# Applying Conservation of Mass & Energy to a Natural Phenomenon

Instructor: Matthew Bergstresser

Matthew has a Master of Arts degree in Physics Education

Einstein derived an equation that relates mass to energy. In this lesson, we will investigate what this equation means and we will go through a few examples using it.

## Energy and Mass

Dr. Albert Einstein derived one of the most famous equations known. It is a very simple equation, but its meaning is mind blowing. Let's dig deeper into what this equation is, what it means, and work a few examples using it.

## Missing Mass

One of the fundamental laws of the universe is the conservation of mass and energy. Basically, mass can't be created, it can't be destroyed and the same goes with energy. Einstein equated mass and energy with the equation:

E = mc2

where

• E is energy in joules (J).
• m is mass in kilograms (kg).
• c is the speed of light, which is 3.0 x 108 m/s.

Notice in the equation that the speed of light is squared! Think about that - 300 million squared! This is an enormous number, so it won't take much mass to equal a gigantic amount of energy. In other words, mass is an extremely concentrated form of energy.

The nucleus of the atom is composed of positively charged protons and neutral particles called neutrons. Positive charges repel other positive charges and the proximity of the protons in the nucleus should cause an extreme repulsion, but that doesn't happen. Something strange is going on. It turns out that the mass of the protons and neutrons have different masses inside of the nucleus compared to outside the nucleus. The difference between the mass of the individual nucleons (protons and neutrons) outside of the nucleus compared to inside the nucleus is called the mass defect. The missing mass is turned into the energy that keeps the nucleus together. It is called the binding energy. Table 1 includes the masses of the nucleons in the nucleus.

Table 1: Masses of Nucleons
Particle Mass in AMU (atomic mass unit = 1.66056 x 10-27 kg)
Proton 1.007825
Neutron 1.008665

To determine the binding energy, we take the following steps:

1. Sum the mass of the reactants (left side of the reaction), and subtract from it the mass of products (right side of the reaction).
2. Plug in the result from step 1 into E = mc2.

Let's calculate the binding energy of hydrogen-2. The number after the element is the mass number, which equals the number of protons and neutrons. Since hydrogen's atomic number is 1, it has 1 proton, and 1 neutron.

Step 1: Determining the mass defect.

Reactants:

1 proton(1.007825 AMU/proton) + 1 neutron(1.008665 AMU/neutron) = 2.01649 AMU

Product:

From a chart of elemental mass, the given mass of H-2 is 2.014102 AMU.

The change in mass is:

2.01649 AMU - 2.014102 AMU = 0.002388 AMU

Converting this mass defect into kilograms, we get:

0.002388 AMU (1.66056 x 10-27 kg / AMU) = 3.96542 x 10-30 kg

Step 2: Using Einstein's energy-mass equation using the mass defect.

E = mc2

E = (3.96542 x 10-30 kg)(3.0 x 108 m/s)2

E = 3.57 x 10-13 joules.

This may not seem like a large amount of energy, but just compare the magnitude of the missing mass.

(10-30) to the magnitude of the energy it equals (10-13). The energy quantity is 1017 times larger!

Let's see how this equation is applied in terms of a natural nuclear reaction.

## Fusion

A fusion reaction combines relatively light elements into heavier elements. This is a natural process that occurs naturally in the cores of stars. Let's look at the transformation of mass into energy in a fusion reaction.

The most basic fusion reaction involves two hydrogen-2 atoms combining to form helium-4.

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