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This article states that the average human has a mass energy equivalence of 7.8 septillion joules of energy

Converting this into a temperature using $E=k_B T$, we have a temperature on the order of $10^{41} K$, which is obviously not the case.

  • I’m obviously being dumb, but what am I missing?
  • This is an awful lot of energy, is it usable?
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    $\begingroup$ Can you explain why you think relativistic rest energy contributes to temperature? $\endgroup$ May 17, 2020 at 11:37
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    $\begingroup$ @BioPhysicist Using $E=k_B T$? $\endgroup$
    – code noob
    May 17, 2020 at 11:39
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    $\begingroup$ That's just an equation. Why do you think $E$ in that equation is the same $E$ in $E=mc^2$? $\endgroup$ May 17, 2020 at 11:41
  • $\begingroup$ @BioPhysicist I have nothing to make me think otherwise. I’ve informally been told that mass and energy are equivalent and seen my particle physics lecturer convert between energy and temperature. I know I’m obviously misunderstanding something, but not sure what, hence the question $\endgroup$
    – code noob
    May 17, 2020 at 11:43

2 Answers 2

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Your use of the Boltzmann constant in this fashion implicitly assumes that all of the energy in the system (human) is contained in a single internal degree of freedom.

The concept of thermal energy assumes that a system has a large number of internal degrees of freedom. In an ideal monoatomic gas under typical conditions this is limited to the KE of the gas atoms, but it is important to understand that in general thermal energy is not only due to kinetic energy of the molecules but includes all internal degrees of freedom. The idea that thermal energy is internal kinetic energy is a mistaken over-generalization from the case of an ideal gas.

In other materials the thermal energy is also found in other internal degrees of freedom, such as rotational and vibrational modes, excited electron orbital modes, and at high enough temperatures particle modes. The equipartition theorem says that thermal energy is evenly distributed throughout these internal degrees of freedom.

The Boltzmann constant converts between temperature and the amount of energy in each of these internal degrees of freedom. Due to quantum mechanics, these modes have a minimum energy required to excite it above the ground state. If the energy per degree of freedom is less than this minimum energy then the degree of freedom is said to be “frozen out” and does not participate in the thermal exchange. In particular, at human body temperature the particle modes are well frozen out. This means that the mass energy of the particles in the body does not participate in thermal exchange.

At the large temperature you calculated each degree of freedom (e.g. the KE of a single particle) would possess enough energy to produce a human! At our lower temperature most of our energy is in frozen out degrees of freedom. Very little of our mass energy is therefore available to contribute to thermal exchange.

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    $\begingroup$ Technically, $E=k_BT$ assumes two such DOFs (if their contribution to the KE is quadratic). $\endgroup$
    – J.G.
    May 17, 2020 at 13:48
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    $\begingroup$ @J.G. You are absolutely correct, I glossed over the missing factor of 1/2. I cannot see a way to include it that doesn’t distract from the main message $\endgroup$
    – Dale
    May 17, 2020 at 14:19
  • $\begingroup$ Wouldn't rotation and vibration also count as kinetic energy, since they are forms of movement? I guess it's just a matter of terminology $\endgroup$
    – gardenhead
    May 17, 2020 at 20:02
  • $\begingroup$ -1 from me. You didn't say that the energy is from $mc^2$ rather than temperature. $\endgroup$
    – MaxW
    May 17, 2020 at 21:30
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    $\begingroup$ @gardenhead it would depend on the molecule how many degrees of freedom but if it is vibrating in one mode (eg aa molecule of two identical atoms stretching closer and further) then it is one degree of freedom where the energy in that one degree of freedom sloshes back and forth between KE and PE $\endgroup$
    – Dale
    May 21, 2020 at 21:37
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Temperature as we know it is mostly due to the kinetic energy of the atoms of any object (the rotational and vibrational energy of the atoms also accounts in the temperature). Now the energy in $E = mc^2$ is not kinetic energy, rather it is the energy stored by the mass of the object itself. So, while the 'mas energy' of a human may be 7.8 septillion joules, that is not the kinetic energy that we perceive as temperature. And that's why we don't have a temperature of $10^{41}$ K.

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    $\begingroup$ this is wrong, you should read/study the answer below from @Dale $\endgroup$
    – hyportnex
    May 17, 2020 at 15:00
  • $\begingroup$ Yeah, thanks for noting, fixed it a bit, mentioned other DOFs. $\endgroup$
    – PNS
    May 17, 2020 at 15:55

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