Problem 17 - General Chemistry: Principles and Modern Applications

Question

Calculate the wavelengths, in nanometers, of the first four lines of the Balmer series of the hydrogen spectrum, starting with the longest wavelength component.

Step-by-Step Solution

Answer

The wavelengths in nanometers of the first four lines of the Balmer series, from longest to shortest wavelengths, are calculated by substituting each transition into the Rydberg formula.

Step 1: Identify Energy Levels

Identify the energy levels for each line in the Balmer series. Since the Balmer series corresponds to transitions to the n=2 level, the four lines represent transitions from levels n=3, n=4, n=5, and n=6 (since we start from the longest wavelength).

Step 2: Calculate Wavelengths using the Rydberg Formula

Plug the energy levels into the Rydberg formula for each transition and calculate the resulting wavelength. Use R=1.097373 x 10⁷ m⁻¹ and Z=1. Convert the resulting wavelengths from meters to nanometers by multiplying by 1 x 10⁹.

Step 3: Transition from n=3 to n=2

Calculate the wavelength for the transition from n=3 to n=2 by substituting into the Rydberg formula: 1/λ = R (1/2² - 1/3²). Solve for λ and convert the result to nanometers.

Step 4: Transition from n=4 to n=2

Calculate the wavelength for the transition from n=4 to n=2: 1/λ = R (1/2² - 1/4²). Solve for λ and convert to nanometers.

Step 5: Transition from n=5 to n=2

Calculate the wavelength for the transition from n=5 to n=2: 1/λ = R (1/2² - 1/5²). Solve for λ and convert to nanometers.

Step 6: Transition from n=6 to n=2

Calculate the wavelength for the transition from n=6 to n=2: 1/λ = R (1/2² - 1/6²). Solve for λ and convert to nanometers.

Key Concepts

hydrogen emission spectrum

The hydrogen emission spectrum is an incredible phenomenon that offers valuable insight into the structure of atoms. When an electron in a hydrogen atom transitions from a higher energy level to a lower one, it emits energy in the form of light. This light can be broken down into distinct lines that make up the emission spectrum, each corresponding to a specific wavelength. The Balmer series is one of these spectral lines. It is observed when an electron falls to the second energy level (n=2) from higher levels like n=3, n=4, n=5, etc.
These visible lines are important because they were among the first pieces of evidence supporting quantum theories in physics. By analyzing these lines, we can better understand the quantized nature of energy in hydrogen atoms.

Rydberg formula

The Rydberg formula is a powerful tool used to calculate the wavelengths of spectral lines of hydrogen. This formula is specifically designed to predict the wavelengths emitted by electrons dropping to lower energy levels. The basic form of the Rydberg formula is:

• \( \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \)

Here, \( \lambda \) is the wavelength, \( R \) is the Rydberg constant (approximately 1.097373 x 10⁷ m⁻¹), and \( n_1 \) and \( n_2 \) are the lower and higher energy levels, respectively. For the Balmer series, \( n_1 \) is always 2 because the transitions end at the n=2 level.
By using this formula, we can precisely calculate the wavelengths corresponding to different electron transitions and thus map out the hydrogen emission spectrum accurately.

wavelength calculations

Wavelength calculations involve using the Rydberg formula to find the exact wavelengths of light emitted when electrons in hydrogen atoms transition between energy levels. For the Balmer series, we focus on transitions that fall to the n=2 level from higher levels.
To perform these calculations:

• Identify the initial and final energy levels. For the Balmer series, these transitions end at n=2, coming from n=3, n=4, n=5, and n=6.
• Substitute these values into the Rydberg formula. For instance, for a transition from n=3 to n=2, the formula would be \( \frac{1}{\lambda} = R \left( \frac{1}{2^2} - \frac{1}{3^2} \right) \).
• Solve for \( \lambda \) to find the wavelength in meters, then convert it to nanometers by multiplying by 1 x 10⁹.

These calculations allow us to predict the specific wavelengths of visible light that result from electron transitions in the hydrogen atom. Understanding these concepts is key to exploring more complex atomic models and spectral analysis.