List of formulas related to atomic nucleus
Nuclear is made up of
Z: number of protons, N: number of neutrons, A: mass number
$$ ^{A}_{Z} \text{X} $$ $$ A = \text{N + Z} $$Nuclear Reaction
$$ ^{A_1}_{Z_1} \text{A} + ^{A_2}_{Z_2} \text{B} -> ^{A_3}_{Z_3} \text{C} + ^{A_4}_{Z_4} \text{D} $$ $$ A_1 + A_2 = A_3 + A_4 $$ $$ Z_1 + Z_2 = Z_3 + Z_4 $$ $$ m_o = m_A + m_B $$ $$ m = m_C + m_D $$m < mo: nuclear reaction releases energy
$$ E = (m_o - m)c^2 $$m < mo: nuclear reaction absorbs energy
$$ E = (m_o - m)c^2 + E_s $$Radioactive Decay
No: initial amount of active substance, N: quantity that still remains and has not yet decayed after a time t, ΔN: amount of decayed substance, mo: initial mass, m: mass of remaining, λ: decay constant, t: time, T: half life
$$ \lambda = \frac{l_{n2}}{T} = \frac{0, 693}{T} $$ $$ N = N_o e^{- \lambda t } = \frac{N_o}{2 \frac{t}{T}} $$ $$ m = m_o e^{- \lambda t } = \frac{m_o}{2 \frac{t}{T}} $$ $$ \Delta N = N_o - N $$ $$ N_o(1 - \frac{1}{2^{ \frac{t}{T}}}) = N_o (1 - e^{- \lambda t}) $$ $$ N_o = \frac{m_o}{A} 6,022.10^{23} $$ $$ N = N_o .2^{- \frac{t}{T}} = \frac{m_o}{A} 6,022.10^{23}.2^{- \frac{t}{T}} $$ $$ \Delta N = N_o - N $$ $$ = N_o(1 - 2^{- \frac{t}{T}}) $$ $$ = \frac{m_o}{A} 6,022.10^{23}.(1 - 2^{- \frac{t}{T}}) $$Half Life of some elements
| Element | Half Life-T |
|---|---|
| Radon (Rn-219) | 4 seconds |
| Oxygen (O-15) | 122 seconds |
| Iodine (I-131) | 8 days |
| Polonium (Po-210) | 138.4 days |
| Radium (Ra-226) | 1600 years |
| Carbon (C-14) | 5730years |
| Uranium (U-235) | 7.108 |
Amount Of Radioactivity
Ho: initial amount of radioactivity, H: amount of radioactivity at time t
$$ H_o = \lambda . N_o = \frac{ln_2}{T}. \frac{m}{A} .6,022.10^{23} $$ $$ H = \frac{H_o}{2^{\frac{t}{T}}} = H_o e^{- \lambda t} $$