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40.42 - Hydrogen emits radiation with four prominent visible wavelengths - one red, one cyan, one blue, and one violet. The respective frequencies are \(656 \mathrm{nm}, 486 \mathrm{nm}, 434 \mathrm{nm},\) and \(410 \mathrm{nm} .\) We can model the hydrogen atom as an electron in a one-dimensional box, and attempt to match four adjacent emission lines in the predicted spectrum to the visible part of the hydrogen spectrum. (a) Determine the photon energies associated with the visible part of the hydrogen spectrum. (b) The electron-in-a-box emission spectrum is \(E_{n_{i} \rightarrow n_{\mathrm{f}}}=\left(n_{\mathrm{i}}^{2}-n_{\mathrm{f}}^{2}\right) \epsilon\) where \(n_{\mathrm{i}}\) and \(n_{\mathrm{f}}\) are the initial and final quantum numbers of the electron when it drops to a lower energy level and \(\epsilon\) is the energy determined by the Schrödinger equation. What is the smallest possible value of \(n_{\mathrm{i}}\) that can accommodate four emission lines? (c) Using the value from part (b) for \(n_{i}\), estimate the order of magnitude of \(\epsilon\) by dividing the four photon energies by the relevant differences \(n_{\mathrm{i}}^{2}-n_{\mathrm{f}}^{2}\) for transitions in the possible sets of \(\left(n_{i}, n_{f}\right)\) pairings. (d) Using Eq. (40.31) to identify \(\epsilon\), and using the mass of the electron, use your result from part (c) to estimate the length \(L\) of the box. (e) What is the ratio of your estimate of \(L\) to twice the Bohr radius? (Note: The hydrogen atom is better modeled using the Coulomb potential rather than as a particle in a box.)

Step by step solution

01

Calculating photon energies

The energy of a photon is given by the product of Planck's constant (h) and the speed of light (c) divided by the wavelength (\(\lambda\)) of the light. Apply this equation to each wavelength to find the four energy values. The equation is as follows: \(E = \frac{hc}{\lambda}\) where \(h = 6.626 × 10^{-34} Js\) and \(c = 3.0 × 10^8 ms^{-1}\).

02

Derive and solve for the quantum number

Given the equation for the electron-in-a-box emission spectrum and the condition that there must be four emission lines, we can deduce that the smallest possible value for \(n_{i}\) is five. We can conclude this since \(n_{i}\) needs to have four distinct values of \(n_{f}\) that are less than \(n_{i}\) to provide four unique emission lines.

03

Estimating the energy value

Next, we are asked to estimate the order of magnitude of ε by dividing the four photon energies by the relevant differences \(n_{i}^2 - n_{f}^2\) for transitions in the possible sets of pairs \((n_{i}, n_{f})\). These pairs are: (5, 4), (5, 3), (5, 2), and (5, 1). Calculate the photon energy for each pair and take an average to estimate \(ε\).

04

Estimating the length of the box

Using the particle in a box model, the energy epsilon can be represented by \(\epsilon = \frac{\pi^2 h^2}{2 m L^2}\), where \(m\) is the mass of the electron and \(L\) is the length of the box. Solving for \(L\), we get \(L = \sqrt{\frac{\pi^2 h^2}{2m \epsilon}}\). Substitute the values for h, m and epsilon to find \(L\).

05

Finding the ratio of length to Bohr radius

Finally, we are asked to compute the ratio of the length \(L\) of the box to twice the Bohr radius, which has a value \(a_0 = 5.29 × 10^{-11} m\). Ratio is thus \(L / (2*a_0)\).

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Key Concepts

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

Photon Energy Calculation

When we talk about photon energy, it means determining the energy carried by a photon, which is a particle of light. To do this, we use a simple formula; the energy of a photon (E) is given by the equation:

• \[E = \frac{hc}{\lambda}\]

Here, \( h \) represents Planck's constant \( (6.626 \times 10^{-34} Js) \), \( c \) is the speed of light \( (3.0 \times 10^8 ms^{-1}) \), and \( \lambda \) is the wavelength.
This formula helps us calculate the energy for the different visible wavelengths of hydrogen emission: red, cyan, blue, and violet. Each color corresponds to a specific wavelength. For instance, red has a longer wavelength and lower energy, while violet has a shorter wavelength and higher energy. Applying this equation to each given wavelength allows us to find the energy values corresponding to each color.

Particle in a Box Model

In quantum mechanics, the particle in a box model is a foundational concept used to understand the behavior of electrons confined in a small, one-dimensional space. This model simplifies the hydrogen atom as an electron trapped in a theoretical box, simulating the atom's electron cloud.
This model gives us the equation for energy levels: the electron-in-a-box emission spectrum is

• \[E_{n_{i} \rightarrow n_{f}} = (n_{i}^2 - n_{f}^2) \epsilon\]

where \( n_{i} \) and \( n_{f} \) are the initial and final quantum numbers. In this equation, \( \epsilon \) denotes energy derived from the Schrödinger equation.
This model helps us predict various emission lines' energies. By using this, we can match the predicted spectrum energy of an electron transitioning from higher to lower energy levels with the actual observed hydrogen spectrum.

Quantum Numbers in Atomic Transitions

Quantum numbers play a crucial role in understanding atomic transitions. These numbers indicate the energy levels electrons transition between within an atom. They can be compared to labels for energy states that an electron can occupy.
In our exercise, to generate four emission lines, we need four different initial-final quantum number pairs, such as (5, 4), (5, 3), (5, 2), and (5, 1). The number 5, our \( n_{i} \), is the lowest number satisfying the requirement for four distinct values of \( n_{f} \).

• \( n_{i} \): Initial quantum number when the electron is at a higher energy state.
• \( n_{f} \): Final quantum number when the electron transitions to a lower energy state.

In this framework, we analyze atomic emission, as each decrement in quantum numbers results in an energy emission that falls within the photon wavelengths corresponding to the visible spectrum.

Bohr Radius Comparison

The Bohr radius is an essential concept when discussing atomic scale measurements. It is the distance between the proton and the electron in a hydrogen atom at its ground state. This radius is represented as \( a_0 \), with a value of approximately \( 5.29 \times 10^{-11} m \).
In our original exercise, we calculated the length \( L \) of the electron's theoretical box, using the equation:

• \[L = \sqrt{\frac{\pi^2 h^2}{2m \epsilon}}\]

where \( m \) is the mass of the electron.
To gain further insight, we compare this length \( L \) to twice the Bohr radius \( 2a_0 \). Knowing this ratio helps us understand how our theoretical model aligns or differs with known atomic properties. Comparing these values illustrates the particles' confinement inside the box relative to atomic-scale distances.