Two values of Rydberg Constant

Question

In my textbook I came across two values of Rydberg Constant: \begin{align} R&= 109\,677\ \mathrm{cm}^{-1} \\ R_h&= 2.18 × 10^{-18}\ \mathrm J \end{align} when we are calculating energy of the stationary state.

So can anyone clear this confusion of mine?

Answer 1

The Rydberg constant can be expressed in two different units depending on the context and what quantity it is being used to calculate:

1. $\mathrm{R = 109677~cm^{-1}}$. This value is used when calculating wavenumbers (1/wavelength) of photons emitted or absorbed during electron transitions in hydrogen-like atoms using the Rydberg formula:

$$\frac{1}{\lambda} = R\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$$

Where $n_1$ and $n_2$ are the principal quantum numbers of the initial and final states. The units work out to be ${cm^{-1}}$ (or ${m^{-1}}$).

2. $\mathrm{R_h = 2.18 × 10^{-18}~J}$. This is the Rydberg unit of energy and represents the ionization energy of the hydrogen atom in the ground state. It is used when calculating the energy levels of hydrogen-like atoms:

$$E_n = -\frac{R_h}{n^2}$$

Where n is the principal quantum number. The negative sign indicates the electron is bound to the nucleus.

The two values are related by:

$$R_h = hcR$$

Where h is Planck's constant and c is the speed of light. Multiplying the value in ${cm^{-1}}$ by $hc$ converts it to units of energy (J).

Use $\mathrm{R = 109677~cm^{-1}}$ when working with the Rydberg formula and dealing with wavelengths/wavenumbers of transitions. Use $\mathrm{R_h = 2.18 × 10^{-18}~J}$ when calculating energy levels directly. The constant is the same, just expressed in different units for different purposes. Let me know if this clears up the confusion!