Scientific Understanding

**What is the nuclear binding energy?**

Where does the energy of fusion and fission equations come from? Well, so far we have been treating the masses in nuclear equations as whole numbers. However, if you become super accurate at measuring the masses of different isotopes before and after a nuclear reaction, you will start to notice something strange. If you take two light elements and smash them together to make two heavy elements, there is a tiny bit of difference between the light elements and heavy elements. The light elements total mass will have slightly more mass than the single heavy element. This is what books and teachers will call the mass defect. Where did the mass go? The mass turned into energy and that energy was used to help bind together the new heavier nucleus. This is energy is called the nuclear binding energy. The energy was predicted Albert Einstein’s famous equation E = mc^{2} before it was ever demonstrated to be true. The energy can also some times be released to produce the explosions we see in fusion and fission reactions. Einstein’s equation was truly revolutionary but a lot of people have trouble seeing why. Since it is not something you need to know for any chemistry test, I do not include it in this section but you can click on this link to learn more if you wish. What you will need to know for this class is simply how to use and solve for Albert Einstein’s equation of E = mc^{2}. Let us take a look at it below for explanation.

E = | mc^{2} |

E represents energy usually measured in Joules. The m represents mass, which is usually measured in kilograms. Last, c represents the speed of light, which is a fixed number or what they called a constant at 3.0 * 10^{8} m/s. There are only two variables (E and m) because c will always be the same number. That means you can be given m and asked to solve for E or you could be given E and asked to solve for m. Some of the math in this equation goes back to the section on multiplying and dividing scientific notation. Review that section if you are having problems with the math.

**Examples**: Solve the equations below. Don’t forget that you can always use 3.0 * 10^{8} m/s for your speed of light.

If you convert 2.5kg of mass into pure energy, how much energy will you produce?

Answer: 2.25 * 10^{17} kg

How much mass is need to produce a small explosion of 400 kJ if you were able to able to turn all that mass into energy?

Answer: 4.44 * 10^{-12} kg

In a fusion reaction two atoms of hydrogen each with a mass of 3.34755 * 10^{-27} kg collide to make a helium of mass 6.69400 * 10^{-27} kg. What is the nuclear binding energy of a helium nucleus?

Answer: 9.9 * 10^{-14} J

**VIDEO Nuclear Binding Energy Demonstrated Example 1****:** If you convert 7.3kg of mass into pure energy, how much energy will you produce? Don’t forget that you can always use 3.0 * 10^{8} m/s for your speed of light.

What information are we given?

Answer:

m = 7.3 kg

c = 3.0 * 10^{8} m/s

What does the question ask for?

Answer: E = ?

How do we set up the problem?

Answer: Start with the equation

E = | mc^{2} |

What can we fill in for the equation?

Answer: The information we are given

E = | (7.3 kg) * (3.0 * 10^{8} m/s)^{2} |

Apply the square first

E = | (7.3 kg) * (9 * 10^{16} m/s) |

Multiply

E = | (6.57 * 10^{17} J) |

COMPLETE ANSWER: 6.57 * 10^{17} J

**VIDEO Nuclear Binding Energy Demonstrated Example 2****: **How much mass is need to produce a large explosion of 5000 kJ if you were able to able to turn all that mass into energy? Don’t forget that you can always use 3.0 * 10^{8} m/s for your speed of light.

What information are we given?

Answer:

E = 5000 kJ

c = 3.0 * 10^{8} m/s

What conversions are needed?

Answer:

E = 5000 kJ —> 5 * 10^{6} J

c = 3.0 * 10^{8} m/s

What does the question ask for?

Answer: m = ?

How do we set up the problem?

Answer: Start with the equation

E = | mc^{2} |

What can we fill in for the equation?

Answer: The information we are given

5 * 10^{6} J = |
m * (3.0 * 10^{8} m/s)^{2} |

Apply the square first

5 * 10^{6} J = |
m * (9 * 10^{16} m^{2}/s^{2}) |

Divide both sides by 9 * 10^{16} m/s

5 * 10^{6} J = |
m * (9 * 10^{16} m^{2}/s^{2}) |

9 * 10^{16} m^{2}/s^{2} |
9 * 10^{16} m^{2}/s^{2} |

Cross out 9 * 10^{16} m/s on the right

5 * 10^{6} J = |
m * (9 * 10^{16} m^{2}/s^{2}) |

9 * 10^{16} m^{2}/s^{2} |
9 * 10^{16} m^{2}/s^{2} |

Simplify

5 * 10^{6} J = |
m |

9 * 10^{16} m^{2}/s^{2} |

Divide the left side

5.55 * 10^{-11} kg = |
m |

COMPLETE ANSWER: 5.55 * 10^{-11} kg

**VIDEO Nuclear Binding Energy Demonstrated Example 3****: **In a fusion reaction two atoms of helium each with a mass of 6.69400 * 10^{-27} kg collide to make a Beryllium of mass 1.3375 * 10^{-26} kg. What is the nuclear binding energy of a helium nucleus? Don’t forget that you can always use 3.0 * 10^{8} m/s for your speed of light.

What information are we given?

Answer:

mass of one helium = 6.69400 * 10^{-27} kg

c = 3.0 * 10^{8} m/s

mass of beryllium = 1.3375 * 10^{-26} kg

What does the question ask for?

Answer: E = ?

Since there are two helium atoms in the reaction we need to find a total mass of the both the helium together.

6.69400 * 10^{-27} kg * 2 = 1.3388 * 10^{-26} kg

Since this is a mass defect problem that means the mass we calculated minus the mass of the beryllium is the mass that is turning into energy.

1.3388 * 10^{-26} kg – 1.3375 * 10^{-26} kg = 1.3 * 10^{-29} kg = m

Now we have our mass but we need to solve for our E. Start with the Equation.

E = | mc^{2} |

Start by filling in the equation

E = | (1.3 * 10^{-29} kg) * (3.0 * 10^{8} m/s)^{2} |

Apply the square

E = | (1.3 * 10^{-29} kg) * (9 * 10^{16} m^{2}/s^{2}) |

Multiply the right

E = | 1.17 * 10^{-12} J |

COMPLETE ANSWER: 1.17 * 10^{-12} J

**PRACTICE PROBLEMS**: Solve the nuclear binding energy examples below with. Don’t forget that you can always use 3.0 * 10^{8} m/s for your speed of light.

If you convert 1 g of mass into pure energy, how much energy will you produce?

Answer: 9 * 10^{13} J

How much mass is need to produce a heat a small cup of water at 1200 J if you were able to able to turn all that mass into energy?

Answer: 1.33 * 10^{-14} kg

In a fusion reaction two atoms of Beryllium each with a mass 1.3375 * 10^{-26} kg collide to make an Oxygen atom of mass 2.6742 * 10^{-26} kg. What is the nuclear binding energy of a helium nucleus?

Answer: 7.2 * 10^{-13} J