Atomic nucleus

1. What properties does the atomic nucleus have?

Solution:
The atomic nucleus forms the central part of the atom. It accounts for almost all of its mass. The atomic nucleus contains protons and neutrons—nucleons.

Atomic constant:
$m_u, A_r = 1, m = 1.6605 \cdot 10^{-27} \, \text{kg} = 931.5 \, \text{MeV} \cdot c^{-2}$

Proton: $p, A_r = 1.0072, m = 1.673 \cdot 10^{-27} \, \text{kg} = 938.275 \, \text{MeV} \cdot c^{-2}$
Neutron: $n, A_r = 1.0086, m = 1.673 \cdot 10^{-27} \, \text{kg} = 939.565 \, \text{MeV} \cdot c^{-2}$

$^A_Z X$ , $Z =$ proton number (number of protons in the nucleus, electrons in the shell, the order of the element in the Mendeleev table)
$N =$ neutron number (number of neutrons in the nucleus)
$A =$ nucleon number $A = N + Z$ (number of nucleons – protons and neutrons in the nucleus)

Isotopes are nuclides of the same element (same $Z$ ) that differ in neutron number $N$ (and thus also in number $A$ ).

The mass of the nucleus $m_j$ is always smaller than the mass of $Z$ protons and $N$ neutrons.

$B_j =$ mass defect of the nucleus:

m_j < Z \cdot m_p + N \cdot m_n

m_j + B_j = Z \cdot m_p + N \cdot m_n

B_j = (Z \cdot m_p + N \cdot m_n) - m_j

m_j = A_r \cdot m_u

Binding energy of the nucleus:

E_j = B_j c^2

Binding energy per nucleon:

\varepsilon_j = \frac{E_j}{A}

New unit of mass:

1 \, \text{MeV} \cdot c^{-2} = 1.7825 \cdot 10^{-30} \, \text{kg}

1 \, \text{kg} = 0.561 \cdot 10^{30} \, \text{MeV} \cdot c^{-2}

2.Lead has four isotopes. Determine the number of protons and neutrons in the nucleus.

Solution:

3. Calculate:

a.) the rest mass of the nitrogen nucleus, given A_r(N) = 14,0067
b.) the relative mass of the chlorine nucleus, given m(Cl) = 5,885·10^–26 kg, 1 MeV·c^–2 = 1,7825·10^–30 kg, m_u = 1,6605·10^–27 kg

Solution:

The rest mass of the nitrogen nucleus is m = 13 050 MeV·c^-2.
The relative mass of the chlorine nucleus is A_r = 35.44.

4.Determine the mass defect, the binding energy, and the energy per nucleon for a radium nucleus. Express the energy in MeV.

Solution:

Analysis:

The mass defect is B_j = 1708.9 MeV·c^-2, the binding energy is E_j = 1708.9 MeV, the energy per nucleon is ε_j = 7.56 MeV.

5.Determine the percent composition of chlorine with a relative atomic mass of 35.5, which is a mixture of isotopes

Solution:

Analysis:

Chlorine contains 75% of the first isotope and 25% of the second isotope.

6.In the atomic bomb dropped by the Americans on Hiroshima, the charge contained 44.5 kg of uranium. What energy was released?

Solution:

7.How long would it take a power plant, whose two blocks have a power output of P = 2,440 MW, to produce the energy from problem no. 6 (E = 3.24·10¹⁶ J)?

Solution:

Analysis:

E = 3.24·10¹⁶ J, P = 2,440 MW = 880·10⁶ J·s^-1, t = ?

The power plant would produce this energy in 1.167 years.

8.A nucleus with mass number 100 and binding energy ε_j1 = 7.4 MeV splits into two nuclei with binding energy per nucleon ε_j2 = 8.2 MeV. What energy is released in the reaction?

Solution:

Analysis:

A = 100, ε_j1 = 7.4 MeV, ε_j2 = 8.2 MeV, E_j = ?

The reaction releases 80 MeV of energy.

9. Determine the energy obtainable by fission of 1 kg of uranium if fission of one uranium nucleus releases 200 MeV of energy. What mass of hard coal with a heating value of 3·10⁷ J·kg would be needed to obtain the same energy?

Solution:

The mass of hard coal would need to be M = 2,733 tons.

10.A block of a nuclear power plant that converts nuclear energy into electrical energy with 40% efficiency has an electrical output of 600 MW. Determine the mass of uranium consumed in 24 hours if fission of a single nucleus releases energy of 200 MeV.

Solution:

Analysis:

The mass of uranium is 1.58 kg.

Physics

Atomic nucleus

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