Mass and energy are two concepts that are deeply related in the context of particle physics and modern physics. The mass-energy relation is expressed in a formula by the famous physicist Albert Einstein. It expresses that any change in the energy of an object at rest also produces a change in its mass.
The relation between mass and energy
The relation between mass and energy was studied by many scientists before Einstein. Isaac Newton speculated about matter and light being convertible to each other, and several attempts to relate matter to kinetic energy were made in the following centuries. J. J. Thompson and Oliver Heaviside observed changes in the mass of an object when it has an electrical charge, a phenomenon that has been described as electromagnetic mass.
But it was Einstein who proposed that when an object emits energy E in the form of electromagnetic radiation, it loses mass equal to the energy divided by the square of the velocity of light. This is expressed as:
\[m = \frac{E}{c^2}\]
Here, m is the mass lost in kg, E is the energy in Joules, and c is the velocity of light in a vacuum, which is equal to \(3.00 \cdot 10 ^ 8 m/s\).
The Einstein derivation
Albert Einstein derived his famous equation in 1905 in an article entitled ‘Does the inertia of a body depend upon its energy content’. Einstein describes how the emission of energy in the form of light reduces the kinetic energy of the body. His results include two important points:
The energy emitted is independent of the body’s characteristics. The mass of a body is a measure of its energy content.
Einstein’s theory only speculated that energy emission in the form of light reduced the amount of the object’s mass. However, the mass-energy equivalence can be used with respect to all forms of energy.
Mass defect and binding energy
An example of the energy-mass relationship is found in nuclear physics. Atoms are composed of particles that occur in their nucleus. These are electrons, neutrons, and protons. Each particle has its own properties, such as charge and mass. Elements that have more particles in the nucleus are, therefore, more massive and have more charge.
There is, however, a curiosity. If you take the masses of each individual particle that forms the nucleus of an element, its combined weight is larger than the mass of the element’s nucleus. The lost mass is stored in the nucleus as energy.
Mass defect
The difference between the total mass of the individual elements in the atom and the atom’s real mass is known as the mass defect. It can be expressed as follows:
\[\sum^{A-Z}_{n=1} m_n + \sum^{Z}_{p=1} m_p > \text{nuclear mass}\]
Here, mn and mp are the individual masses of the protons and neutrons that compose a particular atom. The total mass of the nucleus will be the sum of all neutrons and protons. The total mass of all neutrons will be the sum of the mass of each neutron mn from n=1 to A-Z. The total mass of the protons will be the sum of the mass of each proton, mp from n=1 to Z will be the sum of the mass of all neutrons from n=1 to A-Z. A is the mass number, while Z is the atomic number. The nuclear mass is just the mass of the atom nucleus.
A stable helium atom has two protons and two neutrons. The mass of an atom can be expressed by a unit called atomic mass unit (amu), which is the mass of 1/12th of the mass of a neutral, non-bonded carbon. Using this unit, we can say that the helium atom has a mass of 4.0026 amu.
The proton has a mass of 1.00727 amu, while the neutron has a mass of 1.00866 amu. What is the mass defect of the helium?
To calculate this, we add the masses of the protons and neutrons together.
\[\text{mass of particles} = 2.0145 amu + 2.0173 amu = 4.0318 amu\]
The mass defect will be equal to the difference between the calculated mass and the actual mass, as expressed in the following formula:
\[\text{mass defect = calculated mass - mass of the atomic nucleus}\]
Applied to the helium atom, we get:
\[\text{mass defect} = 4.0318 amu - 4.0026 amu = 0.0292 amu\]
As we can see, mass is lost when the nucleus is formed and stored in the form of energy, which is known as binding energy.
Figure 2. The mass of the Helium nucleus is smaller than the mass of the individual particles that form it. This is due to the mass defect.
Binding energy
When an atom is formed, several forces are present. As we know, the particles have an electrostatic charge, and the protons are positive. Equal charges repel each other, so two protons exert a repulsive force that seeks to separate them. However, in the nucleus, protons gather alongside neutrons, which have a neutral charge.
But there is another force that binds the particles in the nucleus together. It is known as the ‘strong nuclear force’, and it acts against the electrostatic force. It is the balance between these forces in the nucleus that keeps the atom together.
The energy formed by the bonding of the particles is the binding energy, which is equal to the mass lost by the particles when the atom is formed. Thus, the mass is reduced by being converted into energy.
Reconsidering the example of the helium atom, we can calculate how much energy is equivalent to this mass by using the mass-energy equivalence equation.
If the mass defect of a helium atom nucleus is equal to 0.0303 amu, how much energy is this equivalent to?
To calculate this, we need to use the following equation:
\[E = mc^2\]
First, we need to convert the 0.0303 amu to kilograms.
\[1 amu = 1.66054 \cdot 10^{-27} kg\]
This gives us:
\(0.0303 amu = 0.0303 \cdot (1.66054 \cdot 10^{-27}kg) = 5.03144 \cdot 10^{-29} kg\)
This gives us m if we multiply by the square of the light velocity in the vacuum.
\(E = (5.03144 \cdot 10^{-29}kg) \cdot (299792 \cdot 10^3 m/s)^2 = 4.52202 \cdot 10^{-12}j\)
Although this is a small amount of energy, it is several orders of magnitude larger than the mass that is lost.
It is normal for these reactions to be expressed as electron volts, which makes calculations easier, as 1 amu is equal to 931.5 MeV.
Applications of the mass-energy equivalence
One of the main applications of the mass-energy equivalence is in nuclear energy and nuclear reactions. During the process of fission, heavy atoms break into smaller particles and elements. This process is known as nuclear decay, and it releases energy from the atom itself.
Nuclear Fission
Nuclear fission is an atomic reaction during which heavy elements break away, releasing neutrons that are trapped by other heavy elements. The breakaway produces smaller elements and releases free particles as well as large amounts of energy, which can be used to heat a working fluid (usually water) and produce high-pressure steam that is directed towards a steam turbine connected to an electrical generator.
The process in which a fission reaction occurs is a chain reaction. In this kind of reaction, many atoms of heavy unstable elements are packed together. Some of these elements decay naturally and release energy while breaking into smaller pieces, some of which are neutrons. The free neutrons impact other atoms, inducing more breakaways and increasing the amount of energy released in this chain reaction.
The process is controlled by another material that can absorb electrons, reducing their number and making the fission reaction decay.
Some uses of fission include:
- Production of energy by nuclear reactors.
- Production of isotopes used in medicine and tracing technologies.
- Production of isotopes used in space applications, such as plutonium.
Mass and Energy - Key takeaways
- Mass and energy equivalence states that when an object releases energy, it reduces its mass by an amount of \(\frac{E}{c ^ 2}\), where E is the energy released, and c is the light’s velocity in a vacuum.
- Energy-mass equivalence was first envisaged by Isaac Newton, but it was not demonstrated until experiments conducted by Heaviside, Thompson, and others observed that mass and energy were two related concepts.
- Einstein formulated the equation of mass-energy equivalence as \(E = m \cdot c ^ 2\).
- The binding energy and the mass defect are clear examples of mass-energy equivalence.
- Mass-energy equivalence applies to any form of energy.
- Nuclear energy is an application of the concept of mass-energy equivalence and nuclear decay.
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