Problem 111 - Chemistry Matter and Change

Question

Hydrogen Atom The hydrogen atom's energy is \(-6.05 \times 10^{-20} \mathrm{J}\) when the electron is in the \(n=6\) orbit and \(-2.18 \times 10^{-18} \mathrm{J}\) when the electron is in the \(n=1\) Calculate the wavelength of the photon emitted when the electron drops from the \(n=6\) orbit to the \(n=1\) orbit. Use the following values: \(h=6.626 \times 10^{-34} \mathrm{J} \bullet \mathrm{s}\) and \(c=3.00 \times 10^{8} \mathrm{m} / \mathrm{s}\)

Step-by-Step Solution

Answer

The wavelength of the photon emitted when the electron drops from the n=6 orbit to the n=1 orbit in a hydrogen atom is approximately \(9.38 \times 10^{-8} \: \mathrm{m}\).

Step 1: Calculate the energy difference

We are given the energy of the hydrogen atom in the n=6 orbit and the n=1 orbit. To find the energy of the photon emitted when the electron drops from the n=6 orbit to the n=1 orbit, we need to find the energy difference between the two orbits. To calculate this, subtract the energy of the n=1 orbit from the energy of n=6 orbit: Energy difference = \((-6.05 \times 10^{-20} \: \mathrm{J}) - (-2.18 \times 10^{-18} \: \mathrm{J})\)

Step 2: Calculate the energy of the photon

Now that we have the energy difference, we can use this to find the energy of the emitted photon. Since the energy difference is equal to the energy of the photon emitted, we can write: Energy of the photon = \((2.18 \times 10^{-18} \: \mathrm{J}) - (6.05 \times 10^{-20} \: \mathrm{J}) = 2.12 \times 10^{-18} \: \mathrm{J}\)

Step 3: Calculate the frequency of the photon

Using Planck's constant (h) and the energy of the photon, we can find the frequency (f) of the photon emitted: \(f = \frac{Energy\:of\:the\:photon}{h}\) \(f = \frac{2.12 \times 10^{-18} \: \mathrm{J}}{6.626 \times 10^{-34} \: \mathrm{J\: s}} = 3.20 \times 10^{15} \: \mathrm{Hz}\)

Step 4: Calculate the wavelength of the photon

Now that we have the frequency of the emitted photon, we can use the speed of light (c) to find the wavelength (λ) of the photon using the equation: \(λ = \frac{c}{f}\) \(λ = \frac{3.00 \times 10^{8} \: \mathrm{m/s}}{3.20 \times 10^{15} \: \mathrm{Hz}} = 9.38 \times 10^{-8} \: \mathrm{m}\) So, the wavelength of the photon emitted when the electron drops from the n=6 orbit to the n=1 orbit in a hydrogen atom is approximately \(9.38 \times 10^{-8} \: \mathrm{m}\).

Key Concepts

Bohr Model

The Bohr model is an early atomic model that describes the hydrogen atom. It was proposed by Niels Bohr in 1913.

• According to this model, electrons orbit the nucleus in fixed paths called orbits, much like planets orbit the sun.
• Each orbit corresponds to an energy level. Electrons can jump between these levels by absorbing or emitting energy, typically in the form of photons.

When an electron moves to a lower energy orbit, it emits a photon. The energy of this photon corresponds to the energy difference between the initial and final orbits.
The Bohr model was revolutionary because it introduced quantized energy levels, which explained the discrete spectral lines observed in hydrogen.

Photons

Photons are the basic units of light. They are particles of electromagnetic radiation and carry energy.

• In the context of the Bohr model, when an electron moves between energy levels, a photon is either absorbed or emitted.
• The energy of this photon is equal to the difference in energy between the two levels.

This concept of photon emission and absorption is crucial in understanding atomic spectra and the interaction of light with matter.
Photons have no mass and always move at the speed of light, which is roughly \(3.00 \times 10^8 \:\mathrm{m/s}\).
Understanding photons helps explain a wide range of phenomena such as how solar panels work and how we perceive colors.

Planck's Constant

Planck's constant, denoted by \( h \), is a fundamental constant in physics that relates the energy of a photon to its frequency.
This constant is a core concept in quantum mechanics and is used in the formula:
\[ E = h \times f \]
where \( E \) is the energy of the photon and \( f \) is its frequency.

• In our context, you use Planck's constant to calculate the frequency of a photon given its energy.
• This helps us find the wavelength of light when the frequency is known, linking energy levels and spectroscopic data.

Understanding and using Planck's constant allows for precise calculations in quantum mechanics and is fundamental to the study of light and matter.

Wavelength Calculation

Wavelength calculation is an important step in understanding how energy transitions relate to photonic emissions.
When an electron transitions between energy levels, the wavelength of the emitted photon can be calculated using the following relation:
\[ \lambda = \frac{c}{f} \]
where \( \lambda \) is the wavelength, \( c \) is the speed of light, and \( f \) is the frequency of the emitted photon.

• This relationship shows the inverse correlation between frequency and wavelength; as the frequency of a photon increases, its wavelength decreases.
• Wavelength helps us categorize different types of electromagnetic radiation, from radio waves to gamma rays.

Accurate calculation of wavelengths is essential not just in physics, but also in applications like telecommunications and medical imaging.