Formula Bank - Nuclei
Searchable database of all formulas with dimensional analysis
Quick reference for all Nuclei formulas. Use search to find specific formulas instantly. Each formula includes units and typical applications.
Nuclear Structure
Nuclear Radius
R = R₀ A^(1/3)
R₀ = 1.2 × 10⁻¹⁵ m, A = mass number
Dimension: [L]
Dimension: [L]
Nuclear Density
ρ = 3M/(4πR³) ≈ 2.3 × 10¹⁷ kg/m³
Constant for all nuclei
Dimension: [ML⁻³]
Dimension: [ML⁻³]
Neutron Number
N = A - Z
N = number of neutrons, A = mass number, Z = atomic number
Nuclear Volume
V = (4/3)πR³ = (4/3)π(R₀A^(1/3))³ ∝ A
Volume proportional to mass number
Dimension: [L³]
Dimension: [L³]
Mass-Energy & Binding Energy
Mass-Energy Equivalence
E = mc²
c = 3 × 10⁸ m/s
Dimension: [ML²T⁻²]
Dimension: [ML²T⁻²]
Energy Unit Conversion
1 u = 931.5 MeV/c² = 1.66 × 10⁻²⁷ kg
MOST IMPORTANT conversion in Nuclei
Mass Defect
Δm = [Zmₚ + Nmₙ] - M_nucleus
mₚ = 1.007276 u, mₙ = 1.008665 u
Dimension: [M]
Dimension: [M]
Binding Energy (Total)
BE = Δm × c² = Δm × 931.5 MeV
If Δm in u, multiply by 931.5 to get MeV
Dimension: [ML²T⁻²]
Dimension: [ML²T⁻²]
Binding Energy per Nucleon
BE/A = (Total BE) / A
Measure of nuclear stability. Max at Fe-56 (8.8 MeV)
Dimension: [ML²T⁻²]
Dimension: [ML²T⁻²]
Common Error: Forgetting to convert u to MeV using 931.5 factor. Always check units in your answer!
Radioactive Decay Laws
Decay Rate (Activity)
dN/dt = -λN
λ = decay constant, N = number of nuclei
Dimension of λ: [T⁻¹]
Dimension of λ: [T⁻¹]
Exponential Decay Law
N(t) = N₀ e^(-λt)
N₀ = initial nuclei at t = 0
Most frequently used formula
Most frequently used formula
Number Decayed
N_decayed = N₀(1 - e^(-λt))
N_remaining = N₀ e^(-λt)
N_decayed + N_remaining = N₀
N_decayed + N_remaining = N₀
Half-Life
T½ = (ln 2)/λ = 0.693/λ
Time for N to become N₀/2
Dimension: [T]
Dimension: [T]
After n Half-Lives
N = N₀/(2^n) where n = t/T½
Quick calculation method
After 3 half-lives: N = N₀/8
After 3 half-lives: N = N₀/8
Mean Life (Average Life)
τ = 1/λ = T½/(ln 2) = 1.44 T½
τ always longer than T½
Dimension: [T]
Dimension: [T]
Activity
A = λN = A₀ e^(-λt)
Number of decays per second
Dimension: [T⁻¹]
Dimension: [T⁻¹]
Activity in terms of T½
A = (0.693 N)/T½
Alternative form when T½ is given
Unit: Becquerel (Bq) = 1 decay/s
Unit: Becquerel (Bq) = 1 decay/s
Activity Units
1 Ci = 3.7 × 10¹⁰ Bq
Curie (Ci) = older unit
Becquerel (Bq) = SI unit
Becquerel (Bq) = SI unit
Relation: λ, T½, τ
λ = (ln 2)/T½ = 1/τ
All three parameters are interrelated
Pro Tip: In exam, if T½ is given, use N = N₀/(2^n) for faster calculation. If λ is given, use N = N₀e^(-λt).
Radioactive Decay Equations
Alpha (α) Decay
ᴬ_Z X → ᴬ⁻⁴_{Z-2} Y + ⁴_2 He
A decreases by 4, Z decreases by 2
Beta Minus (β⁻) Decay
ᴬ_Z X → ᴬ_{Z+1} Y + ⁰_{-1} e + ν̄
A unchanged, Z increases by 1
n → p + e⁻ + ν̄
n → p + e⁻ + ν̄
Beta Plus (β⁺) Decay
ᴬ_Z X → ᴬ_{Z-1} Y + ⁰_{+1} e + ν
A unchanged, Z decreases by 1
p → n + e⁺ + ν
p → n + e⁺ + ν
Gamma (γ) Decay
ᴬ_Z X* → ᴬ_Z X + γ
No change in A or Z
Only energy state changes
Only energy state changes
Nuclear Reactions
Q-Value (Energy Released)
Q = [Σm_reactants - Σm_products] × c²
Q = (Δm) × 931.5 MeV (if Δm in u)
Dimension: [ML²T⁻²]
Dimension: [ML²T⁻²]
Q-Value Sign Convention
Q > 0 → Exothermic (energy released)
Q < 0 → Endothermic (energy absorbed)
Q < 0 → Endothermic (energy absorbed)
Exothermic reactions are spontaneous
Conservation of Mass Number
Σ A_reactants = Σ A_products
Total nucleon number conserved
Conservation of Charge
Σ Z_reactants = Σ Z_products
Total charge (proton number) conserved
Energy in Fission
E_fission ≈ 200 MeV per fission
For U-235 fission
Released as KE of fragments and neutrons
Released as KE of fragments and neutrons
Energy in Fusion (D-T)
²H + ³H → ⁴He + n + 17.6 MeV
Deuterium-Tritium fusion
Requires ~10⁷ K temperature
Requires ~10⁷ K temperature
Q-value calculations appear in 40% of NEET/JEE numericals. Master the mass-to-energy conversion (931.5 MeV/u).
Carbon Dating & Applications
Age Calculation (Method 1)
t = (T½/0.693) ln(N₀/N)
For C-14: T½ = 5730 years
N₀ = initial C-14, N = current C-14
N₀ = initial C-14, N = current C-14
Age Calculation (Method 2)
t = (2.303/λ) log₁₀(N₀/N)
Using log base 10
λ = 0.693/T½
λ = 0.693/T½
Activity Ratio Form
t = (T½/0.693) ln(A₀/A)
Can also use activity ratio
A ∝ N (both decay exponentially)
A ∝ N (both decay exponentially)
C-14 Half-Life
T½ = 5730 years
Suitable for dating up to ~50,000 years
Only works for organic materials
Only works for organic materials
Key Constants (Memorize)
Proton Mass
mₚ = 1.007276 u
= 1.6726 × 10⁻²⁷ kg
= 1.6726 × 10⁻²⁷ kg
Use in mass defect calculations
Neutron Mass
mₙ = 1.008665 u
= 1.6749 × 10⁻²⁷ kg
= 1.6749 × 10⁻²⁷ kg
Slightly heavier than proton
Electron Mass
mₑ = 0.000549 u
= 9.109 × 10⁻³¹ kg
= 9.109 × 10⁻³¹ kg
Much lighter than nucleons
Atomic Mass Unit
1 u = 931.5 MeV/c²
= 1.66 × 10⁻²⁷ kg
= 1.66 × 10⁻²⁷ kg
MOST USED constant
Speed of Light
c = 3 × 10⁸ m/s
Use in E = mc²
Fermi Constant
R₀ = 1.2 × 10⁻¹⁵ m
= 1.2 fm
= 1.2 fm
Use in R = R₀A^(1/3)
Avogadro Number
Nₐ = 6.022 × 10²³ mol⁻¹
Use when converting moles to atoms
ln 2
ln 2 = 0.693
Appears in half-life formulas
Curie Unit
1 Ci = 3.7 × 10¹⁰ Bq
Activity unit conversion
Formula Mastery Tips:
- Don't just memorize—understand derivations (helps in JEE Advanced)
- Practice unit conversions daily (1 u = 931.5 MeV is CRITICAL)
- Create formula flashcards for quick revision
- Always check dimensional consistency in your answers
- Know when to use which form (exponential vs half-life)