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Magazine Chapter 12: Problem 13
Obtain an expression for the frequency of radiation emitted when a hydrogen atom de-excites from level \(n\) to level \((n-1) .\) For large \(n\), show that this frequency equals the classical frequency of revolution of the electron in the orbit.
Short Answer
Expert verified
Frequency of transition: \( \nu \approx \frac{13.6 \text{ eV}}{h n^3} \), matches classical frequency \( \nu_{classical} \).
Step by step solution
01
Understanding the Initial State
We start with a hydrogen atom in an excited state with a principal quantum number \( n \). This electron can transition to a lower energy level \( n-1 \), emitting radiation in the process.
02
Energy of Hydrogen Levels
The energy of an electron in a hydrogen atom at level \( n \) is given by the formula \( E_n = - \frac{13.6 \text{ eV}}{n^2} \). Similarly, the energy at level \( n-1 \) will be \( E_{n-1} = - \frac{13.6 \text{ eV}}{(n-1)^2} \).
03
Calculate Energy Expression for Transition
The energy difference between these levels represents the energy of the photon emitted during the transition: \[ \Delta E = E_{n-1} - E_n = - \frac{13.6 \text{ eV}}{(n-1)^2} + \frac{13.6 \text{ eV}}{n^2} \].
04
Simplify Delta Energy for Large n
For large \( n \), \( n-1 \approx n \). The expression becomes \[ \Delta E \approx 13.6 \text{ eV} \left(\frac{1}{n^2} - \frac{1}{n^2}\right) \approx \frac{13.6 \text{ eV}}{n^3} \].
05
Relate Energy Difference to Frequency
The frequency \( u \) of the emitted radiation is related to the energy by the relation \( \Delta E = h u \), where \( h \) is Planck鈥檚 constant. Thus, \[ u = \frac{\Delta E}{h} \approx \frac{13.6 \text{ eV}}{h n^3} \].
06
Classical Frequency for Large n
The classical frequency \( u_{classical} \) for an electron in orbit is given by \( u_{classical} = \frac{v}{2\pi r} \), where \( v \) is the velocity and \( r \) is the orbit radius. For large \( n \), \( u_{classical} \approx \frac{13.6 \text{ eV}}{h n^3} \), which matches our previous expression.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Energy levels in hydrogen atom
In the hydrogen atom, electrons occupy distinct energy levels, which are determined by the principal quantum number, denoted as \(n\). This quantum number is an integer that specifies the electron's orbit in the atom. For each energy level, the energy is negative, indicating that the electron is bound to the nucleus. The formula for the energy levels is given by:
\[E_n = - \frac{13.6 \, \text{eV}}{n^2}.\]
This formula implies that as \(n\) increases, the energy of the level becomes less negative, meaning the electron is less tightly bound. When an electron transitions from a higher energy level to a lower one, it emits a photon whose energy corresponds to the difference between these two levels. Understanding these energy levels helps explain phenomena like spectral lines, which were historically critical in the development of quantum mechanics.
\[E_n = - \frac{13.6 \, \text{eV}}{n^2}.\]
This formula implies that as \(n\) increases, the energy of the level becomes less negative, meaning the electron is less tightly bound. When an electron transitions from a higher energy level to a lower one, it emits a photon whose energy corresponds to the difference between these two levels. Understanding these energy levels helps explain phenomena like spectral lines, which were historically critical in the development of quantum mechanics.
Photon emission
Photon emission is the process by which an excited electron drops to a lower energy level, releasing a photon鈥攁 particle of light鈥攊n the process. The energy of the emitted photon is equal to the energy difference between the initial and final levels of the electron. Mathematically, this can be expressed as:
\[\Delta E = E_{n-1} - E_n.\]
If we substitute the expressions for \(E_n\) and \(E_{n-1}\) from the hydrogen atom energy levels, it leads to:
\[\Delta E = -\frac{13.6 \, \text{eV}}{(n-1)^2} + \frac{13.6 \, \text{eV}}{n^2}.\]
This energy difference corresponds to the photon's energy. The formula illustrates the concept of quantized energy鈥攎eaning the energy can only change in discrete amounts鈥攁n idea central to quantum mechanics. These transitions are what give rise to the unique spectral lines seen in hydrogen and other atoms.
\[\Delta E = E_{n-1} - E_n.\]
If we substitute the expressions for \(E_n\) and \(E_{n-1}\) from the hydrogen atom energy levels, it leads to:
\[\Delta E = -\frac{13.6 \, \text{eV}}{(n-1)^2} + \frac{13.6 \, \text{eV}}{n^2}.\]
This energy difference corresponds to the photon's energy. The formula illustrates the concept of quantized energy鈥攎eaning the energy can only change in discrete amounts鈥攁n idea central to quantum mechanics. These transitions are what give rise to the unique spectral lines seen in hydrogen and other atoms.
Classical frequency of electron
In classical physics, the frequency of an electron orbiting a nucleus is defined differently than in quantum mechanics. Classically, if we imagine the electron as a tiny planet orbiting the nucleus, we can think of it having a frequency of revolution given by the formula:
\[u_{\text{classical}} = \frac{v}{2\pi r},\]
where \(v\) is the orbital velocity and \(r\) is the radius of the orbit. As we move to larger quantum numbers (higher energy levels), quantum and classical descriptions of electron behavior converge. In the case of large \(n\), the frequency of radiation due to de-excitation of the electron matches this classical frequency. This insight beautifully illustrates the correspondence principle, where quantum mechanics aligns with classical mechanics in the limit of large quantum numbers.
\[u_{\text{classical}} = \frac{v}{2\pi r},\]
where \(v\) is the orbital velocity and \(r\) is the radius of the orbit. As we move to larger quantum numbers (higher energy levels), quantum and classical descriptions of electron behavior converge. In the case of large \(n\), the frequency of radiation due to de-excitation of the electron matches this classical frequency. This insight beautifully illustrates the correspondence principle, where quantum mechanics aligns with classical mechanics in the limit of large quantum numbers.
Planck's constant
Planck's constant \(h\) is a fundamental constant in quantum mechanics, representing the 'quantum of action.' It plays a crucial role in the energy-frequency relationship for radiation. The formula that links the energy of a photon to its frequency is given by:
\[E = hu.\]
In the context of hydrogen atom de-excitation, this relationship allows us to express the frequency of emitted light as:
\[u = \frac{\Delta E}{h}.\]
Planck鈥檚 constant has a value of approximately \(6.63 \times 10^{-34} \, \text{Js}\). Its introduction marked a significant departure from classical physics, allowing the development of quantum theory. This constant ensures that energy changes in atoms occur in discrete amounts, leading to observations such as quantized energy levels and spectral lines, foundational concepts in understanding atomic and molecular structures.
\[E = hu.\]
In the context of hydrogen atom de-excitation, this relationship allows us to express the frequency of emitted light as:
\[u = \frac{\Delta E}{h}.\]
Planck鈥檚 constant has a value of approximately \(6.63 \times 10^{-34} \, \text{Js}\). Its introduction marked a significant departure from classical physics, allowing the development of quantum theory. This constant ensures that energy changes in atoms occur in discrete amounts, leading to observations such as quantized energy levels and spectral lines, foundational concepts in understanding atomic and molecular structures.
