Question

Electrons are de-exciting from the fifth orbit in hydrogen atoms but the first orbit is not available for them. The maximum number of spectral lines should be (a) 10 (b) 6 (c) 15 (d) 3

Short answer

The maximum number of spectral lines is 3.

Step-by-step solution

Understand the concept of spectral lines due to electronic transitions

When electrons de-excite in a hydrogen atom, they move from higher energy levels to lower energy levels, emitting radiation at specific frequencies, each corresponding to a spectral line. The number of spectral lines produced is determined by the number of possible transitions between the energy levels involved.

Use the formula of calculating spectral lines

The formula for calculating the maximum number of spectral lines when an electron de-excites from a higher energy level is given by the formula: \( n(n - 1) / 2 \), where \( n \) is the number of energy levels the electrons can transition between.

Calculate the number of transitions for the given problem

Since the electrons are de-exciting from the fifth orbit and the first orbit is not available, they can transition to the second, third, and fourth orbits only. Therefore, there are \( n = 3 \) energy levels available for the electrons to transition.

Apply the formula to find the answer

Using the formula \( n(n - 1) / 2 \), where \( n = 3 \), we calculate the maximum number of spectral lines: \[ 3(3 - 1) / 2 = 3(2) / 2 = 6 / 2 = 3 \]. So, the maximum number of spectral lines that can be observed is 3.

Key concepts

Understanding Electronic Transitions in Hydrogen

In a hydrogen atom, electrons are arranged in various orbits or energy levels. An electronic transition occurs when an electron moves between these levels, either climbing to higher levels by absorbing energy (excitation) or dropping to lower levels by releasing energy (de-excitation). For example, as electrons return to lower energy states, they release photons of light, which correspond to the spectral lines that we can observe.

Energy levels in hydrogen atoms are quantized, meaning electrons can only occupy specific orbits with fixed amounts of energy. The energy difference between these levels determines the wavelength and frequency of the emitted light. Since each element has a unique set of energy levels, these spectral lines can act like fingerprints to identify elements in space through a process called spectroscopy.

Energy Levels in Hydrogen

The energy levels in a hydrogen atom are determined by the principal quantum number, commonly denoted as 'n'. This number can have integer values starting from 1, which corresponds to the atom's ground state, up to infinity representing free electrons. Higher energy levels have higher 'n' values and are less stable, so an electron in such an orbit is more likely to lose energy (de-excite) to return to a lower energy state.

These energy levels are extremely important because they dictate the possible electronic transitions in an atom, which in turn produce the spectral lines we can observe. When we know the number of energy levels electrons can transition between, we can predict the possible transitions and thus, the number and type of spectral lines that may result.

De-excitation of Electrons

The process of de-excitation in electrons involves a transition from a higher to a lower energy level within an atom. When an electron de-excites, it emits a photon with energy equal to the difference in energy between the two levels involved in the transition. This emitted photon is what we see as a line in a spectrum.

For hydrogen atoms, as mentioned in the example of the original exercise, if the electron descends from the fifth orbit and the first orbit is not available, it has fewer options for this transition, limiting the number of spectral lines produced. These transitions are what give rise to emission spectra, specific types of light emitted depending on the element and energy change involved.

Spectral Lines Calculation Formula

The spectral lines calculation formula is a mathematical relation used to determine the maximum number of spectral lines resulting from electronic transitions between different energy levels in an atom. The formula, given by \( n(n - 1) / 2 \), where \( n \) is the number of available energy levels for transition, is based on combinations of different level transitions.

For example, applying the formula to an electron transitioning from the fifth energy level in a hydrogen atom with the first level unavailable (\