Question

Match the constants given in column 1 with their values given in column Ii and mark the appropriate choice. $$ \begin{array}{|l|l|l|l|} \hline {\text { Column I }} && {\text { Column II }} \\ \hline \text { (A) } & \text { Rydberg constant } & \text { (i) } & 6.626 \times 10^{-34} \mathrm{~J} \mathrm{~s} \\ \hline \text { (B) } & \text { Planck's constant } & \text { (ii) } & 3.00 \times 10^{8} \mathrm{~m} \mathrm{~s}^{-1} \\ \hline \text { (C) } & \text { Velocity of light } & \text { (iii) } & 750 \times 10^{-9} \mathrm{~m} \\ \hline \text { (D) } & \begin{array}{l} \text { Wavelength of red } \\ \text { light } \end{array} & \text { (iv) } & 109,677 \mathrm{~cm}^{-1} \\ \hline \end{array} $$

Short answer

A-iv, B-i, C-ii, D-iii

Step-by-step solution

Identify Each Constant

Match each physical constant (Rydberg constant, Planck's constant, Velocity of light, Wavelength of red light) with its measurement value from the provided data.

Match Planck's Constant

Planck's constant is universally recognised by its value of approximately \(6.626 \times 10^{-34} \mathrm{J} \cdot \mathrm{s}\), which corresponds to Column II (i). Therefore, match (B) with (i).

Match Velocity of Light

The velocity of light in a vacuum is a fundamental physical constant with the value of about \(3.00 \times 10^{8} \mathrm{m} \cdot \mathrm{s}^{-1}\), which correlates to Column II (ii). Hence, pair (C) with (ii).

Match Rydberg Constant

The Rydberg constant is known to have a value of approximately \(109,677 \mathrm{cm}^{-1}\), which matches with Column II (iv). Therefore, link (A) with (iv).

Match Wavelength of Red Light

The wavelength range for red light is approximately between \(620 \) and \(750 \times 10^{-9} \mathrm{m}\), corresponding to Column II (iii). So, we need to connect (D) with (iii).

Key concepts

Rydberg Constant

The Rydberg constant is a physical constant important in quantum physics, particularly when dealing with the spectroscopy of hydrogen and other atoms. Named after the Swedish physicist Johannes Rydberg, this constant signifies the highest wavenumber (inverse wavelength) that can be emitted from an atom.

It plays a crucial role in the Rydberg formula, which predicts the wavelengths of the spectral lines of many chemical elements. The Rydberg constant is approximately valued at \(109,677 \mathrm{cm}^{-1}\), referring to the reciprocal meter as the unit of energy in spectroscopy. This value corresponds to the energy difference between the infinitely high energy level and the ground state of a hydrogen atom.

Understanding this constant allows for a deeper comprehension of atomic structures and the energy transitions that occur within an atom when an electron jumps between energy levels, emitting or absorbing energy in the form of light.

Planck's Constant

Planck's constant is a fundamental constant that plays a pivotal role in quantum mechanics. It is denoted by \(h\) and has a value of approximately \(6.626 \times 10^{-34} \mathrm{J} \cdot \mathrm{s}\).

Max Planck, a German physicist, introduced this constant in 1900, leading to the birth of quantum theory. It relates the energy of a photon to its frequency, as expressed by the equation \(E = hu\), where \(E\) is energy, \(h\) is Planck's constant, and \(u\) is the frequency of the photon.

Planck's constant is essential in understanding phenomena at atomic and sub-atomic scales, such as the photoelectric effect, where light can be used to eject electrons from a material. It bridges the gap between the macroscopic classical physics and the microscopic quantum world, allowing scientists to calculate the energy levels and behavior of particles at the quantum level.

Velocity of Light

The velocity of light, often symbolized as \(c\), is a universal physical constant that is crucial in the fields of physics and astronomy. In a vacuum, light travels at an incredible speed of roughly \(3.00 \times 10^{8} \mathrm{m} \cdot \mathrm{s}^{-1}\), or about 299,792 kilometers per second.

This speed serves as the cosmic speed limit for the propagation of all forms of electromagnetic radiation, including light, and it's a foundational pillar in the theory of relativity proposed by Albert Einstein. He showed that the laws of physics are the same for all non-accelerating observers and that the speed of light within a vacuum is the same no matter the speed at which an observer travels.

As such, the constancy of the speed of light has profound implications on our understanding of space and time, leading to the realization that time can dilate and lengths can contract depending on the relative speeds of observers and objects in motion.

Wavelength of Red Light

The wavelength of red light refers to the distance between two consecutive peaks or troughs in the light wave, specifically within the red portion of the visible spectrum. In general, red light wavelengths fall within the range of approximately \(620\) to \(750 \times 10^{-9} \mathrm{m}\), which is its approximate value represented in nanometers (nm).

This range fits into the broader spectrum of visible light, which extends from about 380 nm to 750 nm, and is the part of the electromagnetic spectrum that can be perceived by the human eye. Red light, with its longer wavelength, sits at one end of the visible spectrum, opposite the shorter-wavelength violet light.

Wavelength is a key concept in understanding various optical phenomena such as diffraction, interference, and the color properties of materials. For example, red light's longer wavelengths allow it to pass through atmospheric obstacles more easily than blue light, which is why the sky is blue during the day and tends toward red hues during sunrise and sunset when the light travels through a greater thickness of the atmosphere.