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To complete some of the details of this chapter, the Rydberg constant can be calculated from more basic units. It is

(1.6)

where m_e is the mass of the electron (m_e = 9.1 × 10⁻³¹ kg), e is the electronic charge (e = 1.6 × 10⁻¹⁹ C), ε is the permittivity of the material, that is the product of the relative permittivity of the material (e.g. for hydrogen ε_r = 253.8) times the permittivity of free space (ε₀ = 8.85 × 10⁻¹² m⁻³ kg⁻¹ s² C²), h is Planck's constant (h = 1.06 × 10⁻³⁴ J s), and c is the speed of light (c = 3 × 10⁸ m s⁻¹). All of these quantities are fundamental physical constants.

One thing that we often forget or do not check is the fact that any measurement is composed of a numerical value and a unit. In Eq. (1.6) I show several quantities, each of which has a number and unit(s). The result has to agree with the numerical calculation and the units. Very few people check the units when they perform an operation, which can result in mistakes (remember the fiasco when the Mars orbiter failed because of the confusion between metric and English units). So, let me do this with Eq. (1.6). First the units:

Look at the three terms in the denominator. There are meter (m) terms in the denominator with exponents −6 + 6 + 1 = 1 m, so only one meter unit remains in the denominator. Similarly, with the exponents of the kilograms, there are −2 + 3 = 1 in the denominator. There are four Qs in the denominator, and the seconds in the denominator cancel out (4 – 3 – 1 = 0). Now the kilograms and the coulombs in the numerator cancel those in the denominator, leaving only the reciprocal of a meter as the remaining unit, in agreement with Eq. (1.6).

Now let's do the numbers.

This agrees with the published result.