**Nuclear energy**comes either from spontaneous nuclei conversions or induced nuclei conversions. Among these conversions (nuclear reactions) belong for example nuclear fission, nuclear decay and nuclear fusion. Conversions are associated

**with mass and energy changes**. One of the striking results of

**Einstein’s theory of relativity**is that

**mass and energy are equivalent and convertible**, one into the other. Equivalence of the mass and energy is described by Einstein’s famous formula:

, where M is the small amount of mass and C is the speed of light.

What that means? If the nuclear energy is generated (splitting atoms, nuclear fussion), a small amount of mass (saved in the **nuclear binding energy**) transforms into the pure energy (such as kinetic energy, thermal energy, or radiant energy).

**Example:**

The energy equivalent of one gram (1/1000 of a kilogram) of mass is equivalent to:

**89.9 terajoules****25.0 million kilowatt-hours (≈ 25 GW·h)****21.5 billion kilocalories (≈ 21 Tcal)****85.2 billion BTUs**

or to the energy released by combustion of the following:

**21.5 kilotons of TNT-equivalent energy (≈ 21 kt)****568,000 US gallons of automotive gasoline**

Any time energy is generated, the process can be evaluated from an **E = mc ^{2}** perspective.

## Nuclear Binding Energy – Mass Defect

**absoluteness**. One of the striking results of

**Einstein’s theory of relativity**is that

**mass and energy are equivalent and convertible**one into the other.

**Equivalence**of the mass and energy is described by Einstein’s famous formula

**E = mc**. In words,

^{2}**energy**equals

**mass**multiplied by the

**speed of light squared**. Because the speed of light is a very large number, the formula implies that any small amount of matter contains a very large amount of energy. The mass of an object was seen to be equivalent to energy, to be interconvertible with energy, and to increase significantly at exceedingly high speeds near that of light. The

**total energy**of an object was understood to comprise its

**rest mass**as well as its

**increase of mass**caused by

**increase in kinetic energy**.

**In special theory of relativity** certain types of **matter may be created or destroyed**, but in all of these processes, the mass and energy associated with such matter **remains unchanged in quantity**. It was found the **rest mass an atomic nucleus is measurably smaller than the sum of the rest masses of its constituent protons, neutrons and electrons**. Mass was no longer considered unchangeable in the closed system. The difference is a measure of the nuclear binding energy which holds the nucleus together. According to the Einstein relationship (**E = mc ^{2}**) this binding energy is proportional to this mass difference and it is known as the

**mass defect**.

**E=mc ^{2}** represents the new conservation principle – the conservation of mass-energy.

If the splitting releases energy and the fusion releases the energy, so where is the breaking point? For understanding this issue it is better to relate the binding energy to one nucleon, to obtain **nuclear binding curve**. The binding energy per one nucleon is not linear. There is a peak in the binding energy curve in the region of stability near** iron** and this means that either the breakup of heavier nuclei than iron or the combining of lighter nuclei than iron will yield energy.

The reason the trend reverses after iron peak is the growing positive charge of the nuclei. The electric force has greater range than strong nuclear force. While the strong nuclear force binds only close neighbors the electric force of each proton repels the other protons.

**mass defect**of a

**nucleus if the actual mass of**

^{63}Cu^{63}Cu in its

**nuclear ground state is 62.91367 u.**

^{63}Cu nucleus has 29 protons and also has (63 – 29) 34 neutrons.

The mass of a proton is **1.00728 u** and a neutron is **1.00867 u**.

The combined mass is: 29 protons x (1.00728 u/proton) + 34 neutrons x (1.00867 u/neutron) = **63.50590 u**

**The mass defect** is Δm = 63.50590 u – 62.91367 u = **0.59223 u**

**Convert the mass defect into energy (nuclear binding energy).**

(0.59223 u/nucleus) x (1.6606 x 10^{-27} kg/u) = **9.8346 x 10 ^{-28} kg/nucleus**

ΔE = (9.8346 x 10^{-28} kg/nucleus) x (2.9979 x 10^{8} m/s)^{2} = **8.8387 x 10 ^{-11} J/nucleus**

The energy calculated in the previous example is the **nuclear binding energy**. However, the nuclear binding energy may be expressed as kJ/mol (for better understanding).

Calculate the nuclear binding energy of 1 mole of ^{63}Cu:

(8.8387 x 10^{-11} J/nucleus) x (1 kJ/1000 J) x (6.022 x 10^{23} nuclei/mol) = **5.3227 x 10 ^{10} kJ/mol of nuclei.**

One mole of ^{63}Cu (~63 grams) is bound by the nuclear binding energy (5.3227 x 10^{10} kJ/mol) which is equivalent to:

**14.8 million kilowatt-hours (≈ 15 GW·h)****336,100 US gallons of automotive gasoline**

**mass defect**of the

**3000MW**reactor core after one year of operation.

_{th}It is known the average recoverable energy per fission is about **200 MeV**, being the total energy minus the energy of the energy of antineutrinos that are radiated away.

The **reaction rate** per entire **3000MW _{th}** reactor core is about

**9.33×10**.

^{19}fissions / second**The overall energy release** in the units of joules is:

200×10^{6} (eV) x 1.602×10^{-19} (J/eV) x 9.33×10^{19} (s^{-1}) x 31.5×10^{6} (seconds in year) = **9.4×10 ^{16} J/year**

The mass defect is calculated as:

Δm = ΔE/c^{2}

**Δm** = 9.4×10^{16} / (2.9979 x 10^{8})^{2} = **1.046 kg**

That means in a typical **3000MWth** reactor core about 1 kilogram of matter is **converted** into pure energy.

Note that, a typical annual uranium load for a **3000MWth **reactor core is about **20 tonnes** of **enriched uranium **(i.e. about **22.7 tonnes of UO _{2}**). Entire reactor core may contain about 80 tonnes of enriched uranium.

### Mass defect directly from E=mc^{2}

The mass defect can be calculated directly from the Einstein relationship (**E = mc ^{2}**) as:

Δm = ΔE/c^{2}

Δm = 3000×10^{6} (W = J/s) x 31.5×10^{6} (seconds in year) / (2.9979 x 10^{8})^{2 }= **1.051 kg**

## Nuclear Energy and Electricity Production

Today we use the nuclear energy to generate useful heat and electricity. This electricity is generated in nuclear power plants. The heat source in the nuclear power plant is a nuclear reactor. As is typical in all conventional thermal power stations the heat is used to generate steam which drives a steam turbine connected to a generator which produces electricity. In 2011 nuclear power provided 10% of the world’s electricity. In 2007, the IAEA reported there were 439 nuclear power reactors in operation in the world, operating in 31 countries. They produce base-load electricity 24/7 without emitting any pollutants into the atmosphere (this includes CO2).

## Nuclear Energy Consumption – Summary

**Consumption of a 3000MWth (~1000MWe) reactor (12-months fuel cycle)**

*It is an illustrative example, following data do not correspond to any reactor design.*

- Typical reactor may contain about
**165 tonnes of fuel**(including structural material) - Typical reactor may contain about
**100 tonnes of enriched uranium**(i.e. about 113 tonnes of uranium dioxide). - This fuel is loaded within, for example,
**157 fuel assemblies**composed of over**45,000 fuel rods.** - A common fuel assembly contain energy for approximately
**4 years of operation at full power**. - Therefore about
**one quarter of the core**is yearly removed to**spent fuel pool**(i.e. about 40 fuel assemblies), while the remainder is rearranged to a location in the core better suited to its remaining level of enrichment (see Power Distribution). - The removed fuel (
**spent nuclear fuel**) still contains about**96% of reusable material**(it must be removed due to decreasing**k**of an assembly)._{inf}

**Annual natural uranium consumption**of this reactor is about**250 tonnes of natural uranium**(to produce of about 25 tonnes of enriched uranium).

**Annual enriched uranium consumption**of this reactor is about**25 tonnes of enriched uranium**.

**Annual fissile material consumption**of this reactor is about**1 005 kg**.

**Annual matter consumption**of this reactor is about**1.051 kg**.

- But it corresponds to
**about 3 200 000 tons**of**coal burned in**per year.**coal-fired power plant**

See also: Fuel Consumption