Problem 49 - A Complete Resource Book in Chemistry for JEE Main

Question

When the \(\mathrm{e}^{-}\)of a \(\mathrm{H}\)-atom moves from \(\mathrm{n}=4\) to \(\mathrm{n}=1\) the number of spectral lines emitted is (a) 3 (b) 6 (c) 9 (d) 15

Step-by-Step Solution

Answer

The number of spectral lines emitted is 6.

Step 1: Understanding the Concept

The spectral lines emitted in a hydrogen atom are a consequence of electrons transitioning between different energy levels. When an electron moves from a higher energy level, such as \( n=4 \), to a lower energy level, such as \( n=1 \), it can take multiple paths. Each path represents a different spectral line.

Step 2: Determine Possible Transitions

The electron starting from \( n = 4 \) can move directly to \( n = 1 \), or it can transition to \( n = 3 \) or \( n = 2 \) before finally reaching \( n = 1 \). The general formula to determine the number of spectral lines is given by the combination formula \( \frac{n(n-1)}{2} \), where \( n \) is the initial energy level.

Step 3: Applying the Formula

Using the formula \( \frac{n(n-1)}{2} \) where \( n = 4 \), we calculate the number of possible spectral lines: \[ \frac{4(4-1)}{2} = \frac{4 \times 3}{2} = 6 \]

Step 4: Conclusion

The number of spectral lines emitted when the electron moves from \( n = 4 \) to \( n = 1 \) is 6, which corresponds to option (b).

Key Concepts

Spectral Lines

Spectral lines are like the unique fingerprints of atoms. They are the distinct lines you see in a spectrum, resulting from the emission or absorption of light. Each line corresponds to a specific transition between energy levels within an atom. In simpler terms, when electrons in an atom move from a higher energy level to a lower one, they emit energy. This energy appears as a spectral line. This process is crucial in astronomy for identifying the composition of stars and galaxies far away.

These lines fall into different series, named after the scientists who discovered them, such as Lyman, Balmer, and Paschen series in the hydrogen atom. Each series corresponds to transitions ending at specific energy levels. For example, in the hydrogen atom, transitions ending at the first energy level (n=1) are part of the Lyman series.

Energy Levels

Energy levels in an atom are like different floors in a building, where electrons can "live". Each level corresponds to a specific amount of energy. Higher levels mean more energy. In a hydrogen atom, these are quantized, meaning electrons can only occupy specific levels, not in-between. Each move from one level to another involves absorbing or releasing energy.

This quantization is described by principal quantum numbers, represented by n. For example, n=1 is the lowest and most stable energy level for an electron. Higher numbers indicate higher energy levels. When electrons jump to these levels and drop back, they emit photons of light, which we see as spectral lines. Understanding these levels helps explain many atomic phenomena and is vital in quantum physics.

Electron Transition

Electron transitions are like magical leaps between energy levels. An electron in an atom can jump from one energy level to another. This jump happens when the electron gains or loses the right amount of energy. If it gains energy, it moves to a higher level. When it loses energy, it falls back to a lower level. This is what produces light in the form of spectral lines.

For students, it's key to know that each transition pathway can emit a unique spectral line. In our hydrogen atom example, when an electron goes from the fourth level (n=4) to the first level (n=1), there are several possible paths involving intermediate levels (n=3 or n=2). Each path produces a unique spectral line. The total number of possible transitions can be calculated using the formula \( \frac{n(n-1)}{2} \), where n is the initial energy level. In this case, calculating for n=4 reveals 6 possible transitions, hence 6 spectral lines.