Question

(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

(a) The speed is \(\frac{Z}{n} \cdot v_1\). (b) Largest \(Z\) is 13.

Step-by-step solution

Understanding the Problem

We have a one-electron ion with a nuclear charge Z. We are asked to find the speed of an electron in a Bohr-model orbit labeled with quantum number \(n\). We also need to compare this speed with the orbital speed for hydrogen when \(n = 1\), denoted as \(v_1\). Additionally, we must find the largest value of \(Z\) for which this speed at \(n = 1\) is less than 10\% of the speed of light \(c\).

Applying Bohr's Model for Speed

In the Bohr model, the speed of an electron at an orbit for a hydrogen-like atom with nuclear charge \(Z\) and orbit \(n\) is given by: \[ v = \frac{Z}{n} \cdot v_1 \] where \( v_1 \) is the speed of the electron in the \(n = 1\) orbit for hydrogen, which is approximately \(2.18 \times 10^6 \, \text{m/s}\).

Solving Part (a)

We need to express the speed \(v\) of an electron in a more general term using \(v_1\). Using the expression \( v = \frac{Z}{n} \cdot v_1 \), we find that: \[ v = \frac{Z}{n} \cdot v_1 \] Thus, the speed of the electron at an orbit labeled \(n\) for a nucleus with charge \(Z\) is \( \frac{Z}{n} \cdot v_1.\)

Solving Part (b)

We want \(v\) for \(n = 1\) to be less than 10\% of the speed of light. Hence, \( v_1 \cdot Z < 0.1 \, c \). Using \( c = 3 \times 10^8 \, \text{m/s} \), we have: \[ v_1 \cdot Z < 0.1 \times 3 \times 10^8 \] \[ Z < \frac{0.1 \times 3 \times 10^8}{2.18 \times 10^6} \] Simplifying this inequality gives: \[ Z < 13.76 \]

Final Calculation and Conclusion

Since \(Z\) must be a whole number, the largest integer value satisfying the inequality is \(Z = 13\). Thus, the largest value for the nuclear charge \(Z\) is 13 that keeps the \(n = 1\) orbital speed under 10\% of the speed of light.

Key concepts

Understanding One-Electron Ions

One-electron ions are fascinating entities in the study of atomic structures and spectra. These ions contain only a single electron in orbit around the nucleus, which makes them relatively simple to describe compared to multi-electron atoms. Common examples of one-electron ions include hydrogen-like atoms, where hydrogen
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.

Understanding these ions plays a critical role in fields like quantum mechanics and spectroscopy, enabling scientists to predict energy levels and spectral lines.

Exploring Nuclear Charge (Z)

The nuclear charge, denoted by \(Z\), is a fundamental concept when studying ions and atoms. It represents the total positive charge of the nucleus, which is determined by the number of protons present. The greater the nuclear charge, the stronger its pull on orbiting electrons.

• 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.

In the context of Bohr's model, the nuclear charge is central to predicting outcomes like spectral line positions and electron velocities. This makes \(Z\) crucial in the theoretical and experimental analysis of atomic behaviors.

Defining Electron Speed in Bohr's Model

Electron speed in the Bohr model is critical for understanding atomic behavior, especially in one-electron systems. The model provides a simplified view that calculates the speed of an electron based on its orbit and the nuclear charge \(Z\).
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.

Understanding electron speed helps scientists predict how atoms interact with light and other electromagnetic phenomena, an essential aspect of quantum theory.

Significance of Quantum Numbers

Quantum numbers are numerical values that provide essential insights into the properties of electrons within an atom. In the Bohr model, the principal quantum number \(n\) identifies the energy level of the given electron, which determines its orbit size and energy.

• 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.

Understanding quantum numbers is critical for delving into atomic structures, allowing a clear depiction of how electrons behave under various conditions.