Ch 38: Quantization

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 38: Quantization

Problem 34

Chapter 38, Problem 34

Find the radius of the electron’s orbit, the electron’s speed, and the energy of the atom for the first three stationary states of He+.

Verified step by step guidance

Understand the problem: The He+ ion is a hydrogen-like atom with a single electron orbiting a nucleus of charge +2e. The Bohr model can be used to calculate the radius of the electron's orbit, its speed, and the energy of the atom for the first three stationary states (n = 1, 2, 3). The formulas for these quantities depend on the principal quantum number (n) and the nuclear charge (Z).

Step 1: Write the formula for the radius of the electron's orbit in the Bohr model:

r = \frac{n^{2}}{a}

₀/Z2, where

a

₀ is the Bohr radius (approximately 5.29 × 10⁻¹¹ m), n is the principal quantum number, and Z is the atomic number (Z = 2 for He+). Substitute the values of n = 1, 2, 3 to calculate the radii for the first three stationary states.

Step 2: Write the formula for the speed of the electron in the Bohr model:

v = \frac{n}{e} ....

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Bohr Model of the Atom

The Bohr model describes the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits. It quantizes the angular momentum of the electrons, leading to specific energy levels. This model is particularly useful for hydrogen-like atoms, such as He+, where the electron's orbit radius and energy can be calculated using simple formulas.

Quantization of Energy Levels

In the context of the Bohr model, energy levels are quantized, meaning that electrons can only occupy certain discrete energy states. For He+, the energy levels can be calculated using the formula E_n = -Z² * 13.6 eV / n², where Z is the atomic number and n is the principal quantum number. This quantization explains why electrons do not radiate energy continuously and instead exist in stable orbits.

Centripetal Force and Electron Speed

The speed of an electron in its orbit can be derived from the balance of centripetal force and electrostatic force. The centripetal force required to keep the electron in circular motion is provided by the electrostatic attraction between the positively charged nucleus and the negatively charged electron. This relationship allows for the calculation of the electron's speed in each stationary state.

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