Question

Number of possible spectral lines in the bracket series in hydrogen spectrum, when electrons present in the ninth excited state return to the ground state, is (a) 36 (b) 45 (c) 5 (d) 6

Short answer

None of the given options (a) 36, (b) 45, (c) 5, (d) 6 is correct. The number of possible spectral lines is 21.

Step-by-step solution

Understanding the Brackett Series

The Brackett series in the hydrogen spectrum corresponds to the transitions of electrons from higher energy levels to the energy level with principal quantum number 4 (n=4). Since we need to calculate the number of spectral lines when an electron drops from the ninth excited state, we first need to recognize that the ninth excited state refers to the principal quantum number, n=10 (since n=1 is the ground state).

Using the Formula for Number of Spectral Lines

The number of spectral lines that can be observed when an electron drops from a higher energy level to a lower energy level is given by the formula: Number of spectral lines = \( \frac{n(n-1)}{2} \), where n is the number of energy levels involved in the transition. Here, transitions are from n=10 down to n=4.

Calculating the Number of Spectral Lines

Substitute n=7 into the formula to calculate the number of spectral lines, as the levels from n=10 to n=4 (inclusive) are involved: Number of spectral lines = \( \frac{7(7-1)}{2} \) = \( \frac{7 \times 6}{2} \) = 21.

Identifying the Correct Answer

None of the given options matches the calculated number of 21. There seems to be an error in the given options or a misunderstanding of the Brackett series range. For the Brackett series, electron transitions should end at n=4, and the electron is initially at the ninth excited state (n=10), so we consider levels from 10 to 4.

Key concepts

Spectral Lines Calculation

Understanding how to calculate the number of spectral lines in atomic transitions is fundamental in quantum chemistry. The appearance of spectral lines is due to the movement of electrons between different energy levels within an atom. When an electron transitions from a higher energy level to a lower one, it emits energy in the form of light, which we observe as a spectral line. The Brackett series in the hydrogen spectrum is a specific case of such transitions, where electrons fall to the fourth energy level.

The formula for computing the number of spectral lines is given by \( \frac{n(n-1)}{2} \), where \( n \) is the total number of energy levels involved, including both the initial and final levels. This allows us to predict the number of possible lines seen in a spectrum when an electron falls from any given level to a lower level in an atom. Considering the electron transitions within the Brackett series specifically, there is a clear process to determine the correct number of lines, which is crucial for interpreting spectral data accurately.

Principal Quantum Number

The principal quantum number, designated as \( n \), describes the size of the electron orbitals and therefore the energy levels of an atom. It can take any positive integer value, with each value corresponding to a different energy level or 'shell'. In the context of the hydrogen atom, where \( n = 1 \) is the ground state, increasing values of \( n \) represent higher energy levels or excited states.

For the ninth excited state of hydrogen, we refer to \( n = 10 \) as excited states are counted from the first state above the ground state, which is \( n = 2 \) for the first excited state. The principal quantum number is essential for predicting the electron transitions that result in spectral lines as well as understanding the atom's electronic structure. When dealing with spectral lines calculations, recognizing the correct values of \( n \) for the transitions in question is vital.

Electron Transitions in Hydrogen Atom

In the hydrogen atom, electron transitions are movements that occur from one energy level to another. These transitions emit or absorb energy, which matches the energy difference between the two levels. In spectroscopy, these transitions are represented as lines – absorption lines if energy is absorbed when the electron jumps to a higher energy level, and emission lines if energy is emitted as light when the electron falls to a lower energy level.

The Brackett series of the hydrogen spectrum consists of emission lines that are caused by transitions from higher energy levels down to the fourth principal quantum level (\( n = 4 \) is the Brackett series ground state). An understanding of these transitions is crucial for anyone studying quantum mechanics or interpreting the hydrogen emission spectrum, as each transition level corresponds to a specific spectral line, whose wavelength can be calculated using the Rydberg formula.

Excited States in Quantum Chemistry

When discussing excited states in quantum chemistry, we refer to the state of an atom or molecule with a higher energy level than its lowest energy state, known as the ground state. An electron in an excited state is at one of these higher energy levels, having absorbed energy sufficient to promote it from the ground state.

This concept is key to understanding electron transitions and spectral lines, as the arrangement and behavior of electrons in these excited states determine the unique spectral fingerprints of elements. When an electron returns to a lower energy level from an excited state, it emits a photon with energy characteristic of the transition. The study of these energy transitions and the resulting spectra provide critical information about the structure and characteristics of atoms and molecules in various states of excitation.