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Michelle Mizuno-Wiedner Dept. of Astronomy and Space Physics Uppsala University

Comment on the constant

One of the annoying things about converting between cgs and SI units is that you must be especially careful whenever using a formula involving the , 1 1 7 1 1 = 8.3143 J K− mol− or 8.3143 10 erg K− mol− (1) R × As an example, let us look at the law (or perfect gas law), ρ P = nkT = kT (2) µmu where P is the , T the , n the number of particles, ρ the 27 density, µ the mean molecular weight and mu = 1.66 10− kg the unit. The gas constant is related to the × k by R = kNA (3) R 23 1 where NA = 6.022 10 mol− is Avogadro’s number. The then becomes × P = R ρT (4) µmuNA

Both Eqs. 2 and 4 are valid regardless of whether you use cgs or SI units. In terms of dimensions, if mu is regarded as a mass, then µ is dimensionless.

1 One important point is that muNA = 1 (g mol− ), but only in cgs units. This is because 1 is defined in terms of grams and not . Thus many books write the ideal gas law as P = R ρT [ cgs ] (5) µ

Please note: omitting the factor muNA means that µ is no longer dimensionless 1 but now has the units of g mol− . This could lead to problems if you use SI units but still assume that µ is dimensionless, because then there will be this missing factor 1 of muNA = 0.001 kg mol− . For example, if you plug in the correct SI values for ρ and into Eq. 5 but take the mean molecular weight µ of an element equal to its atomicR weight , then your numerical result will be 1000 times too small. A

The course textbook states the gas constant not as = kNA but as = k/mu, R R again assuming that muNA = 1. This gives the gas constant in terms of kilograms 3 1 1 rather than moles (i.e., = 8.3143 10 J kg− K− ), which yields the correct nu- merical result in cases likeR Eq. 5. Personally,× I find the resulting units somewhat 24 1 inconsistent, e.g., the atomic mass unit mu = 1/NA = 1.66 10− mol− is in inverse moles rather than in (kilo)grams. ×

Of course there may be a simpler way to understand the hassle with the units, and if so, please tell me about it! In the meantime, my advice is: if you want to avoid cgs units, stick with the Boltzmann constant k rather than the gas constant . R