This chapter explains the structure and properties of atomic nuclei, including their components, stability, and phenomena like radioactivity, fission, and fusion.
NUCLEI – Formula & Equation Sheet
Essential formulas and equations from Physics Part - II, tailored for Class 12 in Physics.
This one-pager compiles key formulas and equations from the NUCLEI chapter of Physics Part - II. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
E = mc²
E is energy (J), m is mass (kg), and c is the speed of light (≈ 3 × 10⁸ m/s). This formula illustrates the equivalence of mass and energy, fundamental in nuclear reactions.
R = R₀ A^{1/3}
R is the nucleus radius (m), R₀ = 1.2 x 10⁻¹⁵ m (constant), and A is the mass number. This formula relates nucleus size to the number of nucleons.
A = Z + N
A is mass number, Z is atomic number (protons), and N is neutron number. It defines the total number of nucleons in a nucleus.
ΔM = (Z mᵖ + (A - Z) mⁿ) - M
ΔM is the mass defect, Z is the number of protons, mᵖ is the mass of a proton, mⁿ is the mass of a neutron, and M is the actual mass of the nucleus. It shows the difference between the mass of separated nucleons and the bound nucleus.
E_b = ΔM c²
E_b is the binding energy (J), ΔM is the mass defect, and c is the speed of light. It quantifies the energy required to disassemble a nucleus into its individual nucleons.
E_{bn} = E_b / A
E_{bn} is the binding energy per nucleon (MeV/nucleon), E_b is the total binding energy, and A is the mass number. This value indicates the stability of a nucleus.
A = 2Z + 2N - 2k
This indicates isospin conservation in nuclear reactions where k is the number of emitted particles. It helps maintain nucleon count in reactions.
N_a = N_0 e^{-λt}
N_a is the remaining number of nuclei, N_0 is the initial number, λ is the decay constant, and t is time (s). It illustrates exponential decay in radioactivity.
T_{1/2} = ln(2) / λ
T_{1/2} is the half-life (s) of a radioactive substance, λ is the decay constant. This formula calculates the time required for half of the nuclei to decay.
Q = (m_A + m_B - m_C - m_D)c²
Q is the energy released (MeV) in a nuclear reaction, and m_A, m_B, m_C, m_D are the masses of the participating nuclei. It indicates whether a reaction is exothermic or endothermic.
Equations
F = k(q₁q₂)/r²
F is the electrostatic force (N), k is Coulomb's constant (≈ 9 × 10⁹ N m²/C²), q₁ and q₂ are charges (C), and r is the distance (m) between charges. This foundational equation describes the interaction between charged particles.
E = E_0 + m₀c²
E is the total energy (J), E₀ is kinetic energy (J), m₀ is rest mass (kg), and c is the speed of light. This equation is fundamental in calculating energy in relativistic systems.
m = n M_u
m is the total mass (kg), n is the number of particles, and M_u is the atomic mass unit (kg). Used for calculating mass from the count of nucleons or atoms in chemical calculations.
A = (Z + N) / V
A is the mass number, Z is the atomic number, N is the neutron number, and V is volume of the nucleus (m³). This relates nuclear size and composition.
λ = 0.693/T_{1/2}
λ is the decay constant, and T_{1/2} is the half-life. It calculates how quickly a radioactive substance will decay.
N = N_0 e^{-λt}
N is the number of remaining unstable nuclei, N_0 is the original number, e is the base of the natural logarithm, λ is the decay constant. This is a model of nuclear decay.
c = λν
c is the speed of light (m/s), λ is wavelength (m), and ν is frequency (Hz). This is essential in understanding the properties of electromagnetic radiation emitted during nuclear decay.
A = 4/3 πR³
A is the volume of a sphere, R is the radius. This formula helps in understanding the three-dimensional structure of atomic nuclei.
K.E. = 0.5mv²
K.E. is kinetic energy (J), m is mass (kg), v is velocity (m/s). Used to calculate the motion of nucleons within the nucleus.
Q = E_{initial} - E_{final}
Q is the energy in a nuclear reaction, this principle applies to the conservation of energy and helps determine the energy changes in reactions.
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