Chemistry - Can Rydberg constant be in joules?

Solution 1:

Authors may be sloppy about notation in this matter. I recommend considering $R_\ce{H} \approx \pu{10973 cm-1}$ and $Ry \approx \pu{2.18e-18 J}$, noting $Ry = hc \cdot R_\ce{H}$. Units of wavenumbers $(\pu{cm-1})$ and energy are often considered interchangeable in practice because they are proportional to each other by the constant value $hc$.

In my notes, I would always be sure to write $R_\ce{H}$ or $Ry$ to explicitly remind myself "which" Rydberg constant I was using (in fact I merged the R and y into a single symbol because I didn't like the suggestion of multiplication.)

Note also that there is a unit of energy known as a Rydberg, with $\pu{1 Ry} = Ry = hc \cdot R_\ce{H}$.

Solution 2:

Rydberg constant $R_∞$ is usually given in reciprocal length units historically and because it's determined from hydrogen and deuterium transition frequencies [1]. Current value (in $\pu{m-1}$) is listed at NIST [2] website (accessed 2019-06-05):

$$R_∞ = \pu{10973731.568160(21) m-1}$$

Since it's an energy unit, one can convert it to SI rather trivially via multiplying the value in reciprocal length units by $hc$ ($h$ is the Planck constant; $c$ is the speed of light in vacuum):

$$E = hν = \frac{hc}{λ} \quad\text{or}\quad R_∞[\pu{J}] = hc\cdot R_∞[\pu{m-1}]$$

resulting in the following value:

$$R_∞ = \pu{2.1798723611035(42)e-18 J}$$

References

1. Mohr, P. J.; Newell, D. B.; Taylor, B. N. CODATA Recommended Values of the Fundamental Physical Constants: 2014. Reviews of Modern Physics 2016, 88 (3). https://doi.org/10.1103/RevModPhys.88.035009.
2. Tiesinga E.; Mohr P. J.; Newell D. B.; Taylor, B. N. "The 2018 CODATA Recommended Values of the Fundamental Physical Constants" (Web Version 8.0). Database developed by J. Baker, M. Douma, and S. Kotochigova. Available at http://physics.nist.gov/constants, National Institute of Standards and Technology, Gaithersburg, MD 20899. 2019.

---

Solution 3:

In spectroscopy and related fields it is common to measure energy levels in units of reciprocal centimeters (e.g., IR and Raman spectroscopy). Strictly speaking, these units ($\pu{cm^{−1}}$) are not energy units, but units proportional to energies, with $hc$ being the proportionality constant (Wikipedia). In general, $hc$ can be attributed to the value $\pu{1.986E-23 J cm}$. Hence: $$R_∞ = \pu{109677 cm^{−1}}$$

$$\pu{1 Ry} = \pu{109677 cm^{−1}} \times hc = \pu{109677 cm^{−1}} \times \pu{1.986E-23 J cm} = \pu{2.178E-18 J}$$