Question

Consider an electron in the hydrogen atom. If you are able to excite its electron from the \(n=1\) shell to the \(n=2\) shell with a given laser, what kind of a laser (that is, compare wavelengths) will you need to excite that electron again from the \(n=2\) to the \(n=3\) shell? Explain.

Short answer

Answer: To excite the electron from the n=1 shell to the n=2 shell, a laser with an approximate wavelength of 121.4 nm is needed. To excite the electron from the n=2 shell to the n=3 shell, a laser with an approximate wavelength of 656.2 nm is needed.

Step-by-step solution

Calculate the energy levels of the hydrogen atom's shells

To find the energy differences, start by calculating the energy level at the hydrogen atom's shells using the provided energy level formula: \(E_1 = -\frac{13.6 \text{ eV}}{1^2} = -13.6 \text{ eV}\) \(E_2 = -\frac{13.6 \text{ eV}}{2^2} = -3.4 \text{ eV}\) \(E_3 = -\frac{13.6 \text{ eV}}{3^2} = -1.51 \text{ eV}\)

Calculate the energy differences between shells

Now that you have the energy levels, find the energy differences between the first and the second shell and between the second and the third shell: \(\Delta E_{12} = E_2 - E_1 = -3.4 \text{ eV} - (-13.6 \text{ eV}) = 10.2 \text{ eV}\) \(\Delta E_{23} = E_3 - E_2 = -1.51 \text{ eV} - (-3.4 \text{ eV}) = 1.89 \text{ eV}\)

Convert energy differences to wavelengths

Now you'll use the energy differences to find the required wavelengths for the lasers. To do this, you can use the formula relating energy and wavelength: \(E = \frac{hc}{\lambda}\), where \(h\) is the Planck's constant (\(6.626 x 10^{-34} \text{ Js}\)), \(c\) is the speed of light (\(3.0 x 10^8 \text{ m/s}\)), and \(\lambda\) is the wavelength. Solve the formula for wavelength: \(\lambda = \frac{hc}{E}\) Convert eV to Joules (1 eV = \(1.6 \times 10^{-19} \text{ J}\)): \(\Delta E_{12} = 10.2 \text{ eV} \times \frac{1.6 \times 10^{-19} \text{ J}}{1 \text{ eV}} = 1.63 \times 10^{-18} \text{ J}\) \(\Delta E_{23} = 1.89 \text{ eV} \times \frac{1.6 \times 10^{-19} \text{ J}}{1 \text{ eV}} = 3.02 \times 10^{-19} \text{ J}\) Finally, calculate the wavelengths: \(\lambda_{12} = \frac{(6.626 \times 10^{-34} \text{ Js})(3.0 \times 10^8 \text{ m/s})}{1.63 \times 10^{-18} \text{ J}} = 1.214 \times 10^{-7} \text{ m}\) \(\lambda_{23} = \frac{(6.626 \times 10^{-34} \text{ Js})(3.0 \times 10^8 \text{ m/s})}{3.02 \times 10^{-19} \text{ J}} = 6.562 \times 10^{-7} \text{ m}\) Since lasers are usually specified in terms of their wavelength in nanometers (nm), convert the values: \(\lambda_{12} \approx 121.4 \text{ nm}\) \(\lambda_{23} \approx 656.2 \text{ nm}\) So, you will need a laser with a wavelength of approximately 121.4 nm to excite the electron from the \(n=1\) shell to the \(n=2\) shell, and a laser with a wavelength of approximately 656.2 nm to excite the electron from the \(n=2\) shell to the \(n=3\) shell.

Key concepts

Electron Excitation

Electron excitation is a fundamental concept in atomic physics, especially in the context of the hydrogen atom. It refers to the process of an electron absorbing energy and moving from a lower energy level to a higher one. In a hydrogen atom, which consists of a single electron orbiting a proton, each energy level or shell is denoted by the principal quantum number, \( n \). How Excitation Works:

• When an electron absorbs a photon with just the right amount of energy, it jumps to a higher energy level.
• This can be visualized as stepping up a staircase, with each step representing a discrete energy level.
• For instance, an electron can move from the \( n=1 \) level (ground state) to the \( n=2 \) level (first excited state) by absorbing a specific wavelength of light.

Understanding electron excitation lays the groundwork for explaining how transitions between different energy states affect the observed spectral lines of elements.

Energy Differences

Energy differences between the hydrogen atomic levels are crucial for understanding electron transitions. These differences dictate what wavelengths of light are absorbed or emitted.Calculating Energy Differences:

• The energy of each shell in a hydrogen atom is given by the formula \( E_n = -\frac{13.6 \text{ eV}}{n^2} \), where \( n \) is the shell number.
• By subtracting the energies of respective shells, you find the energy difference needed to move an electron from one shell to another.

For example:

• The difference between \( n = 1 \) and \( n = 2 \) is 10.2 eV, requiring this energy for the transition.
• Between \( n = 2 \) and \( n = 3 \), the difference is 1.89 eV, illustrating a much smaller energy requirement for higher shells.

Understanding these differences is key to determining the wavelengths of light needed for electron transitions.

Wavelength Calculation

Wavelength calculation is a critical step in linking energy differences to observable electromagnetic radiation. The relationship between energy and wavelength is established through Planck’s equation \( E = \frac{hc}{\lambda} \).Steps to Calculate Wavelength:

• First, convert electron volt (eV) energy differences into Joules using the conversion factor \( 1 \text{ eV} = 1.6 \times 10^{-19} \text{ J} \).
• Apply Planck’s constant \( h = 6.626 \times 10^{-34} \text{ Js} \) and the speed of light \( c = 3.0 \times 10^8 \text{ m/s} \) into the formula.
• Rearrange to solve for wavelength: \( \lambda = \frac{hc}{E} \).

For instance, the energy difference for a transition from \( n=2 \) to \( n=3 \) converts to 1.89 eV and results in a wavelength of approximately 656.2 nm. This calculation helps identify the type of laser needed for a given transition.

Laser Transition

Laser transitions involve using specific wavelengths of light to induce transitions in an atom's electron configuration. Different transitions require lasers of precise wavelength, determined by the energy differences between the electron energy levels.Understanding Laser Transitions:

• To excite an electron from one shell to another, a laser with a wavelength corresponding to the calculated energy difference is needed.
• For the hydrogen atom, transitioning from \( n=2 \) to \( n=3 \) requires a longer wavelength laser, specifically around 656.2 nm based on calculations.

These principles are not only essential in understanding atomic physics but are also fundamental to laser technology and applications in spectroscopy and quantum mechanics. Real-world lasers use these principles to fine-tune their outputs for specific scientific and industrial purposes.