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 8

Chapter 38, Problem 8

What is the wavelength, in nm, of a photon with energy (a) 0.30 eV, (b) 3.0 eV, and (c) 30 eV? For each, is this wavelength visible, ultraviolet, or infrared light?

Verified step by step guidance

Step 1: Recall the relationship between the energy of a photon and its wavelength. The energy of a photon is given by the equation:

E = \frac{hc}{λ}

, where

E

is the energy of the photon,

h

is Planck's constant (

4.1357 \times 10 ⁻ 15

eV·s),

c

is the speed of light (

3.00 \times 10 ⁸

m/s), and

λ

is the wavelength in meters.

Step 2: Rearrange the formula to solve for wavelength

λ

λ = \frac{hc}{E}

. Substitute the known values of

h

and

c

into the equation.

Step 3: Convert the energy values from eV to joules if necessary, using the conversion factor

1 eV = 1.602 \times 10 ⁻ 19

J. For example, for part (a),

0.30 eV = 0.30 \times 1.602 \times 10 ⁻ 19

Step 4: Calculate the wavelength

λ

for each energy value by substituting the energy (in joules) into the formula

λ = \frac{hc}{E}

. Ensure the result is in meters, then convert to nanometers by multiplying by

10 ⁹

Step 5: Compare the calculated wavelengths to the ranges of visible light (approximately 400–700 nm), ultraviolet light (less than 400 nm), and infrared light (greater than 700 nm) to determine the type of light for each case.

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Key Concepts

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

Photon Energy and Wavelength Relationship

The energy of a photon is inversely related to its wavelength, described by the equation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. This relationship indicates that higher energy photons have shorter wavelengths, while lower energy photons have longer wavelengths.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, categorized by wavelength. Visible light ranges from approximately 400 nm (violet) to 700 nm (red), while ultraviolet light has shorter wavelengths (10 nm to 400 nm) and infrared light has longer wavelengths (700 nm to 1 mm). Understanding this spectrum is crucial for classifying the type of light based on its wavelength.

Energy Units and Conversion

Energy is often measured in electronvolts (eV) in the context of photons. To find the wavelength in nanometers (nm), one must convert the energy from eV to joules and then apply the photon energy-wavelength relationship. This conversion is essential for accurately determining the wavelength corresponding to a given energy value.

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