Subfield: Modern Physics

- Proton mass, $m_{p}=1.67 \cdot 10^{-27} \mathrm{~kg}$
- Neutron mass, $m_{n}=1.67 \cdot 10^{-27} \mathrm{~kg}$
- Electron mass, $m_{e}=9.11 \cdot 10^{-31} \mathrm{~kg}$
- Avogadro's constant, $N_{0}=6.02 \cdot 10^{23} \mathrm{~mol}^{-1}$
- Universal gas constant, $R=8.31 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K})$
- Boltzmann's constant, $k_{B}=1.38 \cdot 10^{-23} \mathrm{~J} / \mathrm{K}$
- Electron charge magnitude, $e=1.60 \cdot 10^{-19} \mathrm{C}$
- 1 electron volt, $1 \mathrm{eV}=1.60 \cdot 10^{-19} \mathrm{~J}$
- Speed of light, $c=3.00 \cdot 10^{8} \mathrm{~m} / \mathrm{s}$
- Universal Gravitational constant,

$$
G=6.67 \cdot 10^{-11}\left(\mathrm{~N} \cdot \mathrm{m}^{2}\right) / \mathrm{kg}^{2}
$$

- Solar Mass

$$
M_{\odot}=1.988 \cdot 10^{30} \mathrm{~kg}
$$

- Acceleration due to gravity, $g=9.8 \mathrm{~m} / \mathrm{s}^{2}$
- 1 unified atomic mass unit,

$$
1 \mathrm{u}=1.66 \cdot 10^{-27} \mathrm{~kg}=931 \mathrm{MeV} / \mathrm{c}^{2}
$$

- Planck's constant,

$$
h=6.63 \cdot 10^{-34} \mathrm{~J} \cdot \mathrm{s}=4.41 \cdot 10^{-15} \mathrm{eV} \cdot \mathrm{s}
$$

- Permittivity of free space,

$$
\epsilon_{0}=8.85 \cdot 10^{-12} \mathrm{C}^{2} /\left(\mathrm{N} \cdot \mathrm{m}^{2}\right)
$$

- Coulomb's law constant,

$$
k=\frac{1}{4 \pi \epsilon_{0}}=8.99 \cdot 10^{9}\left(\mathrm{~N} \cdot \mathrm{m}^{2}\right) / \mathrm{C}^{2}
$$

- Permeability of free space,

$$
\mu_{0}=4 \pi \cdot 10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A}
$$

- Magnetic constant,

$$
\frac{\mu_{0}}{4 \pi}=1 \cdot 10^{-7}(\mathrm{~T} \cdot \mathrm{m}) / \mathrm{A}
$$

- 1 atmospheric pressure,

$$
1 \mathrm{~atm}=1.01 \cdot 10^{5} \mathrm{~N} / \mathrm{m}^{2}=1.01 \cdot 10^{5} \mathrm{~Pa}
$$

- Wien's displacement constant, $b=2.9$. $10^{-3} \mathrm{~m} \cdot \mathrm{K}$
- Stefan-Boltzmann constant,

$$
\sigma=5.67 \cdot 10^{-8} \mathrm{~W} / \mathrm{m}^{2} / \mathrm{K}^{4}
$$

Question: In the cosmic galaxy, the Sun is a mainsequence star, generating its energy mainly by nuclear fusion of hydrogen nuclei into helium. In its core, the Sun fuses hydrogen to produce deuterium $(2 \mathrm{H})$ and tritium $(3 \mathrm{H})$, then makes about 600 million metric tons of helium (4He) per second. Of course, there are also some relatively smaller portions of fission reactions in the Sun's core, e.g. a nuclear fission reaction with Uranium-235 (235U). The Fusion reaction:

$$
{ }^{2} \mathrm{H}+{ }^{3} \mathrm{H} \rightarrow{ }^{4} \mathrm{He}+n+\text { Released Energy }
$$

The Fission reaction:

$$
{ }^{235} U+n+(\text { Initial Energy }) \rightarrow{ }^{144} \mathrm{Ba}+{ }^{90} \mathrm{Kr}+2 n+\text { Released Energy }
$$

Isotope Mass (at rest) 

| Isotope Names | Mass (at rest) $(\mathrm{u})$ |
| :--- | :--- |
| Deuterium $\left({ }^{2} \mathrm{H}\right)$ | 2.0141 |
| Tritium $\left({ }^{3} \mathrm{H}\right)$ | 3.0160 |
| Helium $\left({ }^{4} \mathrm{He}\right)$ | 4.0026 |
| Neutron $(\mathrm{n})$ | 1.0087 |
| Uranium-235 $\left({ }^{235} \mathrm{U}\right)$ | 235.1180 |
| Barium-144 $\left({ }^{144} \mathrm{Ba}\right)$ | 143.8812 |
| Krypton-90 $\left({ }^{90} \mathrm{Kr}\right)$ | 89.9471 |

Calculate the kinetic energy (in $\mathrm{MeV}$ ) released by the products in one fusion reaction.

Solution:
Let the kinetic energy released be

$$
K E_{\text {released }}=-\Delta m c^{2}
$$

Let the mass of helium be $m_{h}$, deuterium be $m_{d}$, tritium $m_{t}$, and mass of neutron $m_{n}$ Therefore.

$$
-\Delta m=m_{d}+m_{t}-m_{h}-m_{n}=3.0160+2.0141-4.0026-1.0087=0.0188 \mathrm{u}
$$

which gives

$$
K E_{\text {released }}=(0.0188 \mathrm{u}) \cdot\left(931.494 \frac{\mathrm{MeV}}{\mathrm{u}}\right)=17.51 \mathrm{MeV}
$$

Final answer: 17.51 MeV