Question

What wavelength of light is needed to excite an electron in a \(0.2\) nanometer ( \(1 \mathrm{~nm}=10^{-9} \mathrm{~m}\) ) box from the ground state to the second excited state? What wavelength of light is emitted when the same electron falls from the second excited state to the first excited state?

Short answer

The wavelength of light needed to excite the electron in a 0.2 nm box from the ground state to the second excited state is \(\lambda_{13} = 6.21 \times 10^{-9} ~m\) or \(6.21 ~nm\). The wavelength of light emitted when the electron falls from the second excited state to the first excited state is \(\lambda_{21} = 2.07 \times 10^{-8} ~m\) or \(20.7 ~nm\).

Step-by-step solution

Energy levels in a one-dimensional box formula

We will calculate the energy levels of the electron in the one-dimensional box using the formula: \(E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}\) Where: \(E_n\) is the energy at level \(n\) \(n\) is the state of the electron (integer values starting at 1) \(\hbar\) is the reduced Planck constant, approximately \(1.054 \times 10^{-34} Js\) \(m\) is the mass of the electron, approximately \(9.11 \times 10^{-31} kg\) \(L\) is the length of the box

Calculate energy differences

We will calculate the energy difference between the ground (n=1) state, the first excited state (n=2), and the second excited state (n=3). To find the energy difference between the ground state and second excited state, we will subtract the energy at the ground state (\(E_1\)) from the energy at the second excited state (\(E_3\)): \(\Delta E_{13} = E_3 - E_1\) Similarly, we will find the energy difference between the first and second excited states: \(\Delta E_{21} = E_2 - E_1\)

Find wavelengths corresponding to energy differences

We will use the energy differences and the Planck's constant (\(h\), approximately equal to \(6.626 \times 10^{-34} Js\)) to find the corresponding wavelengths. The relation between energy and wavelength is given as: \(E = \frac{hc}{\lambda}\) Where: \(E\) is the energy \(h\) is the Planck's constant \(c\) is the speed of light in vacuum, approximately \(3.0\times10^8 \frac{m}{s}\) \(\lambda\) is the wavelength We have to solve for the wavelengths, so we can rewrite the equation as: \(\lambda = \frac{hc}{E}\) Calculate the wavelength of light needed to excite the electron from ground state to second excited state as: \(\lambda_{13} = \frac{hc}{\Delta E_{13}}\) And the wavelength of light emitted when the electron falls from the second excited state to the first excited state as: \(\lambda_{21} = \frac{hc}{\Delta E_{21}}\)

Perform the calculations and write down the answer

Now that we have all the equations, we can plug in the values to find the wavelengths. Remember that the length of the box is given as \(L = 0.2 nm = 2\times10^{-10} m\). 1. Calculate the energy levels: \(E_1 = \frac{1^2\pi^2\hbar^2}{2m(2\times10^{-10})^2}\), \(E_2 = \frac{2^2\pi^2\hbar^2}{2m(2\times10^{-10})^2}\), \(E_3 = \frac{3^2\pi^2\hbar^2}{2m(2\times10^{-10})^2}\). 2. Calculate energy differences: \(\Delta E_{13} = E_3 - E_1\), \(\Delta E_{21} = E_2 - E_1\). 3. Calculate the corresponding wavelengths: \(\lambda_{13} = \frac{hc}{\Delta E_{13}}\), \(\lambda_{21} = \frac{hc}{\Delta E_{21}}\). Plug in the values and solve for the respective wavelengths of light. Make sure to express the answer in nanometers.

Key concepts

Particle in a One-Dimensional Box

The concept of a particle in a one-dimensional box, also known as an 'infinite potential well,' is a critical and foundational model in quantum mechanics. This model helps us understand how quantization of energy levels occurs for a particle - such as an electron - confined to a very small space, comparable to atomic dimensions.

Imagine a particle that is free to move within a one-dimensional box with impenetrable walls. Despite its simplicity, this model allows us to apply the principles of quantum mechanics and derive some important results. The particle is restricted from being found outside the box, thus having a zero probability of being detected there. Inside the box, however, the particle exhibits wave-like properties and can only occupy certain states, which correspond to standing waves. The quantization arises because only waves with specific wavelengths can fit into the box, leading to distinct energy levels.

Quantum Energy Levels

Quantum energy levels are a fundamental aspect of quantum mechanics that describe the discrete energies that a quantum system, such as an electron in an atom, can possess. Unlike classical systems, where a particle can have a continuous range of energies, quantum systems are restricted to certain 'allowed' energy levels.

For the particle in a one-dimensional box, the energy levels are determined by the size of the box and the mass of the particle. The energy levels can be calculated using the formula: \[E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}\].Here, \(n\) is a positive integer representing the quantum number and can only take on whole number values. As \(n\) increases, so does the energy, leading to a hierarchy of energy states the particle can occupy.

Planck's Constant

Planck's constant, symbolized as \(h\), is a fundamental physical constant that plays a pivotal role in quantum mechanics. This constant relates the frequency of a photon to its energy through the simple equation \(E = hf\), where \(E\) is energy and \(f\) is frequency. Its importance is recognized in the quantization of light and other phenomena.

The reduced Planck constant \(\hbar\) (h-bar), which is \(h\) divided by \(2\pi\), appears frequently in quantum mechanics, including in the formula for calculating energy levels in a one-dimensional box. The value of \(h\) is approximately \(6.626 \times 10^{-34} Js\), and it signifies the smallest action that can be observed in the universe, marking the scale at which classical physics transitions to quantum behavior.

Wavelength Calculation

The wavelength of light is inversely proportional to its energy, a relationship that is crucial in understanding various quantum mechanical processes, including electronic transitions. To determine the wavelength of light required to move an electron from one energy level to another, we can rearrange the equation \(E = \frac{hc}{\lambda}\), solving for wavelength (\(\lambda\)):

\[\lambda = \frac{hc}{E}\].Using the calculated energy difference between two quantum states and this formula, you can determine both the wavelength of light necessary to excite an electron, and that emitted when it relaxes to a lower energy state. It is important to note that when photons are absorbed or emitted by an electron moving between energy levels, the wavelength of the photons corresponds precisely to the energy difference between those levels.