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If you have an object at rest, and you measure its mass to a really high accuracy.

Does the energy in $E= mc^2$ equal the internal energy in thermodynamics? Does it measure all the kinetic and potential energy contributions inside the body?

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Energy is the zeroth component of the 4-momentum. The 4-momentum is defined as: $$\tag{1}p_{\mu}=\left(\frac{E}{c},\vec{p}\right)\label{1}$$ The dot product of 4-momentum with itself is: $$\tag{2}p_{\mu}\cdot p^{\mu}=m^2c^2\Rightarrow \frac{E^2}{c^2}-p^2=m^2c^2\label{2}$$ In particular, if the object is at rest, the spatial component of the 4-momentum is zero and in this case, the mass is given by: $$\tag{3}m=\frac{E}{c^2}\label{3}$$

In thermodynamics, the internal energy is given by the sum of all kinetic and potential energies of the particles that form the thermodynamic system. If the gas is ideal, the individual gas molecules do not interact, so there is no potential energy to consider in the system. The internal energy of an ideal gas is given only by the kinetic energy of the molecules.

The mass of an elementary particle is a manifestation of potential energy transferred to the particle when it interacts with the Higgs field. For a composite particle, the mass in the equation (\ref{3}) takes into account all of the kinetic and potential energy of the internal components of the particle. For example, the proton is a hadron, which means that it is a composite particle made of quarks and gluons. The protons are made of two up quarks and a down quark. Quarks are elementary particles. The mass of the proton is given by the mass of the quarks and by the QCD binding energy. In fact, most of the mass of the protons comes from the binding energy between the quarks.

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The thermodynamic notion of "internal energy" was developed long before special relativity, so there is no way the historical definition could include the mass energy. Rather the historical definition includes molecular kinetic energy, intermolecular potential energies, and chemical energies.1

And of course that means that for most problems ignoring the mass energy in your definition of the internal energy is just fine.

That said, mass energy is "an energy present in the system in modes that are not apparent at the macroscopic scale" (one possible way of defining the internal energy), so there is no reason you can't count it in there. It just won't be a very interesting term in most cases because it won't change appreciably between the initial and final states (and you have to watch out for double counting). You could ask if it make sense to consider in in the case of, say the URCA process in neutron star cooling, but even there the matter is usually modeled in terms of chemical potentials.2


1 One could argue that chemical energies manifest as changes in mass, and it wouldn't be wrong but it also wouldn't be very useful.

2 This is basically the same game as saying that chemical energy is distinct from mass energy, but here it is less obvious that thinking in terms of chemical potential is a better than thinking in terms of mass energy.

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