Question

The wavelength of first line of Lyman series of \(\mathrm{H}\) -atom is \(1216 \AA\). What will be the wavelength of first line of Lyman series in 10 time ionized sodium atom \((Z=11)\) (a) \(1216 \AA\) (b) \(12.16 \AA\) (c) \(10 \AA\) (d) \(110 \AA\)

Short answer

The wavelength of the first line of the Lyman series in a 10 times ionized sodium atom is close to \(12.16 \AA\).

Step-by-step solution

Title - Understand the Lyman Series

The Lyman series of the hydrogen spectrum corresponds to electron transitions from higher energy levels to the first energy level. The formula for the wavelength of the spectral lines is given by the Rydberg formula: \[\frac{1}{\lambda} = R_Z \left( \frac{1}{1^2} - \frac{1}{n^2} \right)\] where \(\lambda\) is the wavelength, \(R_Z\) is the Rydberg constant for the atom in question (which depends on the nuclear charge \(Z\)), and \(n\) is the principal quantum number of the energy level to which the electron is transitioning.

Title - Calculate the Adjusted Rydberg Constant

To find the Lyman series for a 10 times ionized sodium atom (11 protons), you must use the corrected Rydberg constant, which accounts for the increased nuclear charge. The modified Rydberg constant, \(R_Z\), can be calculated from the Rydberg constant for hydrogen, \(R_H\), which is approximately \(1.097 \times 10^7 m^{-1}\). Because the sodium atom is 10 times ionized (has lost 10 electrons), it essentially has one electron like a hydrogen atom, but with a nuclear charge of \(Z=11\).\[R_{11} = R_H \times Z^2 = (1.097 \times 10^7 m^{-1}) \times (11)^2\]

Title - Apply the Rydberg Formula for Ionized Sodium

Apply the Rydberg formula for the first line of the Lyman series for ionized sodium (which corresponds to a transition from \(n=2\) to \(n=1\), since that's the transition that gives the first line of the series), using the modified Rydberg constant from step 2.\[\frac{1}{\lambda_{Na^{10+}}} = R_{11} \left( \frac{1}{1^2} - \frac{1}{2^2} \right) \]

Title - Solve for the Wavelength

Solve for the wavelength, \(\lambda_{Na^{10+}}\), using the value of \(R_{11}\) obtained in Step 2.\[\lambda_{Na^{10+}} = \frac{1}{R_{11} \left( \frac{3}{4} \right)} = \frac{1}{(1.097 \times 10^7 m^{-1}) \times (11)^2 \times (\frac{3}{4})} \]Convert the wavelength from meters to angstroms by multiplying by \(10^{10}\) (since \(1 m = 10^{10} \AA\)).

Key concepts

Rydberg Formula

The Rydberg formula is crucial for understanding the spectral lines of elements, especially hydrogen. This mathematical equation explains the wavelength of emitted or absorbed light during electron transitions between energy levels in an atom.

For a hydrogen atom, the formula is given by:\[\begin{equation}\frac{1}{\text{λ}} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)\end{equation}\]where λ is the wavelength of the light, \(R_H\) is the Rydberg constant for hydrogen, and \(n_1\) and \(n_2\) are the principal quantum numbers of the lower and upper energy levels, respectively. In the case of multi-ionized atoms, the formula must be adjusted to account for the higher nuclear charge, using an effective Rydberg constant specific to that atom.

Spectral Lines of Hydrogen Atom

The spectral lines of the hydrogen atom provide a signature pattern unique to hydrogen, resulting from electrons transitioning between energy levels. These transitions release photons of specific wavelengths, thereby creating a series of spectral lines that can be observed using spectroscopy.

For instance, the Lyman series involves transitions to the first energy level (\(n=1\)). The first line in this series, known as the Lyman-alpha line, occurs when an electron falls from the second energy level to the first. The specific wavelengths of the lines in the Lyman series can be predicted by the Rydberg formula.

Nuclear Charge Effect on Spectral Lines

The nuclear charge of an atom has a direct impact on its spectral lines. As the charge increases due to the presence of more protons in the nucleus, the attractive force on the electrons also intensifies. This increment results in energy levels that are more tightly bound.

Consequently, the Rydberg constant must be adjusted to reflect the higher nuclear charge, fundamentally altering the wavelengths of the spectral lines as we see with multi-ionized atoms. In the case of ionized sodium (\(Na^{10+}\)), the effective Rydberg constant, \(R_Z\), would be much larger compared to that of hydrogen, leading to shorter wavelengths for its spectral lines.

Quantum Number Transitions

Quantum number transitions refer to the movement of an electron between energy levels in an atom. These energy levels are described by principal quantum numbers, symbolized by \(n\). The value of \(n\) indicates the 'shell' or 'orbit' the electron is in, with higher numbers representing higher energy levels.

When an electron jumps from a higher energy level (with a larger \(n\) value) to a lower level, it emits a photon with a wavelength that corresponds to the energy difference between those levels. Conversely, to jump up to a higher energy level, an electron must absorb a photon with just the right energy, which also corresponds to the specific difference in energy levels.