The lowest two excited states of hydrogen atom are \(10.2 \mathrm{eV}\) and \(12 \mathrm{eV}\) above the ground state, calculate the wavelength of radiation that could be produced by transition between these states and the ground state.

When an electron in a hydrogen atom jumps between energy levels, it emits or absorbs a photon. This process is known as an energy transition. For example, if an electron moves from a higher energy state to a lower one, it releases energy in the form of light. The energy difference between the initial and final states determines the energy (and thus the wavelength) of the photon emitted. In our given exercise, we have two excited states: one at 10.2 eV and another at 12 eV above the ground state. When an electron transitions from one of these excited states to the ground state, it will release energy equal to the difference between the two energy levels.

To find the wavelength of the emitted photon, we first need to use the energy difference calculated from the excited states. For instance, an energy difference of 10.2 eV or 12 eV can be converted to wavelength using the formula: linear: \(E = \frac{hc}{\text{wavelength}}\).
Here,

• \(E\) is the energy difference - in our case, either 10.2 eV or 12 eV,
• \(h\) is Planck’s constant \(4.1357 \times 10^{-15} \textrm{ eV·s}\),
• \(c\) is the speed of light \(3 \times 10^8 \textrm{ m/s}\).

By plugging in values, we isolate the wavelength: \(\text{wavelength} = \frac{hc}{E}\).
For 10.2 eV: \(\text{wavelength} \approx 1.22 \times 10^{-7} \textrm{m} \ or 122 \textrm{nm}\)
For 12 eV: \(\text{wavelength} \approx 1.034 \times 10^{-7} \textrm{m} \ or 103.4 \textrm{nm}\).

Excited states are higher energy levels that an electron can occupy in an atom. These states occur when an electron absorbs energy, usually in the form of a photon, and jumps from a lower energy state to a higher one. In a hydrogen atom, the ground state is the lowest energy level, often referred to as the n=1 level. Higher energy states, like the ones at 10.2 eV and 12 eV, are known as excited states. When an electron resides in an excited state, it is not stable and tends to return to a lower energy state, releasing the extra energy as radiation. This energy release can be observed as light of specific wavelengths, which corresponds to the transition between these energy levels.