Problem 32 - Chemistry

Question

Calculate the frequency (Hz) and wavelength (nm) of the emitted photon when an electron drops from the \(n=4\) to the \(n=2\) level in a hydrogen atom.

Step-by-Step Solution

Answer

Using the given steps, the typical calculated values are, in Hz for frequency and nm for wavelength, approximately \( 4.57 x 10^{14} Hz \) and \( 656 nm \) respectively. Subsequent calculations might yield slightly different results based on rounding methods and precision of calculations.

Step 1: Apply Rydberg formula

Firstly, use the Rydberg formula to find the wave number \( \overline{\nu} \) of the emitted photon: \( \frac{1}{\lambda} = \overline{\nu} = R_H |\frac{1}{n_{lower}^2} - \frac{1}{n_{upper}^2}|\), where \( R_H \) is the Rydberg constant (\( 1.097373 x 10^7 m^{-1} \)), \( n_{lower} = 2 \), and \( n_{upper} = 4 \).

Step 2: Compute wave number

Input the values into the equation: \( \overline{\nu} = 1.097373 x 10^7 m^{-1} * |\frac{1}{2^2} - \frac{1}{4^2}| \). Calculate the result to get the wave number. This value will be in \(m^{-1}\).

Step 3: Calculate the wavelength in meters

Next, calculate the wavelength in meters using \( \frac{1}{\overline{\nu}} \). The wavelength, \( \lambda \), in meters is found by taking the reciprocal of the wave number.

Step 4: Convert Wavelength into nanometers

Convert the wavelength, \( \lambda \), found in the previous step from meters to nanometers (nm) by multiplying by \( 10^9 \).

Step 5: Calculate the frequency of the photon

Use the speed of light equation, \( c = \nu * \lambda \), to calculate the frequency (\nu). Make sure to use the wavelength in meters from step 3. Since the speed of light, \( c \), is known (approximately \( 3 x 10^8 m/s \)), and \( \lambda \) is calculated in step 3, you can solve for the frequency.

Key Concepts

Frequency of Emitted Photon

When an electron in a hydrogen atom transitions between energy levels, it either absorbs or emits a photon. The frequency of this photon () is directly related to the energy difference between the two levels. To understand this frequency, we use the Rydberg formula which relates the frequency, wavelength, and the quantum numbers of the involved energy levels.

The frequency can be obtained through the relationship between the speed of light (), wavelength (), and frequency: is given by (the speed of light) divided by (the wavelength). The speed of light is a constant (), approximately . Once the wavelength is known, the frequency can simply be calculated using the formula . This is crucial in spectroscopy and quantum mechanics, where the frequency of emitted or absorbed light reveals important information about the atomic structure.

Wavelength Calculation

The calculation of wavelength () of the emitted photon during an electron transition is an important aspect of understanding atomic spectra. Following the Rydberg formula, the inverse of the wavelength () is equal to the Rydberg constant () multiplied by the absolute difference between the reciprocals of the squares of the principal quantum numbers ().

Here's how you do it step by step: First, you apply the Rydberg formula to find the wave number (). Next, you take its reciprocal to find the wavelength in meters. It's essential to remember to convert this value into nanometers () since this unit is more commonly used in spectroscopy. The Rydberg formula is an incredible tool because it allows us to predict the position of the spectral lines solely based on the quantum numbers of the electron transitions.

Hydrogen Atom Electron Transition

Electron transitions in a hydrogen atom refer to the movement of an electron between different energy levels or shells around the nucleus. According to Niels Bohr's model of the atom, these levels are quantized which means electrons can only exist in certain allowed levels. When an electron drops from a higher-level () to a lower-level (), energy is released in the form of a photon, which leads to the emission of light that can be observed in a spectrum.

These transitions are governed by certain rules, one of which is the selection rule that dictates the change in quantum number between the two levels involved in the transition. Hydrogen atom electron transitions are quantitatively described by the Rydberg formula, making it a cornerstone for understanding atomic structure and the underlying principles of quantum mechanics.