Question

What is the colour corresponding to the wavelength of light emitted when the electron in a hydrogen atom undergoes transition from \(n=4\) to \(n=2 ?\) (a) Blue (b) Red (c) Yellow (d) Green

Short answer

The color corresponding to the transition from \(n=4\) to \(n=2\) in a hydrogen atom is typically red.

Step-by-step solution

Understanding the concept of electronic transition

When an electron in an atom moves from a higher energy level (higher principal quantum number, n) to a lower energy level, it emits energy in the form of light. The energy difference between the two levels determines the wavelength and thus the color of the emitted light.

Using the Rydberg formula to calculate the wavelength

The Rydberg formula can be used to calculate the wavelength (\(\backslash lambda\) of the emitted light: \[1/\lambda = R_H * (1/n_{1}^{2} - 1/n_{2}^{2})\] where \(R_H\) is the Rydberg constant (approximately \(1.097 \times 10^7 \textrm{m}^{-1}\)), \(n_1\) is the lower energy level, and \(n_2\) is the higher energy level.

Calculating the wavelength for the given energy levels

Plug in the values for the transition given, with \(n_1 = 2\) and \(n_2 = 4\), into the Rydberg formula to find the wavelength of the emitted photon.

Relating wavelength to color

Once the wavelength is determined, compare it with the known spectral lines for the hydrogen atom to find the corresponding color. Wavelengths around 400-450 nm correspond to the violet/blue end of the spectrum, 450-495 nm to blue, 495-570 nm to green, 570-590 nm to yellow, 590-620 nm to orange, and 620-750 nm to red.

Key concepts

Electronic Transition

In the context of an atom, an electronic transition refers to the movement of an electron from one energy level to another within an atom. This concept is a crucial piece in understanding the color of light emitted or absorbed by an atom. When an electron jumps from a higher energy level (a state with a higher principal quantum number, denoted by n), to a lower energy level, it releases energy. Conversely, when it moves to a higher energy level, it absorbs energy. The amount of energy involved in this jump correlates to specific wavelengths and colors of light in the electromagnetic spectrum.

Rydberg Formula

The Rydberg formula is a mathematical equation used to predict the wavelength of light resulting from an electronic transition in hydrogen-like atoms. The formula is expressed as:
\[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_{1}^2} - \frac{1}{n_{2}^2} \right) \]
where \( \lambda \) is the wavelength, \( R_H \) is the Rydberg constant (approximately \( 1.097 \times 10^7 m^{-1} \)), \( n_1 \) is the principal quantum number of the lower energy level, and \( n_2 \) is the principal quantum number of the higher energy level. The Rydberg formula is an essential tool for calculating the wavelengths (and therefore the colors) of light that an electron emits when it transitions between energy levels.

Principal Quantum Number

The principal quantum number, symbolized by n, is a positive integer that signifies the main energy level occupied by an electron in an atom. It is a fundamental term in quantum mechanics that determines the size and energy of an electron's orbit. The principal quantum serves as an address for indicating the location of an electron, with energy increasing as n becomes larger. In simpler terms, the larger the value of n, the further an electron is from the nucleus, and the more energetic it is. It plays a role in the electronic transition as the energy difference between levels – and thus the color of emitted light – is related to changes in this number.

Emission Spectrum

An emission spectrum is a series of bright lines or bands on a dark background. It is produced when the photons emitted by a substance are passed through a prism or diffraction grating. Spectra are like fingerprints for elements, with each element having a unique spectral line pattern. The emission spectrum of hydrogen, for instance, consists of various lines that correspond to different electronic transitions. These lines occur due to electrons falling from higher to lower principal quantum levels and are observed as discrete wavelengths of color. By analyzing the emission spectrum, scientists can determine the composition of distant stars and the motion of galaxies, making it a powerful tool in both chemistry and astrophysics.