Nuclear Energy

Why nuclear energy

• Conventional sources — coal, petroleum, natural gas, firewood — release energy through chemical reactions, take millions of years to form and are limited in supply.
• Alternatives already in use are hydroelectric power, solar energy, wind energy and nuclear energy, nuclear energy being the largest of these.
• Nuclear energy is the energy released when nuclei undergo a nuclear reaction, that is, when one nucleus or a pair of nuclei change their structure to produce new nuclei and generate energy in the process.
• Chemical reactions release a few eV per reaction; nuclear reactions release a few MeV. For the same weight of fuel, the nuclear energy released is about a million times larger. Against this, nuclear energy generation is complex, expensive and can be extremely harmful.
• The energy comes from the binding energy: the mass of a nucleus is smaller than the sum of the masses of its constituents. We never build nuclei from individual nucleons in practice, so the energy is obtained by two other processes — nuclear fission and nuclear fusion.

Nuclear fission

• Definition: the process in which a heavy nucleus breaks into two lighter nuclei with the release of energy.
• Reasoning from the binding-energy curve: middle-weight nuclei with $A$ between $50$ and $80$ have the highest $\dfrac{E_B}{A}$ and are the most stable, while nuclei of higher and lower mass number have smaller values. If a heavy nucleus with $A \approx 230$ breaks into two nuclei with $A$ between $50$ and $150$, the products have higher $\dfrac{E_B}{A}$, so their combined mass is smaller than that of the parent and the mass difference is released as energy.
• The fuel nucleus: ${}^{236}_{92}\text{U}$ has a half-life of $2.3\times10^{7}$ years and an activity of $6.5\times10^{-5}\ \text{Ci}$ per gram; most of it has already decayed and it is not found in nature. More than $99\%$ of natural uranium is ${}^{238}_{92}\text{U}$ and less than $1\%$ is ${}^{235}_{92}\text{U}$. The needed ${}^{236}_{92}\text{U}$ is made by bombarding ${}^{235}_{92}\text{U}$ with slow neutrons:

$${}^{235}_{92}\text{U} + n \longrightarrow {}^{236}_{92}\text{U}$$

• ${}^{236}_{92}\text{U}$ can fission in several ways, producing different pairs of daughter nuclei and different amounts of energy:

$${}^{236}_{92}\text{U} \longrightarrow {}^{137}_{53}\text{I} + {}^{97}_{39}\text{Y} + 2n$$

$${}^{236}_{92}\text{U} \longrightarrow {}^{140}_{56}\text{Ba} + {}^{94}_{36}\text{Kr} + 2n$$

$${}^{236}_{92}\text{U} \longrightarrow {}^{133}_{51}\text{Sb} + {}^{99}_{41}\text{Nb} + 4n$$

• Energy released:

$$Q = \left(\Delta M\right)c^2,\qquad \Delta M = M_{\text{parent}} - M_{\text{products}}$$

• For the barium-krypton channel, $\Delta M = m_U - \left[m_{Ba} + m_{Kr} + 2m_n\right]$. Multiply a mass defect in u by $931.5$ to get MeV.
• Worked check: for ${}^{236}_{92}\text{U} \to {}^{137}_{53}\text{I} + {}^{97}_{39}\text{Y} + 2n$ with masses $236.04557$, $136.91787$, $96.91827$ and $m_n = 1.00865\ \text{u}$, $\Delta M = 0.19011\ \text{u}$ and $Q = 177.09\ \text{MeV}$.
• Some of the daughter nuclei produced are themselves unstable and decay further to more stable nuclei. The energy produced in fission appears as kinetic energy of the products, that is, as heat, which can be collected and converted to other forms.

Chain reaction

• Neutrons are produced in every fission — some reactions give 2, others 3 or 4, the average being $2.7$ neutrons per reaction.
• These neutrons are in turn absorbed by other ${}^{235}_{92}\text{U}$ nuclei to produce ${}^{236}_{92}\text{U}$, which fission and produce a further $2.7$ neutrons each.
• The effect cascades: the number of neutrons, and therefore the number of fissions, increases very quickly. This is a chain reaction.
• Uncontrolled, it leads to a very rapid increase in the energy produced and hence to an explosion.

Uranium nuclear reactor

• A nuclear reactor is an apparatus in which nuclear fission is carried out in a controlled manner to produce energy in the form of heat, which is then converted to electricity.
• Principle: ${}^{235}_{92}\text{U}$ is used as the fuel; it is bombarded with slow neutrons to produce ${}^{236}_{92}\text{U}$, which undergoes fission. Methods are employed to stop the chain reaction from running away, so that fission and energy generation occur in a controlled fashion. The heat generated is carried away and converted to electricity using turbines.
• More than 15 countries use nuclear power. India has 22 nuclear reactors, the largest at Kudankulam in Tamil Nadu; the maximum nuclear power is generated by the USA.

Nuclear fusion

• Definition: the process in which two lighter nuclei fuse together to form a heavier nucleus, accompanied by a release of nuclear energy.
• Reasoning: light nuclei ($A < 40$) have lower $\dfrac{E_B}{A}$ than heavier ones. If two light nuclei come within about $1\ \text{fm}$ of each other they can react and form a heavier nucleus with higher $\dfrac{E_B}{A}$; the product is therefore lighter than the reactants and energy of the order of MeV is released.
• Conditions required. The two nuclei must come within about $1\ \text{fm}$ so that the nuclear force can act. Two whole atoms cannot get that close because of the electrostatic repulsion between their electrons, so the atoms must first be stripped of their electrons — done by heating the gas so that the electrons get energies larger than the ionisation energies. Even then, the bare nuclei repel each other electrostatically, so the gas must be heated to a very high temperature to give the nuclei enough kinetic energy to overcome that repulsion. Since the nuclear charge increases with $Z$, the temperature required increases with $Z$.
• In the Sun, at a central temperature of about $10^{7}\ \text{K}$, four hydrogen nuclei fuse to form a helium nucleus. Because it is extremely unlikely that four protons meet at once, the fusion proceeds in several steps; the effective reaction is

$$4p \longrightarrow \alpha + 2e^+ + \text{neutrinos} + 26.7\ \text{MeV}$$

• Energy value, mass defect being converted into energy:

$$Q = \left[4m_p - \left(m_\alpha + 2m_e\right)\right]c^2$$

• Worked check: with $m_p = 1.00728\ \text{u}$, $m_\alpha = 4.001506\ \text{u}$ and $m_e = 0.00055\ \text{u}$, $\Delta M = 0.026514\ \text{u}$ and $Q = 24.70\ \text{MeV}$, ignoring the energy taken by the neutrinos.
• These reactions have been going on in the Sun for the past $4.5$ billion years and are expected to continue for a similar period.

Fusion in the universe

• Nuclear fusion takes place all the time in the universe, mostly at the centres of stars where temperatures are high enough; it is the source of energy of stars. Most elements heavier than boron and up to iron were produced by fusion inside stars.
• Fusion inside stars can only take place between nuclei with mass number smaller than that of iron, $56$, because iron has the highest $\dfrac{E_B}{A}$. Fusing an iron nucleus into a heavier one would give a product with smaller $\dfrac{E_B}{A}$ and hence larger mass than the reactants, so energy would have to be supplied. Elements heavier than iron are produced by other nuclear reactions during stellar explosions.
• The light elements — deuterium, helium, lithium, beryllium and boron — were not created inside stars but are believed to have been created within the first $200$ seconds in the life of the universe, after which the temperature fell too low for nuclear reactions.

Fission compared with fusion

Harmful effects

• If an uncontrolled chain reaction is set up in a nuclear fuel, an extremely large amount of energy is generated in a very short time. This is used to make atom bombs or nuclear devices, which employ either fission alone or both fission and fusion.
• The first such devices were made towards the end of the Second World War; several countries including India have since made and tested them. Two atom bombs completely destroyed the cities of Hiroshima and Nagasaki in Japan in early August 1945.

Fuel-mass numericals

$$\text{Number of fissions in time }t = \dfrac{P\,t}{E_{\text{one fission}}},\qquad m = \dfrac{P\,t}{E_{\text{one fission}}}\times\dfrac{A}{N_A}$$

• $P$ is the thermal power in watt, $t$ the time in second, and $E_{\text{one fission}}$ must be converted from MeV to joule by multiplying by $1.6\times10^{-13}$. The mass comes out in gram when $A$ is the mass number of the fuel and $N_A = 6.022\times10^{23}$.