For He$$^+$$, a transition takes place from the orbit of radius 105.8 pm to the orbit of radius 26.45 pm. The wavelength (in nm) of the emitted photon during the transition is ____.

[Use: Bohr radius, $$a = 52.9$$ pm; Rydberg constant, $$R_H = 2.2 \times 10^{-18}$$ J; Planck's constant, $$h = 6.6 \times 10^{-34}$$ J s; Speed of light, $$c = 3 \times 10^8$$ m s$$^{-1}$$]

For a hydrogen-like ion, the radius of the $$n^{\text{th}}$$ Bohr orbit is given by
$$r_n = \frac{n^2 a_0}{Z}$$ where $$a_0 = 52.9 \text{ pm}$$ is the Bohr radius and $$Z$$ is the atomic number.

For $$\text{He}^+$$ the nuclear charge is $$Z = 2$$.
First identify the quantum numbers corresponding to the two given radii.

Initial orbit (larger radius):
$$r_i = 105.8 \text{ pm}$$

$$n_i^2 = \frac{r_i Z}{a_0} = \frac{105.8 \times 2}{52.9} = 4$$
$$\Rightarrow \; n_i = 2$$

Final orbit (smaller radius):
$$r_f = 26.45 \text{ pm}$$

$$n_f^2 = \frac{r_f Z}{a_0} = \frac{26.45 \times 2}{52.9} = 1$$
$$\Rightarrow \; n_f = 1$$

The transition is therefore $$n = 2 \rightarrow n = 1$$.

For a hydrogen-like species, the energy of the $$n^{\text{th}}$$ orbit is
$$E_n = -\frac{Z^2 R_H}{n^2}$$ where the Rydberg constant in energy units is $$R_H = 2.2 \times 10^{-18}\,\text{J}$$.

Compute the energy difference:

$$E_2 = -\frac{(2)^2 R_H}{2^2} = -R_H$$
$$E_1 = -\frac{(2)^2 R_H}{1^2} = -4R_H$$

Energy of the emitted photon:
$$\Delta E = E_1 - E_2 = (-4R_H) - (-R_H) = -3R_H$$
Magnitude: $$|\Delta E| = 3R_H = 3 \times 2.2 \times 10^{-18} = 6.6 \times 10^{-18}\,\text{J}$$

Relate the photon energy to its wavelength $$\lambda$$ using $$E = \frac{hc}{\lambda}$$:

$$\lambda = \frac{hc}{E}$$

With $$h = 6.6 \times 10^{-34}\,\text{J s}$$ and $$c = 3.0 \times 10^{8}\,\text{m s}^{-1}$$,
$$hc = 6.6 \times 10^{-34} \times 3.0 \times 10^{8} = 1.98 \times 10^{-25}\,\text{J m}$$

$$\lambda = \frac{1.98 \times 10^{-25}}{6.6 \times 10^{-18}} = 3.0 \times 10^{-8}\,\text{m}$$

Convert metres to nanometres (1 nm = $$10^{-9}$$ m):
$$\lambda = 30\,\text{nm}$$

Hence, the wavelength of the emitted photon is 30 nm.