91Ó°ÊÓ

Which of the following statements about a hydrogen atom is false? a. An electron in the \(n=1\) level of the hydrogen atom is in its ground state. b. On average, an electron in the \(n=3\) level is farther from the nucleus than an electron in the \(n=2\) state. c. The wavelength of light emitted when the electron goes from the \(n=3\) level to the \(n=1\) level is the same as the wavelength of light absorbed when the electron goes from the \(n=1\) level to \(n=3\) level. d. An electron in the \(n=1\) level is higher in energy than an electron in the \(n=4\) level. e. Light of greater frequency is required for a transition from the \(n=1\) level to \(n=3\) level than is required for a transition from the \(n=2\) level to \(n=3\) level.

Step by step solution

01

Analyze Statement a

Statement a: An electron in the \(n=1\) level of the hydrogen atom is in its ground state.The ground state of an atom is the lowest energy state possible. For a hydrogen atom, the \(n=1\) level is the lowest energy level. Therefore, statement a is true.

02

Analyze Statement b

Statement b: On average, an electron in the \(n=3\) level is farther from the nucleus than an electron in the \(n=2\) state.The average distance of an electron from the nucleus increases with the principal quantum number \(n\). Since \(n=3\) is greater than \(n=2\), an electron in \(n=3\) is, on average, farther from the nucleus than in \(n=2\). Hence, statement b is true.

03

Analyze Statement c

Statement c: The wavelength of light emitted when the electron goes from the \(n=3\) level to the \(n=1\) level is the same as the wavelength of light absorbed when the electron goes from the \(n=1\) level to \(n=3\) level.This statement reflects the principle of energy conservation. The energy difference between any two states is the same whether the transition is absorptive (\(n=1\) to \(n=3\)) or emissive (\(n=3\) to \(n=1\)), making the wavelength of light involved identical. Thus, statement c is true.

04

Analyze Statement d

Statement d: An electron in the \(n=1\) level is higher in energy than an electron in the \(n=4\) level.Electrons in higher \(n\) levels (=4) have more energy than those in lower \(n\) levels (\(n=1\)), because energy levels increase with \(n\). Therefore, this statement is false.

05

Analyze Statement e

Statement e: Light of greater frequency is required for a transition from the \(n=1\) level to \(n=3\) level than is required for a transition from the \(n=2\) level to \(n=3\) level.Transitions involving greater energy differences require light of higher frequency. The energy difference \((\Delta E)\) between \(n=1\) and \(n=3\) is larger than that between \(n=2\) and \(n=3\), so the light of greater frequency is indeed required for the \(n=1\) to \(n=3\) transition. Therefore, statement e is true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quantum Energy Levels

In a hydrogen atom, the electron revolves around the nucleus in specific regions known as quantum energy levels, often referred to using the principal quantum number "n". These energy levels can be thought of as "shells" where the electron is most likely to be found.

• Each energy level is associated with a particular quantum number starting from 1, which is the lowest energy state, or "ground state."
• The ground state represents the most stable configuration for the electron.
• As the quantum number increases, so does the energy level and the distance between the nucleus and the electron on average.

The concept of quantum energy levels is fundamental to understanding the behavior of electrons in atoms and helps explain the variety of possible electron configurations. In a hydrogen atom, these levels are visualized as concentric circular paths or orbits around the nucleus, but in reality, they represent probability densities where an electron is likely to be found.

Electron Transitions

Electrons in an atom can move between these quantum energy levels through processes known as transitions. These transitions involve the absorption or emission of energy in the form of photons.

• When an electron absorbs energy, it moves to a higher energy level (away from the nucleus), a process known as an "absorption" transition.
• Conversely, when an electron loses energy, it falls to a lower energy level, emitting a photon in a process called an "emission" transition.
• The energy difference between the initial and final states of the electron determines the energy (and thus the frequency and wavelength) of the photon involved in the transition.

Understanding electron transitions is vital in determining the spectral lines of an atom, which are unique and serve as fingerprints for identifying elements.

Wavelength of Light

The light emitted or absorbed during these electron transitions has a specific wavelength. The relationship between the energy change of the electron and the wavelength of the light is given by the equation:\[\Delta E = h \frac{c}{\lambda}\]where \( \Delta E \) is the energy difference, \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength of the absorbed or emitted light.

• As the energy change increases, the wavelength of the corresponding light decreases, meaning light emitted or absorbed becomes more energetic.
• This is observed as the light shifts towards the ultraviolet end of the spectrum.

The wavelengths of light involved in these quantum transitions correspond to distinct lines in the hydrogen spectrum and can be used to identify the nature of transitions taking place.

Energy Conservation

The principle of energy conservation plays a crucial role in understanding the processes taking place in electron transitions. This principle states that energy cannot be created or destroyed, only converted from one form to another. In the context of an atom:

• When an electron transitions between energy levels, the amount of energy absorbed or emitted as light is exactly equal to the energy difference between those levels.
• This means that the process is balanced: if energy is absorbed to raise an electron to a higher level, the same amount of energy is released when it falls back to its original level.
• This concept is why the wavelength of light absorbed for a transition from a lower to a higher energy level is identical to the wavelength emitted during the reverse transition.

Energy conservation ensures that processes such as absorption and emission are predictable, forming the basis for our understanding of atomic spectra and the quantization of atomic energy levels.