-Life (t₁/₂)

• Time for half the nuclei to decay. Formula: ( t_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda} ) MEMORISE THIS

• Relation to decay constant: ( \lambda = \frac{0.693}{t_{1/2}} )

3. Nuclear Binding Energy (BE)

• Energy needed to split a nucleus into protons and neutrons. Formula: ( BE = \Delta m \cdot c^2 )
• ( \Delta m ) = Mass defect (kg or u)

• ( c ) = Speed of light (3 × 10⁸ m/s) Given on exam sheet, but know how to use it

• Mass defect (Δm): ( \Delta m = Zm_p + (A-Z)m_n - M_{\text{nucleus}} )

• ( Z ) = Atomic number (protons)
• ( A ) = Mass number (protons + neutrons)
• ( m_p ) = Mass of proton (1.007276 u)
• ( m_n ) = Mass of neutron (1.008665 u)
• ( M_{\text{nucleus}} ) = Actual mass of nucleus (u)

4. Fission & Fusion

• Fission: Heavy nucleus splits into lighter nuclei + energy.
• Example: ( ^{235}{92}U + ^1_0n \rightarrow ^{141} )}Ba + ^{92}_{36}Kr + 3^1_0n + \text{Energy
• Fusion: Light nuclei combine to form heavier nucleus + energy.
• Example: ( 4^1_1H \rightarrow ^4_2He + 2^0_{+1}e + \text{Energy} )

STEP-BY-STEP METHOD

Step 1: Identify the Problem Type

• Radioactive decay? → Use ( N = N_0 e^{-\lambda t} ) or half-life.
• Binding energy? → Calculate mass defect, then ( BE = \Delta m \cdot c^2 ).
• Fission/fusion? → Balance nuclear equation, find energy released.

Step 2: Extract Given Data

• Write down all given values (e.g., ( N_0 ), ( t ), ( t_{1/2} ), masses).
• Convert units if needed (e.g., years → seconds, u → kg).

Step 3: Choose the Right Formula

• Decay problems? → Use ( N = N_0 e^{-\lambda t} ) or ( t_{1/2} = \frac{0.693}{\lambda} ).
• Binding energy? → Find mass defect first.
• Energy released? → Use ( E = \Delta m \cdot c^2 ).

Step 4: Solve Step-by-Step

• For decay: Plug into formula, solve for unknown.
• For binding energy: Calculate mass defect, then energy.
• For fission/fusion: Balance equation, find mass difference, then energy.

Step 5: Check Units & Significant Figures

• Energy in MeV (1 u = 931.5 MeV/c²).
• Time in seconds (unless specified otherwise).
• Round to 3 significant figures (NEET standard).

WORKED EXAMPLES

Example 1 – Basic (Half-Life Calculation)

Problem: The half-life of ( ^{14}C ) is 5730 years. What fraction remains after 11460 years?

Step 1: Identify problem type → Half-life decay. Step 2: Given: - ( t_{1/2} = 5730 ) years - ( t = 11460 ) years Step 3: Use ( N = N_0 \left( \frac{1}{2} \right)^{t/t_{1/2}} ) Step 4: ( \frac{t}{t_{1/2}} = \frac{11460}{5730} = 2 ) - ( N = N_0 \left( \frac{1}{2} \right)^2 = \frac{N_0}{4} ) Step 5: Fraction remaining = ( \frac{1}{4} ).

What we did and why: We used the half-life formula directly because the time given was an exact multiple of the half-life. No need for decay constant here.

Example 2 – Medium (Decay Constant & Activity)

Problem: A sample has ( 10^{12} ) radioactive nuclei with ( \lambda = 0.01 ) s⁻¹. What is its activity after 100 s?

Step 1: Identify problem type → Decay law + activity. Step 2: Given: - ( N_0 = 10^{12} ) - ( \lambda = 0.01 ) s⁻¹ - ( t = 100 ) s Step 3: First, find remaining nuclei: - ( N = N_0 e^{-\lambda t} = 10^{12} e^{-0.01 \times 100} = 10^{12} e^{-1} ) - ( e^{-1} \approx 0.3679 ) - ( N \approx 3.679 \times 10^{11} ) Step 4: Activity ( A = \lambda N = 0.01 \times 3.679 \times 10^{11} = 3.679 \times 10^9 ) Bq. Step 5: Final answer: ( 3.68 \times 10^9 ) Bq (rounded to 3 sig. figs).

What we did and why: We first found the remaining nuclei using the decay law, then calculated activity using ( A = \lambda N ). Always check if the question asks for remaining nuclei or activity.

Example 3 – Exam-Style (Binding Energy)

Problem: Calculate the binding energy per nucleon for ( ^{16}_8O ). Given: - Mass of ( ^{16}_8O ) = 15.994915 u - Mass of proton = 1.007276 u - Mass of neutron = 1.008665 u

Step 1: Identify problem type → Binding energy. Step 2: Given: - ( Z = 8 ), ( A = 16 ) - ( M_{\text{nucleus}} = 15.994915 ) u - ( m_p = 1.007276 ) u - ( m_n = 1.008665 ) u Step 3: Calculate mass defect: - ( \Delta m = Zm_p + (A-Z)m_n - M_{\text{nucleus}} ) - ( \Delta m = 8(1.007276) + 8(1.008665) - 15.994915 ) - ( \Delta m = 8.058208 + 8.06932 - 15.994915 = 0.132613 ) u Step 4: Convert to energy: - ( BE = \Delta m \times 931.5 ) MeV/u - ( BE = 0.132613 \times 931.5 = 123.56 ) MeV Step 5: Binding energy per nucleon: - ( \frac{BE}{A} = \frac{123.56}{16} = 7.72 ) MeV/nucleon

What we did and why: We calculated the mass defect first, then converted it to energy using ( 1 \text{ u} = 931.5 \text{ MeV} ). Always divide by A for binding energy per nucleon.

COMMON MISTAKES

1. MISTAKE: Using ( t_{1/2} ) directly in ( N = N_0 e^{-\lambda t} ) without converting to ( \lambda ). WHY IT HAPPENS: Confusing half-life with decay constant. CORRECT APPROACH: Always use ( \lambda = \frac{0.693}{t_{1/2}} ) first.

2. MISTAKE: Forgetting to convert mass defect to kg before using ( E = mc^2 ). WHY IT HAPPENS: Using u (atomic mass units) directly in ( E = mc^2 ). CORRECT APPROACH: Convert u to kg (1 u = 1.66 × 10⁻²⁷ kg) or use 931.5 MeV/u.

3. MISTAKE: Mixing up activity (A) and number of nuclei (N). WHY IT HAPPENS: Not distinguishing between ( A = \lambda N ) and ( N = N_0 e^{-\lambda t} ). CORRECT APPROACH: Activity is decays per second, not remaining nuclei.

4. MISTAKE: Incorrectly balancing nuclear equations (e.g., forgetting neutrons in fission). WHY IT HAPPENS: Not checking mass and atomic numbers on both sides. CORRECT APPROACH: Ensure ( \Sigma A ) and ( \Sigma Z ) are equal on both sides.

5. MISTAKE: Using wrong units for time (e.g., years instead of seconds). WHY IT HAPPENS: Not converting to consistent units. CORRECT APPROACH: Always use seconds for ( \lambda ) and ( t ).

EXAM TRAPS

1. TRAP: Giving half-life in years but asking for decay constant in s⁻¹. HOW TO SPOT IT: Question mentions "per second" but gives time in years. HOW TO AVOID IT: Convert years to seconds (1 year = 3.15 × 10⁷ s).

2. TRAP: Asking for binding energy per nucleon but only giving total binding energy. HOW TO SPOT IT: Question says "per nucleon" but gives BE in MeV. HOW TO AVOID IT: Always divide by A (mass number).

3. TRAP: Fission/fusion questions with missing mass values (e.g., "energy released" but no masses given). HOW TO SPOT IT: No mass defect provided, but energy is asked. HOW TO AVOID IT: Use mass-energy equivalence (1 u = 931.5 MeV) and balance the equation first.

1-MINUTE RECAP (Night Before Exam)

"Listen up—this is your 60-second nuclei crash course for NEET.

1. Radioactive decay? Use ( N = N_0 e^{-\lambda t} ). Half-life? ( t_{1/2} = \frac{0.693}{\lambda} ). Activity? ( A = \lambda N ). Memorise these three.

2. Binding energy? Find mass defect first: ( \Delta m = Zm_p + (A-Z)m_n - M_{\text{nucleus}} ). Then ( BE = \Delta m \times 931.5 ) MeV. Divide by A for per nucleon.

3. Fission/fusion? Balance the equation. Mass lost? Convert to energy with ( E = \Delta m \cdot c^2 ).

4. Units matter! Time in seconds, mass in kg or u, energy in MeV. 1 u = 931.5 MeV.

5. Common traps? Half-life in years? Convert to seconds. Asking for per nucleon? Divide by A. No masses given? Balance the equation first.

You’ve got this. Now go ace those 10-12 marks!