Consider the following energy levels of a hypothetical atom:...

Consider the following energy levels of a hypothetical atom: $E_{4}$ $-1.0 \times 10^{-19} \mathrm{~J}$ $E_{3}$ $--5.0 \times 10^{-19} \mathrm{~J}$ $E_{2}$ $--10 \times 10^{-19} \mathrm{~J}$ $E_{1}$ $-15 \times 10^{-19} \mathrm{~J}$ (a) What is the wavelength of the photon needed to excite an electron from $E_{1}$ to $E_{4} ?$ (b) What is the energy (in joules) a photon must have in order to excite an electron from $E_{2}$ to $E_{3} ?$ (c) When an electron drops from the $E_{3}$ level to the $E_{1}$ level, the atom is said to undergo emission. Calculate the wavelength of the photon emitted in this process.

Consider the following energy levels of a hypothetical atom:
$E_{4}$
$-1.0 \times 10^{-19} \mathrm{~J}$
$E_{3}$
$--5.0 \times 10^{-19} \mathrm{~J}$
$E_{2}$
$--10 \times 10^{-19} \mathrm{~J}$
$E_{1}$
$-15 \times 10^{-19} \mathrm{~J}$
(a) What is the wavelength of the photon needed to excite an electron from $E_{1}$ to $E_{4} ?$ (b) What is the energy (in joules) a photon must have in order to excite an electron from $E_{2}$ to $E_{3} ?$ (c) When an electron drops from the $E_{3}$ level to the $E_{1}$ level, the atom is said to undergo emission. Calculate the wavelength of the photon emitted in this process.

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Key Concepts

Atomic Energy Levels

Atomic energy levels are the discrete energy states that electrons can occupy within an atom. This quantization is a fundamental concept in quantum mechanics, explaining why electrons do not exist in arbitrary energy states but rather in specific, allowed ones. Understanding these levels is essential for analyzing how electrons transition between states, such as when absorbing or emitting photons.

Electron Transitions

Electron transitions refer to the process by which an electron moves from one energy level to another within an atom. These transitions occur when energy is either absorbed or emitted by the electron, corresponding to the difference in energy between the initial and final states. This concept is central to explaining phenomena such as spectral lines in atomic emission and absorption.

Photon Energy-Wavelength Relationship

The photon energy-wavelength relationship is articulated by Planck's equation, E = hc/?, where E is the energy of the photon, h is Planck's constant, c is the speed of light, and ? is the wavelength. This relationship is pivotal in converting the energy difference between electron levels into a measurable wavelength, thereby linking the quantized energy transitions to the electromagnetic radiation that is observed.

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