Question

The radio of minimum to maximum wave length in Balmer series is (A) \((1 / 4)\) (B) \((5 / 36)\) (C) \((3 / 4)\) (D) \((5 / 9)\)

Short answer

The ratio of minimum to maximum wavelength in the Balmer series is \(\frac{5}{9}\).

Step-by-step solution

Identify the equation for the Balmer series

The formula for the Balmer series is given by the Rydberg formula: \[\frac{1}{\lambda} = R_H \left(\frac{1}{2^2} - \frac{1}{n^2}\right)\] where \(\lambda\) is the wavelength, \(R_H\) is the Rydberg constant for hydrogen, and \(n\) is the energy level of the electron. The transitions will occur when the electron falls from higher energy levels to the second energy level where the minimum and maximum wavelengths can be determined.

Determine the minimum wavelength

The minimum wavelength corresponds to the highest possible energy transition, which occurs when the electron falls from an infinitely high energy level (\(n \rightarrow \infty\)) to the second energy level (n=2). Substitute \(n \rightarrow \infty\) into the equation to find the minimum wavelength, \(\lambda_{min}\): \[\frac{1}{\lambda_{min}} = R_H \left(\frac{1}{2^2} - \frac{1}{\infty^2}\right)\] \[\frac{1}{\lambda_{min}} = \frac{R_H}{4}\] \[\lambda_{min} = \frac{4}{R_H}\]

Determine the maximum wavelength

The maximum wavelength corresponds to the lowest possible energy transition, which occurs when the electron falls from the third energy level (n=3) to the second energy level (n=2). Substitute n=3 into the equation to find the maximum wavelength, \(\lambda_{max}\): \[\frac{1}{\lambda_{max}} = R_H \left(\frac{1}{2^2} - \frac{1}{3^2}\right)\] \[\frac{1}{\lambda_{max}} = R_H \left(\frac{1}{4} - \frac{1}{9}\right)\] \[\frac{1}{\lambda_{max}} = \frac{5R_H}{36}\] \[\lambda_{max} = \frac{36}{5R_H}\]

Calculate the ratio of minimum to maximum wavelength

Divide the minimum wavelength by the maximum wavelength to find the ratio: \[\frac{\lambda_{min}}{\lambda_{max}} = \frac{4/R_H}{36/5R_H} = \frac{4}{36}\times\frac{5R_H}{R_H}\] The ratio simplifies to: \[\frac{\lambda_{min}}{\lambda_{max}} = \frac{5}{9}\] Thus, the correct answer is (D) \(\frac{5}{9}\).

Key concepts

Rydberg formula

The Rydberg formula is at the core of understanding the Balmer series and the spectral lines of hydrogen. This formula calculates the wavelengths of the spectral lines in many chemical elements, but it's primarily used in hydrogen spectroscopy. The general form of the Rydberg formula for hydrogen is: \[\frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)\] where \( \lambda \) is the wavelength of the light emitted or absorbed, \( R_H \) is the Rydberg constant specific to hydrogen, \( n_1 \) is the lower energy level, and \( n_2 \) is the higher energy level.

• The Rydberg constant \( R_H \approx 1.097 \times 10^7 \text{ m}^{-1} \)
• This formula helps predict the wavelengths of light in the spectrum based on energy level transitions in atoms.
• The concept of energy levels \( n_1 \) and \( n_2 \) corresponds to the principal quantum number.

For the Balmer series, \( n_1 = 2 \) and \( n_2 \) can be any whole number greater than 2. This formula is crucial for understanding why we see particular lines in hydrogen’s emission spectrum.

wavelength

Wavelength is the distance between two peaks of a wave, such as light or sound, and is a central concept in the world of physics. It is commonly represented by the Greek letter \( \lambda \). In the context of the Rydberg formula and the Balmer series, wavelength refers to the specific distances of light that are emitted when the electron transitions between energy levels in a hydrogen atom.

• Shorter wavelengths correspond to higher energy transitions.
• Longer wavelengths correspond to lower energy transitions.
• The range of visible light has wavelengths between approximately 400 nm (violet) and 700 nm (red).

When calculating the wavelength for the Balmer series, we are determining the light wavelengths emitted due to electron transitions to the second energy level, which produces visible spectral lines that we often associate with hydrogen’s characteristic colors.

energy levels

Energy levels, also known as electron shells, are the fixed distances from the nucleus of an atom where the electrons may be found. In quantum mechanics, these levels are quantized and are described by whole numbers called principal quantum numbers \( n \). The energy levels grow as you move away from the nucleus.

• Each energy level corresponds to a specific amount of energy that an electron can have.
• Electrons transitioning from higher to lower energy levels emit photons—particles of light—with energies equal to the difference between the two levels.
• In the Balmer series, electrons drop from higher energy levels (\( n > 2 \)) to the second energy level (\( n = 2 \)), which makes these transitions particularly important for visible spectroscopy.

Understanding these energy transitions is essential for exploring atomic physics and helps explain phenomena like spectral lines in various atomic spectra. It demonstrates how quantized energy levels translate into the discrete wavelengths of radiation emitted in atomic phenomena.

hydrogen spectrum

The hydrogen spectrum is an excellent example of how quantum mechanics and atomic physics exhibit themselves in the natural world. It is composed of various spectral lines, each representing electron transitions between different energy levels in a hydrogen atom. The Balmer series is part of this spectrum and is notably visible to the human eye.

• The spectral lines are divided into series based on the lower energy level involved in the transition.
• The Balmer series corresponds to visible light transitions falling to the second energy level (\( n = 2 \)).
• Other series include the Lyman series (transitions to \( n = 1 \), ultraviolet), the Paschen series (transitions to \( n = 3 \), infrared), and more.

Spectral lines are crucial as they act like a fingerprint for elements, helping scientists determine composition and behavior of astronomical objects. Each line’s position within the spectrum corresponds to specific electron transitions explained by quantum mechanics, providing a deep insight into atomic structures and interactions.