Mass Energy Equivalence

Albert Einstein may well be one of the most influential and certainly most famous physicists that the world has ever seen. Whilst his work on the photoelectric effect is what he earned the Nobel prize for in 1921, it's his work on Special Relativity, with its mind-bending ideas about space and time that have lingered on in popular culture. If Einstein is one of the most famous physicists of all time, his mass-energy equivalence equation \(E=mc^2\) is surely the most famous equation in all of science. Walk into any physics student's dorm room and you're bound to find a poster of Einstein and his famous equation! So what exactly was Einstein's great discovery about the nature of mass and energy, and why is a simple equation like \(E=mc^2\) so profound? Well, read on to find out exactly that...

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Team Mass Energy Equivalence Teachers

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    Einstein's Mass-Energy Equivalence Postulates

    Before we can get into the core of mass-energy equivalence and its consequences, we'll first need to cover some basic principles of special relativity. The special theory of relativity, special relativity for short, is one of the core theories of modern physics developed by Albert Einstein in 1905, building on the earlier work of Hendrik Lorentz and Henri Poincaré. Special relativity is largely concerned with the relationship between space and time, and the physics of objects moving at speeds close to the speed of light \(c=3\times10^8\,\mathrm{m}\,\mathrm{s}^{-1}\). Many strange and ground-breaking phenomena were predicted and later confirmed, by special relativity. For example, time dilation states that a stationary clock ticks faster than a clock that is moving relative to it. The principle of mass-energy equivalence is also one of the central ideas within special relativity.

    Mass-Energy Equivalence Portrait of Albert Einstein StudySmarterFig. 1 - Special relativity, and the mass-energy equivalence principle, are regarded as the crowning jewels in the work of Albert Einstein, revolutionizing how we understand space and time.

    Special relativity is based upon two fundamental postulates, from which all other effects can be derived. These postulates, and by extension all of special relativity, rely on the idea of an inertial reference frame.

    A reference frame is an abstract set of coordinates used to describe the position of objects relative to some origin or reference point.

    Inertial frames are non-accelerating reference frames, either stationary or moving at a constant velocity relative to some other reference frame. In these frames, objects with no force acting on them move with constant velocity, as per Newton's first law.

    The postulates can be summarised as:

    • Principle of relativity- The laws of physics are equivalent in all inertial reference frames. This means that, in an inertial frame, no experiment can be done to determine the absolute velocity of the frame, only the velocity relative to some other object.
    • Principle of invariant light speed- Light is always propagated through a vacuum at the same finite speed \(c\) independent of the speed of the body which emits it.

    For a body moving with a constant velocity, we can define a specific type of inertial reference known as the body's rest frame, moving at the same velocity as the body. In this reference frame, the body appears to be at rest. In special relativity, the physical quantities associated with a body, such as time traveled, mass, and length, are determined by the velocity of the frame of reference in which those quantities are being measured. The value of these quantities in the rest frame is taken as the 'proper' value. For example, the mass of a body increases with its velocity relative to the observer measuring it. So, we can define the body's rest mass as its intrinsic mass, with the rest of the mass arising due to its relative velocity. It's this rest mass and rest energy that arises in the principle of mass-energy equivalence.

    Principle of Mass-Energy Equivalence

    Special relativity states that the mass of a body is dependent on its velocity relative to the observer measuring it, however, we can define the mass of the body in its rest frame as the intrinsic mass of the body. Special relativity states that whilst a body has no kinetic energy in its rest frame, it still has intrinsic rest energy. For example, when two (approximately) stationary particle-anti-particle pairs annihilate, the light energy emitted is equal to the combined rest energies of the particles. The principle of mass-energy equivalence shows that this rest energy is entirely equivalent to the rest mass of the body, up to a constant.

    The principle of mass-energy equivalence states that the rest mass \(m\) and the rest energy \(E\) of a body are related by the following formula:

    \[E=mc^2,\]

    where \(c\) is the speed of light. This equivalence can be made exact by choosing the units of mass to be \(\frac{\mathrm{J}}{c^2}.\)

    The idea of mass and energy being equivalent is wholly new to special relativity and is absent in any form of Newtonian physics. This is an astonishingly profound piece of physics, and its simple statement belies some extraordinary consequences.

    Mass-Energy Equivalence Formula

    Mass energy equivalence is mathematically defined by Einstein's famous formula

    \[E=mc^2,\]

    with \(E\) and \(m\) being the rest energy and mass of the body respectively. From this equation, we see that the conversion constant between mass and energy is the speed of light squared \(c^2=9\times10^{16}\,\mathrm{m/s^2}\).

    This is a huge number and so even very small rest masses are equivalent to huge amounts of energy. For example, nuclear weapons with only a few kilograms of fissionable material produce as much energy as tens of thousands of tonnes of regular TNT.

    Mass-Energy Equivalence Image of an explosion after the detonation of a nuclear bomb StudySmarterFig . 2 - Huge amounts of energy can be released through the conversion of rest mass to energy such as in a nuclear fission bomb

    When calculating quantities using this formula it's important to take note of which units are being used for mass and energy. In nuclear and particle physics it is more common to use the unit of mega-electron volts \(\mathrm{MeV}\) rather than Joules \(\mathrm J\).

    One electron-volt \(1\,\mathrm{eV}\) is defined as the amount of kinetic energy gained by an electron accelerating through a potential difference of one-volt \(1\,\mathrm{V}\). \[\begin{align}1\,\mathrm{eV}&=1.60\times10^{-19}\,\mathrm{J}\\1\,\mathrm{MeV}&=1.60\times10^{-13}\,\mathrm{J}\end{align}\]

    It is also common to use units of \(\mathrm{MeV}\,\mathrm{c}^{-2}\) for mass when performing calculations with Einstein's equation. These units are equivalent to setting the speed of light \(c=1\) making energy and mass exactly equal in these units.

    If the rest mass of an electron is \(9.11\times10^{-31}\,\mathrm{kg}\) what is it's rest energy in both Joules \(\mathrm{J}\) and mega-electron volts \(\mathrm{MeV}\)?\[\begin{align}E&=9.11\times10^{-31}\,\mathrm{kg}\cdot(3\times10^8)^2\\&=8.2\times10^{-14}\,\mathrm{J}\end{align}\]

    Converting to \(\mathrm{MeV}\)\[\frac{8.2\times10^{-14}\,\mathrm{J}}{1.60\times10^{-13}}=0.51\,\mathrm{MeV}\]

    We can also express the mass of an electron as \(0.51\,\mathrm{MeV}\,\mathrm{c}^{-2}\)

    The mass-energy equivalence formula can be seen as a special case of Einstein's more general energy-momentum equation

    \[E_r^2=(pc)^2+(m_0c^2)^2.\]

    This equation gives the relativistic energy \(E_r\) of a particle with intrinsic rest mass \(m_0\), moving with a momentum \(p\) relative to the reference frame it is being observed in. So we can see the total relativistic energy of a particle is composed of a kinetic component associated with its momentum relative to an observer, as well as a component stemming from the intrinsic rest mass of the particle which is constant in all frames. If we are in the rest frame of the particle, \(p=0\) and the equation becomes Einstein's mass-energy equivalence relation,\[E_r^2=(m_0c^2)^2\Rightarrow E_r=m_0c^2.\]

    Conversely, for massless particles such as photons, the particle's total energy is entirely dependent on the momentum\[E_r=pc\]

    This demonstrates that there is no possible rest frame for photons, as they always travel at a velocity \(c\) no matter the reference frame.

    Mass-Energy Equivalence Meaning

    To get a better grasp of what this mass-energy equivalence means let's take a look at some direct physical consequences of this fact.

    Nuclear Reactions and the Mass-Defect

    The splitting of the atom in the 1930s may well be one of the single most influential experiments in human history. The discovery that huge amounts of energy can be generated from the process of nuclear fission kickstarted the nuclear age, with both nuclear power plants and nuclear weapons proliferating across the globe throughout the 20th century. The source of this awesome power is entirely dependent on mass-energy equivalence.

    Nuclei within atoms are composed of protons and neutrons bound together by the strong nuclear force. The binding force of the strong force means that when bonded within a nucleus, the protons and neutrons have lower energy than they would do as separate particles. Hence, when protons and neutrons first form into nuclei during nuclear fusion, large amounts of energy are released. Due to mass-energy equivalence, this difference in energy, known as the nuclear binding energy, must be accounted for in the mass of the nuclei meaning the mass of a nucleus is lower than the combined masses of the constituent particles. This mass difference is known as the mass defect and can be experimentally measured.

    For example, the atomic mass of a Helium atom, in atomic mass units \(\mathrm{u}\) is \(4.0015084\,\mathrm{u}\), whereas two protons and two neutrons have a total mass of \[\begin{align}2m_p+2m_n&=2\cdot1.000728\mathrm{u}+2\cdot1.008665\,\mathrm{u}\\&=4.03190\,\mathrm{u}\end{align}\]

    So we see the mass defect of helium is

    \[\begin{align}\Delta m&=4.03190\,\mathrm{u}-4.00150\,\mathrm{u}\\&=0.0303\,\mathrm{u}=5.03\times10^{-29}\,\mathrm{kg}\end{align}\]

    \(1\,\mathrm{u}=1.660539\times10^{-27}\,\mathrm{kg}\)

    This difference in mass is due to the binding energy released when the nucleus was first formed, which by the mass-energy equivalence corresponds to a loss of mass. We can calculate the binding energy of the nuclei \(\Delta E\) using the formula for mass-energy equivalence

    \[\Delta E=\Delta m c^2.\]

    Using the mass defect for helium we find that the binding energy for a helium atom is

    \[\begin{align}\Delta E&=5.03\times10^{-29}\cdot\left(3.00\times10^8\right)^2\\&=45.3\times10^{-20}\,\mathrm{J}.\end{align}\]

    It is this mass defect/binding energy that is behind the awesome power of nuclear fission reaction. Whenever fission occurs a heavy unstable nuclei splits into two smaller nuclei. The combined mass of these product nuclei is smaller than the mass of the original nucleus as the binding energy of the product nuclei is greater. The excess binding energy is released as a by-product of the reaction, this energy release is harnessed in nuclear power stations and nuclear weapons.

    Mass-Energy Equivalence Diagram showing a nuclear fission reaction in Uranium-235 StudySmarterFig. 3 - A diagram showing the fission of Uranium-235 into Krypton-89 and Barium-144 due to the addition of a neutron. The combined mass of the product nuclei is lower than the mass of the original nucleus.

    Particle-anti particle annihilation

    The existence of anti-particles was first postulated by Dirac in 1932 and experimentally confirmed shortly after. These anti-particles have almost identical properties to that of 'regular' matter particles except their charge is opposite. For example, a positron or anti-electron has the same mass and spin as an electron but a positive charge. When a particle and anti-particle pair collide they annihilate releasing energy often in the form of photons, though other particles can be produced. If the kinetic energy of the particle and anti-particle is negligible compared with their rest energies, we can think of annihilation as a process by which the total rest mass of the two particles is converted into energy given off as two photons. Hence the energy of the photons given off can be calculated from the rest masses of the particles by using Einstein's mass-energy equivalence relation.

     Mass-Energy Equivalence Feynman diagram of electron-positron annihilation StudySmarterFig. 4 - A Feynman diagram showing an electron-positron pair annihilating one another. The positron is shown moving backward in time, with the photons produced shown as wavy lines labelled \(\gamma\).

    The rest mass of a proton (and an anti-proton) is \(m_e=1.67\times10^{-27}\,\mathrm{kg}\). If an electron-positron pair collide and annihilate with each other producing two photons, what will the momentum of each photon be in the center of momentum frame? Presume that the kinetic energy of the particles is negligible.

    Using the mass-energy equivalence relation we find that the rest energy of a proton/anti-proton is \[\begin{align}E&=1.67\times10^{-27}\,\mathrm{kg}\cdot(3\times10^8)^2\\&=1.50\times10^{-10}\,\mathrm{J}\end{align}\]

    As we are in the center of momentum frame, the momentum of each photon is equal and opposite, hence the energy of each photon is equal to the rest energy of a proton and its momentum is

    \[\begin{align}p&=\frac{E}{c}=\frac{1.5\times10^{-11}\,\mathrm{J}}{3\times10^8\,\mathrm{m}\,\mathrm{s}^{-1}}\\&=0.5\times10^{-20}\,\mathrm{J}\,\mathrm{c}^{-1}\\&=3.012\times10^{-8}\,\mathrm{MeV}\,\mathrm{c}^{-1}\end{align}\]

    Mass Energy Equivalence Examples

    To recap let's take a look at some example calculations associated with mass-energy equivalence.

    When carbon reacts with oxygen to produce carbon dioxide, energy is released as heat. If \(394\,\mathrm{J}\)are produced for every mole of carbon dioxide produced, what will be the mass difference between the carbon dioxide compound produced and the initial carbon and oxygen molecules, when \(0.5\,\mathrm{mol}\) of carbon dioxide is produced?

    If \(0.5\,\mathrm{mol}\) of CO2 are produced then the heat energy released will be \(0.5\,\mathrm{mol}\cdot394\,\mathrm{J}\,\mathrm{mol}^{-1}=197\,\mathrm{J}\), this energy release means a corresponding mass will also be lost during the reaction. This mass is given by

    \[\begin{align}m&=\frac{E}{c^2}=\frac{197\,\mathrm{J}}{(3\times10^8)^2\,\mathrm{m}\,\mathrm{s}^{-2}}\\&=2.19\times10^{-17}\,\mathrm{kg}\end{align}\]

    Particle-Anti Particle pairs can be produced spontaneously from fast-moving photons whose energy is converted into mass. If a photon with a frequency of \(3\times10^{19}\,\mathrm{Hz}\) spontaneously produces a particle-anti-particle pair, is it possible that the pair produced be a proton-antiproton pair?The energy of the photon can be calculated as \[\begin{align}E=hf&=6.62\times10^{-34}\,\mathrm{J}\,\mathrm{s}\cdot3\times10^{19}\,\mathrm{Hz}\\&=19.86\times10^{-15}\,\mathrm{J}\end{align}\]

    We then need to convert this energy into mass using the mass-energy equivalence formula \[\begin{align}m=&\frac{E}{c^2}\\=&\frac{19.86\times10^{-15}\,\mathrm{J}}{(3\times10^8)^2\,\mathrm{m}\,\mathrm{s}^{-2}}\\=&6.62\times10^{-26}\,\mathrm{kg}\end{align}\]As two particles are produced, the maximum possible mass for one particle is \(3.31\times10^{-26}\,\mathrm{kg}\). As the mass of a proton is \(1.67\times10^{-27}\,\mathrm{kg}\), we see that this photon could not produce a proton-anti-proton pair as it is not energetic enough.

    Mass Energy Equivalence - Key takeaways

    • Special Relativity postulates that the laws of physics are constant in every inertial (non-accelerating) reference frame and that the speed of light \(c\) is constant in every reference frame.
    • Einstein's mass-energy equivalence states that the total energy of a particle measured in its rest frame is given by the famous equation.\[E=mc^2\]
    • The mass of an atom is lower than the sum of the masses of its constituent particles, this is due to the binding energy released when the atom first formed. This difference in mass is known as the mass defect.
    • The binding energy \(\Delta E\) of an atom can be calculated from the mass defect \(\Delta m\) of the atom given by\[\Delta E=\Delta m c^2\]
    • The photon energy given off when a particle-anti-particle pair annihilates can be found from the mass-energy equivalence equation.

    References

    1. Fig.1-Albert Einstein Head( https://commons.wikimedia.org/wiki/File:Albert_Einstein_Head.jpg) by Orren Jack Turner is under Public Domain.
    2. Fig.2-Operation Upshot Knothole Badger (https://commons.wikimedia.org/wiki/File:Operation_Upshot-Knothole_-_Badger_001.jpg) by National Nuclear Security Administration is under Public Domain.
    3. Fig.3-Nuclear Fission Reaction(https://commons.wikimedia.org/wiki/File:Nuclear_fission_reaction.svg) by MikeRun is licenced by CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/deed.en)
    4. Fig.4-Mutual Annihilation of a Positron Electron pair (https://commons.wikimedia.org/wiki/File:Mutual_Annihilation_of_a_Positron_Electron_pair.svg) by Manticorp is licenced under CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/deed.en)
    Frequently Asked Questions about Mass Energy Equivalence

    What is the central equation of mass-energy equivalence? 

    The central equation of mass-energy equivalence is Einstein's equation E=mc2, where E is energy, m is mass, and c is the speed of light.

    What is mass energy equivalence? 

    Mass energy equivalence, desribed by the equation E=mc2 gives the relation between a bodies rest mass and its energy. It means that when energy is lost through some reaction such as chemical or nuclear reactions, some of the body's mass will also be lost.

    Which statement correctly describes mass energy equivalence? 

    Mass-energy equivalence can be summed up by the famous equation E=mc2, which explains how much energy the rest mass of a body corresponds to.

    Who discovered mass energy equivalence? 

    Mass-energy equivalence was first put forward as a general principle of special relativity by Albert Einstein in 1905.

    Does mass equal energy? 

    Mass does not precisely equal energy, but is directly proportional as per the equation E=mc2 with c the speed of light. 

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