Physics Nuclei

$\mathrm{}{\lambda }_{\text{electron}}=\frac{h}{\sqrt[]{2mE}}\mathrm{}\mathrm{}\frac{{\lambda }_{\text{electron}}}{{\lambda }_{\text{photon}}}=\frac{1}{c}\sqrt[]{\frac{E}{2m}}$

${\lambda }_{\text{photon}}=\frac{\mathrm{h}\mathrm{c}}{\mathrm{E}}$

A = Activity = λN = (ln (2)/t? /? ) N
N = (1.5*10? ³ / 198) * 6*10²³
A = (0.693 / (2.7*24*3600) * (1.5*10? ³ / 198) * 6*10²³ disintegration/s
A ≈ 1.32 * 10¹³ Bq = (1.32*10¹³ / 3.7*10¹? ) Ci ≈ 357 Ci

N? /N? = e?
For 33% decay, N? /N? = 0.67 ≈ 2/3.
2/3 = e? ⇒ t? = (1/λ)ln (3/2)
For 67% decay, N? /N? = 0.33 ≈ 1/3.
1/3 = e? ⇒ t? = (1/λ)ln (3)
Δt = t? - t? = (1/λ) [ln (3) - ln (3/2)] = (1/λ)ln (2) = T? /? = 20 min

A = A? (1/2)?
A/A? = 1/8 = (1/2)³ ⇒ n = 3
t = n T½
30 = 3 T½
T½ = 10 years

t? /? = 3 days = 72 hours
dN/dt = λN = (ln2/t? /? ) N
= (0.693 * 6.02*10²³ * 2*10? ³) / (72 * 3600 * 198)
= 1.618 * 10¹³
= 16.18 * 10¹² disintegration/second

Binding Energy = (Δm)c²
= [Zmp + (A-Z)mn - MAl]c²
= [ (13*1.00726 + 14*1.00866) - 27.18846] u
= [ (13.09438 + 14.12124) - 27.18846] u
= [27.21562 - 27.18846] u
= 0.02716 u
= 0.02716 x = 27.16x * 10? ³

3N? /4 = N? e?
⇒ e? = 4/3
t? = (1/λ) ln (4/3)
t? = ln2/λ
t? - t? = (1/λ)ln2 - (1/λ)ln (4/3)
= (1/λ)ln (2/ (4/3) = (1/λ)ln (3/2)

$A=A_0{e}^{-\lambda t_1}$ [Radio active decay law]

$\frac{A}{5}=A_0{e}^{-\lambda \left(t_2-t_1\right)}$

$\Rightarrow ln5=\lambda \left(t_2-t_1\right)$

$\Rightarrow Averagelife=\frac{1}{\lambda }=\frac{\left(t_2-t_1\right)}{\mathcal{l}n5}$

Number of half lives of Y = 3

Number of half lives of X = 6 [As half life of X is half of that of Y].

$\frac{N_1}{2^6}=\frac{N_2}{2^3}\Rightarrow \frac{N_1}{N_2}=8$

Apply energy conservation,

$U_i+K_i=U_f+K_f$

$\Rightarrow -\frac{GMm}{R}+K_i=-\frac{GMm}{3R}+\frac{1}{2}m{v}^2$

$\Rightarrow -\frac{GMm}{R}+K_i=-\frac{GMm}{3R}+\frac{1}{2}*m*\frac{GM}{3R}$

$\Rightarrow K_i=-\frac{1}{6}\frac{GMm}{R}+\frac{GMm}{R}$

$K_i=\frac{5}{6}\frac{GMm}{R}$