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Fission and Fusion

When you learned how to calculate the average atomic mass of an element from the exact mass and relative abundance of its isotopes you may have noticed something interesting. The mass number is not the exact mass of an isotope. For example, the exact mass of ^{16}_{ 8}O
is not exactly 16 *u* but rather 15.994915 *u*.The mass of a nucleus is slightly less than you might expect from the total number of protons and neutrons. Why should this be so?

Natural systems follow a consistent pattern in that they always seek to find the lowest possible energy. Atomic nuclei are no exception. As it happens, the nucleus of an atom is thermodynamically more stable than the protons and neutrons that make it up. This can be shown by a relatively simple calculation using the hypothetical situation of making a ^{16}_{ 8}O
nucleus from 8 hydrogen atoms (including electrons) and 8 neutrons.

Exact Masses | |

^{1}_{ 1}H | 1.00782 u1 p ^{+} & 1 e^{-} |

n^{0} | 1.00867 u |

e^{-} | 5.48580 × 10^{-4} u |

8n^{0}+ 8^{1}_{ 1}H = 8(1.00867) + 8(1.00782) = 16.13192uActual Mass of^{16}O = 15.99492uDifference in mass: 16.13192 - 15.99492 = 0.13700uSo an^{16}O nucleus loses 0.13700 g of mass per mole of^{16}O formed. We ignore the electrons because there are as many electrons in 8 H atoms as in one O atom

The mass difference calculated above is called the **mass defect**. Where does this mass go? It is released as energy when the nucleus forms. The process of forming a nucleus is exothermic: the product is more stable than the starting materials. In other words, the formation of an atomic nucleus releases energy. Energy is in fact a form of matter, as Albert Einstein discovered in the development of his theory of relativity. You have all seen the famous equation:

In this equation a statement about the equivalence of matter and energy is made. It says that energy changes are accompanied by a change in mass equal to the amount of energy divided by c^{2}. In ordinary chemical reactions the amount of energy is so small that the change in mass is not detectable. The c in the equation is a universal constant of nature: the speed of light, 3.00 × 10^{8} m/s. The amount of energy represented by the mass defect is called the binding energy of a nucleus and is the amount of energy that would be required to break the nucleus into its component parts. Here is how to caculate the amount of energy:

E = mc^{2}E = (0.13700 g/mol· 1 kg/1000 g)(3.00 × 10^{8}m/s)^{2}= 1.23 × 10^{13}J/mol

In such calculations grams must be converted to kilograms since 1 joule of energy (1 J) is defined as

1 kg· m^{2}/s^{2}. A joule is a scientific unit of energy equal to 4.184 × 10^{-3} dietary Calories. It is related to the watt (a measure of electrical power) because 1 watt of power uses 1 J per second. From these considerations and from the above calculation you can see that the binding energy is truly enormous! Perhaps you now understand why nuclear energy is so interesting: nearly unlimited power at the center of every atom.

**Fission** is the name for the process in which heavy nuclei (such as^{235}_{92}U) split into two nuclei with smaller mass numbers. **Fusion** is the combination of two light nuclei to form a heavier, more stable nucleus. Both of these processes are highly exothermic and involve the release of energy more than a million times larger than the energy released in chemical reactions.

Fission was discovered in the 1930s. Neutron bombardment of uranium-235 atoms resulted in the formation of smaller nuclides:

The amount of energy released per mole of uranium-235 is 2.1 × 10^{13} J. When methane (natural gas) is burned it releases only 8.0 × 10^{5} J/mol. This is a factor of about 26 million times less energy.

Notice that three additional neutrons are produced in the example fission reaction given for uranium-235 (there are actually at least 200 different possible isotopes that can result from such a fission). These neutrons make a **nuclear chain reaction** possible. In a nuclear chain reaction one fission event causes one or more further fission events. When less than one neutron, on average, causes another fission event then the reaction is called **sub-critical**. If exactly one neutron from each reaction causes another fission event then the process is self-sustaining and is called **critical**. Finally, the situation becomes **supercritical** if more than one neutron from each fission event causes another fission event. This results in a violent explosion, as is well known: a bomb made using a supercritical mass of uranium-235 was dropped on Hiroshima, Japan in 1945. Such bombs are made by arranging for subcritical masses of uranium-235 to be brought together suddenly to form a supercritical mass. Peaceful use of nuclear chain reactions is made in nuclear power generators in which controlled fission chain reactions are used to produce heat, boil water and run steam power generators.

Fusion is the source of the power of the stars. Our Sun is made of hydrogen (73% by mass), helium (26%) and other elements (1%). In the core of the Sun hydrogen nuclei fuse together to form helium nuclei. In the process they release enormous amounts of energy. Seven hundred million tons of hydrogen is fused every second in the core of the Sun and nearly 5 million tons of this matter is released as energy. (Use E = mc^{2} to figure out how much energy this is; one metric ton is 1,000 kg). The core of the Sun is 16,000,000 degrees Celsius. This high temperature makes earthly power sources based on nuclear fusion problematic at best.

- The Sun radiates 3.9 × 10
^{23}J into space every second. What is the rate at which mass is lost from the Sun? - The earth receives 1.8 × 10
^{17}J/s of solar energy. What mass of solar material is converted to energy in one day to provide this energy to earth (24 hr)? What mass of coal would have to be burned to give the same amount of energy? (One gram of coal gives 3.2 × 10^{4}J). - The exact mass of potassium-39 is 38.963707
*u*. Find the mass defect in atomic mass units (*u*) for^{39}K. - Given that the exact mass of nitrogen-15 is 15.000108 g/mol, find the mass defect in g/mol for this isotope.
- The most stable nucleus in terms of binding energy is
^{56}Fe. If the exact mass of^{56}Fe is 55.9349*u*calculate the binding energy of^{56}Fe in J/atom. (1 g = 6.02 × 10^{23}*u*) - Calculate the binding energy in J/mol for carbon-12 (12.00000 g/mol) and uranium-235 (235.043922 g/mol).

- The mass defect for a lithium-6 nucleus is 0.03434 g/mol. Calculate the exact atomic mass of
^{6}Li. - The binding energy for a single atom of magnesium-27 is 3.5802 × 10
^{-11}J. Calculate the atomic mass

of^{27}Mg - The easiest fusion reaction to initiate is

^{2}_{ 1}H +^{3}_{ 1}H →^{4}_{ 2}He +^{1}_{ 0}n^{0}

Calculate the energy released per^{4}He nucleus and per mole of^{4}He nuclei. The atomic masses are:^{2}H = 2.01410,^{3}H = 3.01605, and^{4}He = 4.00260. The solution of this problem is based on the difference in mass between the products and the reactants. The mass that is lost (the products will weigh less) is converted into energy. - One possible fission reaction resulting from the bombardment of uranium-235 with a neutron is:

^{1}_{ 0}n^{0}+^{235}_{92}U →^{140}_{54}Xe +^{94}_{38}Sr + 2^{1}_{ 0}n^{0}

Using the following exact masses, calculate the total amount of energy released in this reaction based on the difference in mass between the products and the reactants. The mass that is lost (the products will weigh less) is converted into energy.

^{235}U = 235.04392,^{140}Xe = 139.9216,^{94}Sr = 93.91537. - What mass of uranium-235 is required to produce the same amount of energy as a metric ton (1,000 kg) of coal? Assume that the controlled fission of uranium provides the same amount of energy per mole as found in the previous problem. One ton of coal provides about 3.2 × 10
^{10}J.