Question

The dynamic mass (in \(\mathrm{kg}\) ) of the photon with a wavelength corresponding to the series limit of the Balmer transitions of the \(\mathrm{He}^{+}\) ion is (a) \(4.22 \times 10^{-36}\) (b) \(2.24 \times 10^{-34}\) (c) \(2.42 \times 10^{-35}\) (d) \(4.22 \times 10^{-35}\)

Short answer

The correct dynamic mass of the photon is not directly calculable from the provided data without the exact value for the series limit wavelength of the Balmer transitions for the Helium ion.

Step-by-step solution

Understanding the concept of dynamic mass of a photon

The dynamic mass of a photon is derived from its energy, which in turn can be obtained from the photon's frequency or wavelength using the energy-wavelength relation: \( E = \frac{hc}{\lambda} \), where \( E \) is the energy, \( h \) is Planck's constant (\(6.626 \times 10^{-34} \mathrm{J\cdot s}\)), \( c \) is the speed of light in vacuum (\(3 \times 10^{8} \mathrm{m/s}\)), and \( \lambda \) is the wavelength of the photon. The mass can then be found using Einstein's equation: \( E = mc^2 \), by solving for \( m \).

Calculate the energy of the photon with given wavelength

Use the relation \( E = \frac{hc}{\lambda} \) to calculate the energy of the photon. Since the wavelength corresponds to the series limit of the Balmer transitions of the \(\mathrm{He}^{+}\) ion, we use the Rydberg formula for hydrogen-like ions: \( \frac{1}{\lambda} = R_Z \left( \frac{1}{2^2} - \frac{1}{n^2} \right) \), where \(R_Z\) is the Rydberg constant for \(Z\) protons (\(Z=2\) for \(\mathrm{He}^{+}\)), and \(n\) goes to infinity for the series limit. The Rydberg constant for hydrogen (\(Z=1\)) is approximately \(1.097 \times 10^7 \mathrm{m}^{-1}\), so for helium it is \(4 \times 1.097 \times 10^7 \mathrm{m}^{-1}\) because \(R_Z = R \times Z^2\).

Compute the dynamic mass of the photon

After finding the energy using the relationship from Step 2, use Einstein's equation \( E = mc^2 \) to find the dynamic mass \( m \) by rearranging the equation as \( m = \frac{E}{c^2} \). Substitute the calculated value of \( E \) into this equation to obtain the mass.

Use the data to calculate the mass

First calculate the wavelength at the series limit of the Balmer transitions for the helium ion. With the series limit meaning \( n \) approaches infinity and using \( R_Z \) for helium, we find \( \lambda \). Then, substitute \( \lambda \) into \( E = \frac{hc}{\lambda} \) to calculate the energy, and use that energy in \( m = \frac{E}{c^2} \) to find the dynamic mass of the photon.

Compare the result with the given options

Calculate the dynamic mass and check which of the given options matches the calculated result.

Key concepts

Energy-Wavelength Relation

Discussing the energy-wavelength relation is essential in understanding the dynamics of photons. Energy of a photon, which is a quantum of light, is inversely proportional to its wavelength. This relation can be described by the equation, \( E = \frac{hc}{\lambda} \), known as the Planck-Einstein relation, where \( E \) represents the photon's energy, \( h \) is Planck's constant, \( c \) stands for the speed of light, and \( \lambda \) is the photon's wavelength.

It shows that shorter wavelengths correspond to higher energy photons and vice versa. When working through physics problems, you might encounter situations where you need to calculate the energy of a photon from its wavelength, especially in spectroscopy and the study of atomic transitions, such as the Balmer series in hydrogen-like ions. It is this fundamental principle that allows us to delve deeper into the properties of light and its interactions with matter.

Einstein's Mass-Energy Equivalence

One cannot explore the dynamic mass of a photon without referring to Einstein's mass-energy equivalence, encapsulated in the iconic equation, \( E = mc^2 \). This profound relation tells us that mass (\( m \) ) and energy (\( E \) ) are interchangeable; a particle with mass possesses energy by virtue of its mass and vice versa.

For photons, which are massless particles when stationary, we refer to the concept of 'dynamic mass' to describe the effective mass they exhibit due to their energy when in motion. As the speed of light (\( c \) ) is a constant, the equation emphasizes that a small amount of mass corresponds to a large amount of energy. This equivalence is particularly important in particle physics and cosmology, where energy often manifests as the 'mass' of particles traveling at the speed of light.

Rydberg Formula for Hydrogen-like Ions

The Rydberg formula is fundamental in atomic physics, particularly when examining the spectral lines of hydrogen-like ions. The original formula is used to predict the wavelengths of the photons emitted or absorbed by electrons transitioning between energy levels in a hydrogen atom.

For hydrogen-like ions, the formula is adjusted to account for the difference in the number of protons. The modified formula is expressed as \( \frac{1}{\lambda} = R_Z \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \), where \( R_Z \) is the Rydberg constant adjusted for the atomic number \( Z \) of the ion, \( \lambda \) is the wavelength of the emitted or absorbed light, and \( n_1 \) and \( n_2 \) are the principal quantum numbers of the electron's initial and final energy levels, respectively.

For the case of helium ions (\( \mathrm{He}^{+} \)), the atomic number \( Z \) is 2, and the Rydberg constant becomes \( R_Z = R \times Z^2 \), effectively quadrupling the value of the constant for hydrogen. This formula provides the basis for calculating the wavelengths at the series limit of spectral lines for such ions, as seen in the context of the Balmer series. When \( n_2 \) approaches infinity, it represents the highest possible energy transition, resulting in a series limit that can be used to calculate the corresponding photon's wavelength and, subsequently, its dynamic mass.