Advanced Nuclear Physics Problems

Solve the following problems:

(a) Calculate the energy released during the fusion of two deuterium nuclei: $$^2_1\text{H} + ^2_1\text{H} \rightarrow ^3_2\text{He} + n$$ Given the masses: $$m(^2_1\text{H}) = 2.01 \, \text{u}, \, m(^3_2\text{He}) = 3.02 \, \text{u}, \, m(n) = 1.01 \, \text{u}$$ Use $1 \, \text{u} = 931.5 \, \text{MeV/c}^2$.

(b) The half-life of a radioactive isotope is 20 years. Calculate the time required for its activity to decrease to 50% of the initial activity. Use the decay law: $$A(t) = A_0 e^{-\lambda t}, \, \lambda = \frac{\ln(2)}{T_{1/2}}.$$

(c) A uranium-235 nucleus absorbs a neutron and undergoes fission: $$^{235}_{92}\text{U} + n \rightarrow ^{144}_{56}\text{Ba} + ^{89}_{36}\text{Kr} + 3n$$ Calculate the total energy released, given: $$m(^{235}_{92}\text{U}) = 235.05 \, \text{u}, \, m(n) = 1.01 \, \text{u},$$ $$m(^{144}_{56}\text{Ba}) = 143.92 \, \text{u}, \, m(^{89}_{36}\text{Kr}) = 88.94 \, \text{u}.$$

(d) Calculate the critical mass for a spherical uranium-235 assembly. Assume the average neutron mean free path is $l = 3.30 \, \text{cm}$, and the density of uranium-235 is $\rho = 19 \, \text{g/cm}^3$. Use the formula: $$M_c = \frac{4}{3} \pi R_c^3 \rho, \, R_c = 2l.$$

(e) A nuclear reactor produces 700 MW of thermal power. Calculate the mass of uranium-235 consumed per day if each fission releases $E_f = 200 \, \text{MeV}$. Use: $$1 \, \text{MeV} = 1.602 \times 10^{-13} \, \text{J}, \, \text{Avogadro's number} = 6.022 \times 10^{23}.$$

1. Rješenje jednadžbe $ \frac{1}{3}x + \frac{1}{2}x = 5 $.

2. Broj suprotan rješenju jednadžbe $ 3x - 4 - 2(x - 1) = -14 $.

4. Rješenje jednadžbe $ 2(3 + x) = 3(x - 6) $.

5. Rješenje jednadžbe $ \frac{a + 2}{6} - \frac{a - 4}{3} = -5 $.

6. Sedmerokratnik rješenja jednadžbe $ \frac{1}{5}x - \frac{x}{3} = -2 $.

1. $ -100 \cdot (-6) - 8 \cdot (-5) $

2. Broj tri puta manji od rješenja jednadžbe $ 2x - 1 = 83 $.

3. Rješenje jednadžbe $ \frac{2}{5}x - \frac{3}{4}x = -7 $.

4. Neposredni sljedbenik rješenja jednadžbe $ 2(2x - 3) - 3(x + 1) = 11 $.