Ch 38: Quantization

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 38: Quantization

Problem 27c

Chapter 38, Problem 27c

The allowed energies of a simple atom are 0.00 eV, 4.00 eV, and 6.00 eV. What wavelengths appear in the atom’s absorption spectrum?

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Identify the energy levels of the atom: The given energy levels are 0.00 eV, 4.00 eV, and 6.00 eV. Absorption occurs when the atom absorbs energy and transitions from a lower energy level to a higher energy level.

Determine the energy differences for possible transitions: Calculate the energy differences between the lower and higher energy levels. For example, the transitions are: (1) 0.00 eV to 4.00 eV, (2) 0.00 eV to 6.00 eV, and (3) 4.00 eV to 6.00 eV. The energy differences are ΔE₁ = 4.00 eV, ΔE₂ = 6.00 eV, and ΔE₃ = 2.00 eV.

Convert the energy differences to wavelengths: Use the relationship between energy and wavelength, given by the equation:

E = \frac{hc}{λ}

, where

E

is the energy difference,

h

is Planck's constant (

6.63×10

^-34 J·s),

c

is the speed of light (

3.00×10

⁸ m/s), and

λ

is the wavelength. Rearrange the equation to solve for wavelength:

λ = \frac{hc}{E}

Substitute the values for each energy difference: For each transition, substitute the energy difference (converted to joules: 1 eV =

1.60×10

^-19 J) into the equation for wavelength. For example, for ΔE₁ = 4.00 eV, convert to joules and calculate

λ

. Repeat for ΔE₂ and ΔE₃.

List the wavelengths: After calculating the wavelengths for each transition, list them as the wavelengths that appear in the atom's absorption spectrum. These correspond to the transitions where the atom absorbs energy.

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Key Concepts

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

Energy Levels

In quantum mechanics, energy levels refer to the discrete values of energy that an electron in an atom can have. Electrons can only occupy these specific energy states, and transitions between them involve the absorption or emission of energy in the form of photons. The allowed energies for the atom in the question are 0.00 eV, 4.00 eV, and 6.00 eV, indicating the quantized nature of electron states.

Photon Wavelength

The wavelength of a photon is inversely related to its energy, as 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. When an electron transitions between energy levels, it absorbs or emits a photon with a specific wavelength corresponding to the energy difference between the levels. This relationship is crucial for determining the wavelengths in the absorption spectrum.

Absorption Spectrum

An absorption spectrum is a spectrum of absorbed light that occurs when electrons in an atom absorb specific wavelengths of light to transition to higher energy levels. The resulting spectrum appears as dark lines or bands at wavelengths corresponding to the energy differences between allowed energy levels. In this case, the absorption spectrum will show wavelengths that correspond to the energy transitions from the ground state to the excited states of 4.00 eV and 6.00 eV.

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