[Physics] Boltzmann constant–why units aren’t joules per degree PER MOLECULE

thermodynamicsunits

I didn't think the information I looked at did a very good job of explaining the Boltzmann Constant, kB, but once I did this next simple calculation, it seemed clear that what I didn't understand were the units. So I simplified the units only:

$8.31… = R$ = energy per degree/mole

NA = molecules/mole, Avogadro's constant

kB = R/NA = (energy per degree/mole)/(molecules/moles)

kB = energy per degree/mole * moles/molecules

kB = energy per degree per molecule

Since kB is in units of energy per deg K per molecule, multiplying it by T in units of deg K will arrive at total energy per molecule. Multiplying that by the number of molecules gives total energy of the mass of air, same as PV. Energy per degree per molecule x degrees x molecules = energy.

But every reference I can find states that the Boltzmann constant is in units of energy per temperature, or joules/deg. Kelvin. What am I missing?

It seems to me that kB is joules per degree per molecule, while R is joules per degree per mole.

Best Answer

The correct unit is joules per kelvin oscillator. A molecule might have several modes of vibration. The theory runs numeric constants for oscillators / molecule.

SI is not a coherent system here, because everyone else derives moles in the same way as electron volts are: a unit and constant. The constant is a dalton (currently a unified mass unit: dalton has been derived for the generic name), the unit is M, and a M-mole is M/(1 dalton). SI uses M in grams here :o.

Since $k=R/N_A$ one sees the unit is (J/kg-mol. K) / (dalton/kg) = (J/Da-mol K) A dalton-mole is by definition, 1, so one gets J/K. If one starts to add in a unit "molecule" for dalton-mole, one gives the impression that there is only one oscillator per molecule, or that $k_B$ applies only to molecular matter.

The theory of black body radiation does not call down to molecular matter, but the boltzman constant is found there too. It is better to think of heat as being made of thermions or modes of energy, the actual energy is thermion * temperature, in much the same way that planck's constant gives energy = h * wavelength * photons.

Thermodynamics normally handles 'oscillators/molecule' as a numeric constant, often simply putting in the resolved value (eg 1.4). So the energy per molecule is not $kT$, but some numeric like $1.4kT$ etc. Oscillators are distributed over molecules, not over weight. So all di-atomic gasses, like H_2 or N_2 or O_2, have the same number of oscillators per molecule, or per lb-mole. In general, an equipartition rule holds, so each thermion tends to have a similar energy, but that is a subject of thermodynamics, not of metrology.

Likewise, 'h' is a measure per oscillator, and $N_A h$ molar planck constant is also known. h and its kith $\hbar$ are likewise missing units, (correctly, these are J.s/cycle = J/Hz, and J.s/rad respectively. Crossing the wavelength $\lambda$ converts that to a radian-length, so $E = hc/\lambda$ works when both are in cycles or in radians.

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