Question

Section 38.2 established that an electron, if observed in the ground state of hydrogen, would be expected to have an observed speed of \(0.0073 c .\) For what atomic charge \(Z\) would an innermost electron have a speed of approximately \(0.500 c,\) when considered classically?

Short answer

#Short Answer# The atomic charge Z for which an electron in the innermost orbit would have a speed of approximately 0.500 c is approximately 69. This result is based on classical considerations using the Bohr model of the hydrogen atom.

Step-by-step solution

Calculate the speed of the electron in terms of atomic charge Z

To find the relationship between the speed of the electron and the atomic charge Z, we can use the following equation for the ground state of hydrogen atom (Bohr model): \(v = \frac{Z e^2}{2 \hbar}\) Now, let's rewrite this equation in terms of the speed of light c: \(v = \frac{Z e^2}{2 \hbar} \cdot \frac{c}{c}\) Here, e is the elementary charge, and ħ (h-bar) is the reduced Planck constant.

Rewrite the equation in terms of given information

We are given that the observed speed of an electron in the ground state of hydrogen is 0.0073 c. Let's use this to rewrite our equation as: \(v = \frac{Z e^2}{(2 \hbar)} \cdot \frac{0.0073 c}{(Z_{hydrogen} e^2)}\) Since Z_hydrogen = 1 (atomic charge for hydrogen), \(v = \frac{Z 0.0073 c}{2 \hbar} \cdot \frac{e^2}{(\hbar)}\) We can simplify this by dividing both sides by 0.0073 c: \(\frac{v}{0.0073 c} = \frac{Z e^2}{e^2}\)

Solve for Z when the electron's speed is 0.500 c

Now, we need to find the atomic charge Z when the electron's speed is 0.500 c. Substitute v = 0.500 c into our equation: \(\frac{0.500 c}{0.0073 c} = Z\) Dividing the speeds, we get: \(Z = \frac{0.500}{0.0073} \approx 68.5\) Since atomic charges are integers, we can round up the result to the nearest whole number: \(Z \approx 69\) Thus, according to classical considerations, the innermost electron would have a speed of approximately 0.500 c for an atomic charge Z of approximately 69.

Key concepts

Atomic Charge

The atomic charge, denoted as Z, refers to the number of protons in an atom's nucleus, which also determines the number of electrons surrounding the nucleus when the atom is neutral. It's essential for understanding the structure of atoms and how they bond with other atoms. In the Bohr model of the atom, it is foundational in calculating other properties such as the electron's speed.

In the Bohr model, the electron orbits the nucleus in distinct, quantized levels; these levels are often referred to as shells. The innermost electron is influenced by the atomic charge more than any other, because it is closest to the nucleus. When solving textbook exercises involving the speed of an electron, such as this one, the atomic charge is an integral part of the equations. The greater the atomic charge, the more strongly the electron is attracted to the nucleus, and the higher its speed for the given energy level.

This relationship between atomic charge and electron speed is quantified in equations derived from Bohr's model. As observed in the solution, by knowing the speed of an electron and applying Bohr's formula, we can deduce the atomic charge required to achieve that particular speed. This understanding is vital as it lays the groundwork for more complex topics in atomic physics and quantum mechanics, fitting everything together like a puzzle piece in the grand scheme of atomic behavior.

Electron Speed

The speed of an electron within an atom, particularly when discussing the Bohr model, is a central topic in the study of atomic physics. Electrons whirl around the nucleus at considerable speeds, which are affected by factors such as the electron's energy level and the nuclear charge it experiences.

In the exercise, the speed of an electron is directly measured against the speed of light, represented by c. This comparison offers a compelling indication of just how fast electrons can move, especially when we're considering atoms with high atomic charges. When learning about electron speed, it's important to grasp the relationship between this speed and the forces acting on the electron—namely, the electrostatic force of attraction to the protons in the nucleus.

The Bohr model simplifies this complex interaction into predictable formulas. As seen in the problem, the innermost electron has a certain speed that is proportional to the atomic charge, implying an increase in electron speed with greater nuclear charge. By manipulating the formula given in the solution steps, students can understand the electrodynamic relationship within an atom, providing a clear concept of how atomic charge influences electron speed. This understanding is invaluable for analyzing atomic structures and the behavior of electrons within different elements.

Reduced Planck Constant

The reduced Planck constant, commonly symbolized by ħ (h-bar), is a fundamental physical constant that plays a pivotal role in quantum mechanics and the Bohr model of the atom. It is essentially the Planck constant divided by 2π and is used in the quantization of physical properties such as momentum and energy.

Bohr's model incorporates the reduced Planck constant in describing the quantized orbits of electrons around the nucleus. It is intrinsically linked to the quantification of angular momentum in these orbits, with an electron's angular momentum being an integer multiple of ħ. The constant is a measure of the smallest action – a product of energy and time – that can be observed in the universe, signifying the quantum of action.

In the given solution, ħ appears in the equation describing the quantized speed of an electron in terms of the atomic charge. This inclusion conveys the quantum nature of the electron's motion. It's enlightening for students to see the empirical implications of ħ, as exercises that include constants like the reduced Planck constant demonstrate their practical value in predicting atomic and subatomic phenomena, grounding theoretical concepts in observable reality.