The first three energy levels of the fictitious element X are shown in FIGURE P38.54. What wavelengths are observed in the absorption spectrum of element X? Express your answers in nm.

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Identify the energy levels of the fictitious element X from the given figure. Let the energy levels be denoted as E₁, E₂, and E₃, where E₁ < E₂ < E₃.

Recall that in an absorption spectrum, photons are absorbed when an electron transitions from a lower energy level to a higher energy level. The energy of the absorbed photon corresponds to the difference between the two energy levels: ΔE = E_final - E_initial.

Use the relationship between the energy of a photon and its wavelength: E = h * c / λ, where h is Planck's constant (6.626 × 10⁻³⁴ J·s), c is the speed of light (3.00 × 10⁸ m/s), and λ is the wavelength. Rearrange this equation to solve for the wavelength: λ = h * c / ΔE.

Calculate the energy differences for all possible transitions that result in absorption: ΔE₁₂ = E₂ - E₁, ΔE₁₃ = E₃ - E₁, and ΔE₂₃ = E₃ - E₂. These energy differences correspond to the absorbed photon energies.

Substitute the values of ΔE₁₂, ΔE₁₃, and ΔE₂₃ into the equation λ = h * c / ΔE to calculate the wavelengths for each transition. Convert the resulting wavelengths from meters to nanometers (1 nm = 10⁻⁹ m) to express the final answers in nm.

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Energy Levels

Energy levels refer to the specific energies that electrons can occupy in an atom. In quantum mechanics, these levels are quantized, meaning electrons can only exist in certain states and not in between. The difference in energy between these levels determines the wavelengths of light absorbed or emitted when electrons transition between them.

Absorption Spectrum

An absorption spectrum is a spectrum of absorbed light that occurs when electrons in an atom absorb specific wavelengths of light to move from a lower energy level to a higher one. Each element has a unique absorption spectrum, which can be used to identify the element based on the specific wavelengths absorbed, corresponding to the energy differences between its quantized energy levels.

Wavelength and Energy Relationship

The relationship between wavelength and energy is 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 equation shows that shorter wavelengths correspond to higher energy transitions. Therefore, to find the wavelengths in the absorption spectrum, one must calculate the energy differences between the energy levels and convert these energies into wavelengths.