Question

The emission spectrum of hydrogen is found to satisfy the expression for the energy change \(\Delta E\) (in joules) such that \(\Delta E=2.18 \times 10^{-18}\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right) J\) where \(n_{1}=1,2,3, \ldots .\) and \(n_{2}=2,3,4\). The spectral lines corresponds to Paschen series if (a) \(n_{1}=1\) and \(n_{2}=2,3,4\) (b) \(n_{1}=3\) and \(n_{2}=4,5,6\) (c) \(n_{1}=1\) and \(n_{2}=3,4,5\) (d) \(n_{1}=2\) and \(n_{2}=3,4,5\)

Short answer

(b) is the correct answer because the Paschen series corresponds to transitions where n1 = 3.

Step-by-step solution

Understanding the Paschen Series

The Paschen series in hydrogen is an emission series that occurs when an electron falls from a higher energy level (n2) to n1 = 3. Therefore, the correct answer must have n1 = 3.

Analyze the Given Options

Evaluate each option to see which one corresponds with the condition for the Paschen series. Option (a) is incorrect because it describes the Lyman series, where n1 = 1. Option (b) is correct because it fits the condition n1 = 3 for the Paschen series. Option (c) describes part of the Balmer series for n1 = 1. Option (d) is incorrect because it corresponds with n1 = 2, which is the Balmer series. Only option (b) satisfies the Paschen series condition.

Key concepts

Hydrogen Emission Spectrum

The hydrogen emission spectrum plays a central role in understanding how atoms behave. When hydrogen atoms get excited—typically by receiving energy in some form—they don't stay that way for long. Excited electrons make a beeline back to lower energy levels, releasing light in the process. This light, when passed through a prism or a diffraction grating, separates into a series of lines called the atomic emission spectrum. Each line corresponds to photons of a specific wavelength, and together, they provide a unique signature for hydrogen.

Interesting Facts

• This emission spectrum is how we can tell what stars are made of, even though they are light-years away.
• The hydrogen spectrum consists of lines in the ultraviolet (Lyman series), visible (Balmer series), and infrared (Paschen, Brackett, and Pfund series) regions.

Understanding this spectrum is like unlocking the atomic fingerprint of hydrogen, allowing scientists to study the structure and properties of hydrogen atoms in detail.

Energy Level Transitions

Energy level transitions are at the heart of the phenomena within the emission spectrum. Think of an atom as a tiny, energetic solar system. Electrons orbit the nucleus in set paths, or orbits, each with a specific energy level. The closest orbit to the nucleus has the lowest energy, and levels increase the further out you go.

When electrons absorb energy, they jump from a lower to a higher energy level in what we call 'excitation'. Losing this energy causes them to fall back down, and they emit energy in the form of light.

The Quantum Leap

The fascinating bit is that these jumps are quantized, meaning electrons can only inhabit certain levels, not in-between. The energy difference between these levels is precisely the energy carried by the photon released as visible light or other electromagnetic radiation. The greater the leap, the more energy the photon carries—a principle essential when calculating spectral lines in the hydrogen emission spectrum.

Spectral Lines

Spectral lines are like the universe's barcodes. They provide us with detailed information about the composition and properties of stars and gases in space. When electrons transition between energy levels, they emit light that manifests as spectral lines in a spectrum.

Every element has its own distinct set of spectral lines, acting as its cosmic identifier. For hydrogen, the Paschen series refers to the infrared spectral lines produced when electrons fall to the third energy level from higher levels. Just remember that each series—like the Lyman, Balmer, and Paschen series—is named after the scientist who discovered them, giving us a nod to the rich history of scientific discovery.

Identifying Elements

Using these lines, astronomers can identify which elements are present in a star or nebula, and chemists can ascertain the identity of substances here on Earth. Conclusively, by decoding the message within these lines, we unravel the mysteries of the atomic makeup of the universe.