9.3 Nuclear Binding Energy and Mass Defect

Learning Objectives

After Chapter 9.3, you will be able to:

• Describe key concepts in nuclear binding energy, including strong nuclear force, mass defect, and binding energy
• Recall the four fundamental forces of nature
• Apply the equation E = mc²
  

Until this point, we’ve examined the relationships between electromagnetic radiation and matter—particularly electrons. Now, we’ll shift to the energy that is stored in the nucleus, which can be emitted under specific circumstances. While one would assume that the mass of the nucleus is simply the sum of the masses of all of the protons and neutrons within it, the actual mass of every nucleus (other than hydrogen) is slightly smaller than that. This difference is called the mass defect. Scientists had difficulty explaining why this mass defect occurred until Einstein characterized the equivalence of matter and energy, embodied by the equation

E = mc²

Equation 9.4

where E is energy, m is mass, and c is the speed of light. The mass defect is a result of matter that has been converted to energy. Because of the large exponent on the speed of light—which is squared in the equation—a very small amount of mass will yield a huge amount of energy. For example, the conversion of one gram of mass to energy will produce 89.9 terajoules (1 TJ = 10¹² joules) or 21.5 billion kilocalories.

When protons and neutrons (nucleons) come together to form the nucleus, they are attracted to each other by the strong nuclear force, which is strong enough to more than compensate for the repulsive electromagnetic force between the protons. Although the strong nuclear force is the strongest of the four fundamental forces, it only acts over extremely short distances, less than a few times the diameter of a proton or neutron. The nucleons have to get very close together in order for the strong nuclear force to hold them together. The bonded system is at a lower energy level than the unbonded constituents, and this difference in energy must be radiated away in the form of heat, light, or other electromagnetic radiation before the mass defect becomes apparent. This energy, called binding energy, allows the nucleons to bind together in the nucleus. Given the strength of the strong nuclear force, the amount of mass that is transformed into the dissipated energy will be a measurable fraction of the initial total mass. The binding energy per nucleon peaks at the element iron, which implies that iron contains the most stable nucleus. In general, intermediate-sized nuclei are more stable than very large or small nuclei.

The weak nuclear force also contributes to the stability of the nucleus, but is about one-millionth as strong as the strong nuclear force. The strong and weak nuclear forces constitute two of the four fundamental forces of nature. The other two are electrostatic forces and gravitation.

Example:

Measurements of the atomic mass of a neutron and a proton yield these results:

contains two protons and two neutrons, which should theoretically give a helium nucleus a mass of 2 × 1.00728 + 2 × 1.00867 = 4.03190 amu. However, the true mass of the helium nucleus is 4.00260 amu. What is the mass defect and binding energy of this nucleus? (Note: )

Solution:

The difference 4.03190 − 4.00260 = 0.02930 amu is the mass defect for the helium nucleus. This is the mass that contributed to the binding energy of the nucleus:

MCAT Concept Check 9.3

Before you move on, assess your understanding of the material with these questions.

1. Define the following terms:

   • Strong nuclear force:
     
     
   • Mass defect:
     
     
   • Binding energy:
     

2. What are the four fundamental forces of nature?

3. How does the mass defect relate to the binding energy?