# 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

$$E = (m_o - m)c^2 + E_s$$

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

ElementHalf Life-T
$$H_o = \lambda . N_o = \frac{ln_2}{T}. \frac{m}{A} .6,022.10^{23}$$ $$H = \frac{H_o}{2^{\frac{t}{T}}} = H_o e^{- \lambda t}$$