Question

Which of the following statements is wrong? (1) In the hydrogen spectrum least energetic series is Pfund series. (2) The lines of longest wavelength in the Balmer series correspond to the transition between \(n=3\) and \(n=2\) levels. (3) The dark lines in a spectrum are produced by existing gases to very high energy levels. (4) The wave number of infinity line in Lyman series of hydrogen spectrum is \(9 \mathrm{R} / 3\).

Short answer

Statement 3 is wrong because the dark lines are produced by absorption, not by exciting gases to very high energy levels.

Step-by-step solution

- Understand the Pfund Series (Statement 1)

The Pfund series refers to the transitions of electrons in a hydrogen atom where the final state is the fifth energy level (=5). The series generally has lower energy transitions compared to other series because it involves higher energy levels.

- Analyze the Longest Wavelength in Balmer Series (Statement 2)

The Balmer series corresponds to electron transitions where the final energy level is =2. The line of longest wavelength in this series is the transition from =3 to =2.

- Examine the Dark Lines in a Spectrum (Statement 3)

Dark lines, or absorption lines, appear in a spectrum when light passes through a cooler gas, and atoms or molecules in the gas absorb light at specific wavelengths. This doesn't involve the gas reaching very high energy levels, making the statement false.

- Calculate the Wave Number of the Infinity Line in the Lyman Series (Statement 4)

For the Lyman series, the transitions end at =1. The wave number for the infinity line (=_∞) in the Lyman series is calculated using the formula: \( \text{wave number} = R \left(1 - \frac {1}{n^2}\right) \). For =1, it simplifies to =R. Therefore, \( 9 \text{R} / 3 \) is incorrect as it must be = R.

Key concepts

Pfund series

The Pfund series is one of the spectral lines in the hydrogen spectrum. It occurs when electrons transition to the fifth energy level, denoted as \( n=5 \). This series involves electron transitions from higher energy levels (\( n>5 \)) down to \( n=5 \).

Because these transitions involve higher initial energy levels and higher principal quantum numbers, they result in lower energy emissions compared to other series. Consequently, the photons emitted in the Pfund series have longer wavelengths.

This is why the Pfund series is often referred to as the least energetic series in the hydrogen spectrum.

Balmer series

The Balmer series involves transitions of electrons in a hydrogen atom where the final energy state is the second energy level, \( n=2 \). These transitions are from higher energy levels (\( n>2 \)) down to \( n=2 \).

The Balmer series is unique because it falls in the visible light spectrum, making it observable to the naked eye. The line with the longest wavelength in the Balmer series is due to the transition from \( n=3 \) to \( n=2 \).

Since the energy difference between these two levels is relatively small, the resulting photon has a longer wavelength and appears in the red part of the visible spectrum. This specific line is often referred to as H-alpha.

Lyman series

The Lyman series describes the transitions of electrons in which the final energy state is the first energy level, \( n=1 \). These transitions are from higher energy levels (\( n>1 \)) down to \( n=1 \).

The Lyman series is in the ultraviolet region of the electromagnetic spectrum, making it invisible to the naked eye. The formula for the wave number \( u \) for the Lyman series is given by: \[ u = R \times \left(1 - \frac{1}{n^2}\right) \] where \( R \) is the Rydberg constant, and \( n \) is the initial energy level.

For the transition from an infinitely high level (\( n=\infty \)) to \( n=1 \), the equation simplifies to \( u = R \), meaning that the wave number of the infinity line in the Lyman series is simply \( R \).

Absorption lines

Absorption lines appear in the spectrum of a light source when light passes through a cooler gas. The gas absorbs certain wavelengths, causing dark lines to appear in the continuous spectrum.

These lines are characteristic of the elements present in the gas because each element absorbs light at specific wavelengths. This relates to the energy needed to excite electrons from a lower energy level to a higher one within the atoms or molecules in the gas.

It is important to note that the gas does not need to be at very high energy levels to produce absorption lines. Instead, it absorbs just enough energy to move electrons to higher states temporarily.