Einstein’s mass-energy equation

Let us postulate that we live in a 3D hypersurface that slides along the u axis with speed v°u = ca, where the u axis coincides with the arrow of time. The 4-velocity is then a (row or column) vector 1 a = ( ca,vx,vy,vz). The plus (resp. minus) sign corresponds to the speed of preons that enter (resp. leave) our 3D world, parallel (resp. antiparallel) to the time arrow. It will be seen below that this constant ca is the one that enters Einstein s mass-energy equation, and corresponds to the speed of our 3D world along the time axis (interpretation 2 in Fig. 1). The speed of electromagnetic radiation in free space is a different constant c. The value of the latter may be either identical or numerically close to c , depending of whether one adopts a relativistic or an emission theory for photons, respectively (see Section V). 

Einstein s mass-energy equation E = nuP the relationship between mass and energy. [ 18.12] electrode the cathode or anode in an electrochemical cell (see cathode and anode). (17.6) electrolysis The process whereby electrical energy is used to bring about a chemical change. [ 17.6] electrolyte A substance whose aqueous solution conducts electricity. [15.5]... 

It is the first one that will be emphasized, and can be broken into conservation of mass and energy, which are coupled with Einstein s mass-energy equivalence (E=mc ). As such, the accumulation terms of the conservation of mass are not affected. Also, we could neglect forced convection effects in the system. The resulting mass diffusion equation would be similar to that in Eq. (1.5.2), except that a so-called elastic strain energy could be added to the potential function to take into account crystal lattice differences between solid phases (De Fontaine, 1967). 

This set of equations connects Planck s photon energy Ep with Einstein s mass/en-ergy equivalence, with Boltzmann s kinetic energy, with the kinetic energy of a particle and with the kinetic energy of an electron in an electric field of a voltage U of 1 V. The most important conversion factors used in photochemistry and photophysics are collected in Tab. 3-2. 

In your study of chemical reactions, you learned that mass is conserved. For most practical situations this is true—but, in the strictest sense, it is not. It has been discovered that energy and mass can be converted into each other. Mass and energy are related by Albert Einstein s most famous equation. 

The splitting of the atom was accompanied by the release of neutrons and the conversion of a small amount of mass to energy. The amount of energy could be calculated using Einstein s famous 1905 equation E = me2, and a new source of power was born. 

We now turn to Einstein s full gravitational equation. There being ten metric components, there are ten partial differential equations to determine them. One is a fanciful elaboration of Poisson s equations with the relativistic energy density—as opposed to rest mass density—as source. Pressure and energy fluxes become the sources of the others. If we are mostly interested in the external gravitational field of a spherically symmetric body, then the sources can be dropped and the unique exact solution is Schwarzschild s metric (not Martin Schwarzschild but his dad Karl Schwarzschild, also the father of photographic photometry) ... 

In the special case of a particle at rest p = 0, we obtain Einstein s famous mass-energy equation E = mc. The alternative root E = — mc is now understood to pertain to the corresponding antiparticle. For a particle with zero rest mass, such as the photon, we obtain p = E/c. RecaUing that kv = c, this last four-vector relation is consistent with both the Planck and de Broglie formulas E = hv and p = h/X. 

Within the linear response approximation, the rate of transport (mass, momentum, or energy) through a system is proportional to the gradient (of concentration, velocity, and temperature), with the transport coefficient being the proportionality constant. This proportionality constant can be computed using equilibrium description of the system through the so-called fluctuation dissipation theorems. One such equation, relating equilibrium fluctuations to the diffusion constant, is Einstein s well-known equation ... 

Albert Einstein s most famous equation relates mass and energy. 

STRATEGY If we know the mass loss, we can find the energy released by using Einstein s equation. Therefore, we must calculate the total mass of the particles on each side of the nuclear equation, take the difference, and substitute the mass difference into Eq. 6. Then we determine the number of nuclei in the sample from N = m(sample)/m(atom) and, finally, multiply the energy released from the fission of one nucleus bv that number to find the energy released by the sample. 

Einstein showed that mass and energy are equivalent. Energy can be converted into mass, and mass into energy. They are related by Einstein s equation ... 

Arts. The data from Table 22-3 are used. The sum of the masses of the products minus the mass of the neutron, converted to energy with Einstein s equation, yields the energy produced. 

Nuclear fusion processes derive energy from the formation of low-mass nuclei, which have a different binding energy. Fusion of two nuclear particles produces a new nucleus that is lighter in mass than the masses of the two fusing particles. This mass defect is then interchangeable in energy via Einstein s equation E = me2. Specifically, the formation of an He nucleus from two protons and two neutrons would be expected to have mass ... 

Ever since Albert Einstein devised his famous equation, E = mc, we have known that mass and energy are interconvertible. In Einstein s equation, E is energy in kg m /s (J), m is the mass in kg, and c is the square of the speed of light. 

Nuclear mass is always lower than the sum of proton and neutron masses because of mass-energy conversion, which transforms part of the mass into binding energy. For example, He has mass 4.002604, whereas the total mass of two protons and two neutrons is 4.032981. Based on Einstein s equation... 

According to Einsteins equation, this newly acquired energy reveals itself as an increase in the nucleon s mass—the mass of a nucleon outside a nucleus is greater than the mass of the same nucleon locked inside a nucleus. For example, a carbon-12 atom—the nucleus of which is made up of six protons and six neutrons—has a mass of exactly 12.00000 atomic mass units. Therefore, each proton and each neutron contributes a mass of 1 atomic mass unit. However, outside the nucleus, a proton has a mass of 1.00728 atomic mass units and a neutron has a mass of 1.00867 atomic mass units. Thus we see that the combined mass of six free protons and six free neutrons—(6 X 1.00728) + (6 X 1.00867) = 12.09570—is greater than the mass of one carbon-12 nucleus. The greater mass reflects the energy that was required to pull the nucleons apart from one another. Thus, what mass a nucleon has depends on where the nucleon is. 

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