Chapter 39: Problem 26
(a) For one-electron ions with nuclear charge Z, what is the speed of the electron in a Bohr-model orbit labeled with \(n\)? Give your answer in terms of \(v_1\), the orbital speed for the \(n\) = 1 Bohr orbit in hydrogen. (b) What is the largest value of Z for which the \(n\) = 1 orbital speed is less than 10\(\%\) of the speed of light in vacuum?
Short Answer
Step by step solution
Understanding the Problem
Applying Bohr's Model for Speed
Solving Part (a)
Solving Part (b)
Final Calculation and Conclusion
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding One-Electron Ions
loses its only electron, or a helium atom with only one electron remaining.
- Hydrogen-like ions share similarities with the hydrogen atom, the simplest atom in the universe, but with different nuclear charges denoted by the symbol \(Z\).
- The Bohr model often applies to such ions, offering insight into the behavior of the remaining electron as it encircles the nucleus.
Exploring Nuclear Charge (Z)
- In a hydrogen-like system or one-electron ion, \(Z\) affects the electron's speed and energy levels significantly.
- A higher \(Z\) implies a stronger attraction between the electron and the nucleus, resulting in higher speeds for the electron in its orbit.
- By knowing \(Z\), it is possible to calculate various physical properties such as the likelihood of an electron transitioning between different energy levels.
Defining Electron Speed in Bohr's Model
Using the formula:\[ v = \frac{Z}{n} \cdot v_1 \]where \( v_1 \) is the base speed in the first orbit for hydrogen, we can derive how fast an electron moves
depending on its energy level \(n\) and \(Z\).
- Electron speed increases with higher nuclear charges \(Z\) and decreases at higher quantum numbers \(n\).
- This intricate relationship outlines why heavier elements (with greater \(Z\)) show different spectra than lighter ones.
Significance of Quantum Numbers
- The principal quantum number \(n\) affects both the radius of the electron's orbit and its speed, directly influencing its energy state.
- Higher values of \(n\) correspond to orbits further from the nucleus, implying lower velocities and energies compared to smaller \(n\) values.
- Quantum numbers help distinguish between different energy levels and predict spectral line appearances in atomic spectroscopy.