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 31a

Chapter 38, Problem 31a

What quantum number of the hydrogen atom comes closest to giving a 100-nm-diameter electron orbit?

Verified step by step guidance

Step 1: Understand the problem. The question asks for the quantum number (n) of a hydrogen atom that corresponds to an electron orbit with a diameter of approximately 100 nm. This involves using the Bohr model of the hydrogen atom, which relates the radius of the electron orbit to the quantum number.

Step 2: Recall the formula for the radius of the electron orbit in the Bohr model:

r_n = n^2 × a_0

, where

a_0

is the Bohr radius (approximately 0.0529 nm) and

n

is the principal quantum number.

Step 3: Convert the given diameter of the orbit (100 nm) into the radius. Since the diameter is twice the radius, the radius is

r = 100/2 = 50

nm.

Step 4: Rearrange the formula to solve for the quantum number

n

. Using

r_n = n^2 × a_0

, substitute

r_n = 50

nm and

a_0 = 0.0529

nm. Solve for

n

using

n = sqrt(r_n / a_0)

Step 5: Perform the substitution and simplify the expression. This will give the approximate quantum number

n

that corresponds to the given orbit diameter. Note that

n

must be an integer, as quantum numbers are discrete.

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Key Concepts

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

Quantum Numbers

Quantum numbers are values that describe the unique quantum state of an electron in an atom. They include the principal quantum number (n), which indicates the energy level and size of the orbital, the azimuthal quantum number (l), which describes the shape of the orbital, and the magnetic quantum number (m_l), which specifies the orientation of the orbital in space.

Bohr Model of the Hydrogen Atom

The Bohr model provides a simplified representation of the hydrogen atom, where electrons orbit the nucleus in defined paths or orbits. According to this model, the radius of the electron's orbit is quantized and depends on the principal quantum number (n), allowing for the calculation of the size of the orbit based on the energy levels.

Electron Orbit Diameter

The diameter of an electron orbit in the hydrogen atom can be calculated using the formula derived from the Bohr model, which relates the radius of the orbit to the principal quantum number. Specifically, the radius increases with the square of n, meaning that larger values of n correspond to larger orbits, which is essential for determining the quantum number that results in a specific diameter.

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