Otdfbt

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: $$E_n = -R_mathrm Hleft(frac1n^2right)$$ and there is a paragraph below it with the following text:

where $R_mathrm H$ is called Rydberg constant and its value is $2.18times10^-18 textJ$.

There is another section with the expression for the wavenumber ($overlinenu$):
$$overlinenu=109,677 left(frac1n_1^2 - frac1n_2^2right) textcm^-1$$
with a paragraph with the following text:

The value $109,677 spacetextcm^-1$ 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 $109,677 spacetextcm^-1$. But when I searched for its value in joules, I found this website with the value of Rydberg constant $= 2.18times10^-18 textJ$.

How can Rydberg constant be written in joules?

Authors may be sloppy about notation in this matter. I recommend considering $R_ceH approx pu10973 cm-1$ and $Ry approx pu2.18e-18 J$, noting $Ry = hc cdot R_ceH$. Units of wavenumbers $(pucm-1)$ 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 $R_ceH$ 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 $pu1 Ry = Ry = hc cdot R_ceH$.

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 $pum-1$) is listed at NIST [2] website (accessed 2019-06-05):

$$R_∞ = pu10973731.568160(21) m-1$$

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):

$$E = hν = frachcλ quadtextorquad R_∞[puJ] = hccdot R_∞[pum-1]$$

resulting in the following value:

$$R_∞ = pu2.1798723611035(42)e-18 J$$

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 Ramon spectroscopy). Strictly speaking, these units ($pucm^−1$) 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 $pu1.986times 10^-23 Jcm$. Hence:
$$R_∞ = pu109677 cm^−1 $$

$$R_y= pu109677 cm^−1 times hc = pu109677 cm^−1 times pu1.986times 10^-23 Jcm = pu2.178times 10^-18 J $$