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 refer to the fixed, quantized energy states that electrons can occupy within an atom. These discrete levels mean that electrons can only exist at specific energy amounts and the difference between these levels determines the energy that must be absorbed or emitted during transitions.

Electron Transitions

Electron transitions involve the movement of electrons between different energy levels. When an electron absorbs energy, it jumps from a lower energy level to a higher one (excitation); conversely, when it loses energy, it falls from a higher level to a lower one (emission). These processes are governed by conservation of energy, where the energy difference between the initial and final states is released or absorbed as a photon.

Photon Energy-Wavelength Relationship

The energy of a photon is directly related to its wavelength by the equation E = hc/?, where E is the photon energy, h is Planck's constant, c is the speed of light, and ? is the wavelength. This relationship is fundamental in spectroscopy and quantum mechanics, as it allows for the calculation of the energy change involved in electron transitions by measuring the corresponding photon wavelength.

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