Question

If the mass of the clectron is reduced to half, the Rydberg constant (1) Remains unchanged (2) Becomes half (3) Becomes double (4) Becomes one-fourth \(\left[\right.\) Hint \(\left.: R_{H}=\frac{2 \pi^{2} m e^{4}}{c h^{3}}\right]\)

Short answer

The Rydberg constant becomes half.

Step-by-step solution

Understanding the given hint

The hint provides the formula for the Rydberg constant: \[ R_{H} = \frac{2 \pi^{2} m e^{4}}{ch^{3}} \] where \(m\) is the mass of the electron, \(e\) is the charge of the electron, \(c\) is the speed of light, and \(h\) is Planck's constant.

Identify the variable of interest

In this context, the mass of the electron \(m\) is being considered. According to the problem, the mass of the electron is reduced to half.

Plug in the new mass into the formula

Substitute \( m' = \frac{m}{2} \) into the formula: \[ R_{H}' = \frac{2 \pi^{2} \left(\frac{m}{2}\right) e^{4}}{ch^{3}} \] Simplify the expression: \[ R_{H}' = \frac{2 \pi^{2} m e^{4}}{2ch^{3}} = \frac{1}{2} \times \frac{2 \ pi^{2} m e^{4}}{ch^{3}} = \frac{R_{H}}{2} \]

Conclusion

From the simplification, it is evident that the new Rydberg constant \( R_{H}' \) is half the original Rydberg constant \( R_{H} \). Therefore, the Rydberg constant becomes half when the mass of the electron is reduced to half.

Key concepts

Electron Mass

Electron mass is a fundamental property of electrons. It’s a measure of how much matter is in an electron. The symbol for the mass of an electron is usually denoted as 'm'.
The standard mass of an electron is approximately 9.109 x 10^-31 kilograms. But why does mass matter in physics? Because it directly affects the behavior and characteristics of particles.
In the context of the Rydberg constant, which is used to predict the wavelength of light from electron transitions in atoms, the electron mass plays a crucial role. When the mass changes, it affects the energy levels of the electron and thus the emitted and absorbed wavelengths of light.
Understanding the mass and its influence helps grasp how electronic transitions can be predicted and measured accurately.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of particles at the atomic and subatomic levels. It explains phenomena that classical physics cannot, such as the dual nature of particles, being both waves and particles simultaneously.
In quantum mechanics, particles like electrons have quantized energy levels. This means that electrons can only occupy specific energy levels and the energy transition between these levels emits or absorbs photons of specific wavelengths. This is fundamentally important in understanding why the Rydberg constant is so reliable in atomic physics.
The formulas and principles of quantum mechanics are essential in predicting the properties and behaviors of tiny particles, including how they interact within an atom.

Atomic Physics

Atomic physics focuses on the structure of the atom and how the electrons within an atom behave. It is the study of the electromagnetic forces that bind the electrons to the nucleus and how these electrons transit between different energy levels.
The Rydberg constant comes into play in atomic physics when dealing with the spectra of emitted or absorbed light from electron transitions. It's a derived constant that helps in calculating the wavelengths of spectral lines for hydrogen and other elements.
One must grasp the atomic structure and how electron transitions emit quantifiable wavelengths of light to strictly understand how the Rydberg constant serves in spectral analysis.

Formulas in Physics

Formulas in physics are like recipes for understanding the natural world. They help us calculate and predict various physical phenomena accurately. The Rydberg formula, given by \( R_{H} = \frac{2 \pi^{2} m e^{4}}{ch^{3}} \), is one such crucial formula in atomic physics.
This formula connects several constants and properties, including the mass of the electron (m), the charge of the electron (e), the speed of light (c), and Planck's constant (h). By manipulating these variables, one can explain and predict the wavelengths of spectral lines.
Changes in any of these values, like the electron mass, make understanding and utilizing the Rydberg constant even more exciting and important for physicists. Formulas provide a straightforward method to decode and solve complex physical interactions in the universe.