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

Consider the following energy levels of a hypothetical atom: $$\begin{aligned} &E_{4}:-1.0 \times 10^{-19} \mathrm{J}\\ &E_{3}:-5.0 \times 10^{-19} \mathrm{J}\\ &E_{2}:-10 \times 10^{-19} \mathrm{J}\\ &E_{1}:-15 \times 10^{-19} \mathrm{J} \end{aligned}$$ (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:
$$\begin{aligned}
&E_{4}:-1.0 \times 10^{-19} \mathrm{J}\\
&E_{3}:-5.0 \times 10^{-19} \mathrm{J}\\
&E_{2}:-10 \times 10^{-19} \mathrm{J}\\
&E_{1}:-15 \times 10^{-19} \mathrm{J}
\end{aligned}$$
(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

Quantum Energy Levels

Atoms have discrete, quantized energy states where electrons reside. This concept arises from quantum mechanics and explains why electrons can only occupy specific energy values rather than any arbitrary level, forming the basis for understanding atomic structure.

Electron Transitions

Electron transitions refer to the movement of an electron between different energy levels. When an electron jumps from one level to another, it must absorb or emit a photon with an energy exactly equal to the difference between the initial and final states, underscoring the quantized nature of atomic energy changes.

Photon Absorption and Emission

Photon absorption occurs when an electron gains energy by moving to a higher energy level, while photon emission happens when an electron loses energy and drops to a lower level. Both processes are central to the study of spectroscopy and atomic behavior, demonstrating how atoms interact with electromagnetic radiation.

Energy-Wavelength Relationship

The energy of a photon is directly related to its wavelength through the equation E = hc/?, where h is Planck's constant and c is the speed of light. This relationship provides a critical link between quantum energy transitions within an atom and the properties of the electromagnetic radiation involved in those transitions.

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