# Can Rydberg constant be in joules?

In my textbook (Chemistry Part - I for Class XI published by NCERT), there is an equation for the energy of an electron in an energy state: $$E_n = -R_\mathrm H\left(\frac{1}{n^2}\right)$$ and there is a paragraph below it with the following text:

where $$R_\mathrm H$$ is called Rydberg constant and its value is $$2.18\times10^{-18}\ \text{J}$$.

There is another section with the expression for the wavenumber ($$\overline{\nu}$$): $$\overline{\nu}=109\,677 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)\ \text{cm}^{-1}$$ with a paragraph with the following text:

The value $$109\,677 \space\text{cm}^{-1}$$ is called the Rydberg constant for hydrogen.

I checked online and found that in most (all) websites (incl. Wikipedia), the value of Rydberg constant is $$109\,677 \space\text{cm}^{-1}$$. But when I searched for its value in joules, I found this website with the value of Rydberg constant $$= 2.18\times10^{-18}\ \text{J}$$.

How can Rydberg constant be written in joules?

• How can a price of a cheeseburger be written in ¥? Well, just like that. Rydberg constant is energy, and Joule is energy, so what's the problem? – Ivan Neretin Jun 5 at 7:14
• Look at the two different equations. Consider the dimensions in each case, and why that means in one case the Rydberg constant must have dimensions of Energy, and in the other it must have dimensions of wave number. Now what relations do you know relating Energy and wave number? – Ian Bush Jun 5 at 7:16
• Although wavenumbers are not strictly energy you will see from the value in joules that it is much more convenient to use wavenumbers. The conversion is $1 \,\mathrm{cm^{-1}}\equiv 1.986\cdot 10^{-23}$ J. – porphyrin Jun 5 at 7:18

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 $$\pu{1 Ry} = Ry = hc \cdot R_\ce{H}$$.

• My notation would have called for your Ry to be R subscript y. Assuming I get my mathjax right it would look like $R_y$. This was clearly distinct from $Ry$ even in my bad writing because the joint of the y was below the line. – Joshua Jun 5 at 17:38

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.

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: $$R_∞ = \pu{109677 cm^{−1}}$$

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