Question

Of the following transitions in hydrogen atom the one which gives an absorption line of lowest frequency is (1) \(n=1\) to \(n=2\) (2) \(n=3\) to \(n=8\) (3) \(n=2\) to \(n=1\) (4) \(n=8\) to \(n=3\)

Short answer

The transition from \( n=3 \) to \( n=8 \) (option 2) gives an absorption line of lowest frequency.

Step-by-step solution

- Understand Absorption Transition

An absorption line occurs when a photon is absorbed by an electron causing it to jump from a lower energy level to a higher energy level. Hence, consider only transitions where the energy level increases.

- Identify Valid Absorption Transitions

From the given options, only the transitions (1) and (2) are valid for absorption, since these involve moving from a lower energy level to a higher energy level.

- Recall Energy of Photon Formula

The energy of the photon absorbed is given by the formula \[ E = h u = h c \frac{1}{\lambda} \], where \( u \) is the frequency of the photon, \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength.

- Use the Energy Level Difference Formula

The difference in energy levels in an atom is given by \[ \Delta E = 13.6 \, \text{eV} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \].

- Compare Energy Differences for Options (1) and (2)

Calculate the energy difference for: **(1)\( n=1 \text{ to } n=2 \):** \[ \Delta E_1 = 13.6 \, \text{eV} \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = 13.6 \, \text{eV} \left( 1 - \frac{1}{4} \right) = 13.6 \, \text{eV} \times \frac{3}{4} = 10.2 \, \text{eV} \]**(2)\( n=3 \text{ to } n=8 \):** \[ \Delta E_2 = 13.6 \, \text{eV} \left( \frac{1}{3^2} - \frac{1}{8^2} \right) = 13.6 \, \text{eV} \left( \frac{1}{9} - \frac{1}{64} \right) = 13.6 \, \text{eV} \left( \frac{64 - 9}{576} \right) = 13.6 \, \text{eV} \times \frac{55}{576} = 1.3 \, \text{eV} \]

- Determine the Lowest Frequency

Lower energy difference corresponds to a lower frequency since \( E = h u \). Comparing \( 10.2 \, \text{eV} \) and \( 1.3 \, \text{eV} \), option (2) has the lowest energy difference and hence the lowest frequency.

Key concepts

Absorption Lines

Absorption lines are dark lines observed in the spectrum of a light source. They occur because certain wavelengths of light are absorbed by electrons in an atom. When an electron in a hydrogen atom absorbs a photon, it jumps from a lower energy level to a higher energy level. These absorption lines help us learn about the structure of atoms and the composition of distant stars. In the context of the given exercise, we are interested in the transitions which cause these absorption lines.

Energy Levels in Atoms

Atoms have distinct energy levels where their electrons reside. These levels are quantized, meaning electrons can only exist in specific energy states. For hydrogen atoms, these levels are labeled by a principal quantum number, n (e.g., n=1, n=2, etc.). The further the energy level from the nucleus, the higher its energy. When an electron jumps from a lower (n=1) to a higher energy level (n=2), it absorbs energy, which corresponds to a specific photon. Practically, only certain transitions (like n=1 to n=2 or n=3 to n=8) result in absorption lines, as detailed in the problem's step-by-step solution.

Photon Energy Calculation

The energy of a photon absorbed or emitted during any transition can be calculated using the formula: where: • E is the energy in electron volts (eV) • h is Planck's constant • 𝜐 (nu) is the frequency of the photon • c is the speed of light • λ is the wavelength. We also use: which simplifies to: The energy difference between two levels corresponds to for hydrogen, we use: E = 13.6 𝑒𝑣 This formula is used to calculate the energy of the photon for transitions such as n=1 to n=2 and n=3 to n=8.

Frequency and Wavelength Relationship

The frequency (𝜐) and wavelength (λ) of light are related by the speed of light (c): A higher frequency means a shorter wavelength and vice versa. In the context of the exercise, lower energy transitions (like n=3 to n=8) result in photons of lower frequency, thus longer wavelength. Understanding this relationship helps us determine the appearance of absorption lines in the hydrogen spectrum. For instance, the transition n=3 to n=8, having a lower energy difference, leads to a lower frequency absorption line compared to n=1 to n=2.