Question

The sodium D-lines, two very distinct lines of yellow light emitted by heated sodium metal, are often used in the calibration of spectrometers. The wavelength of one of these lines is \(5890 \AA\). How much energy does an electron emit (or absorb) in undergoing the electronic transition associated with this line?

Short answer

The energy emitted or absorbed by an electron during the electronic transition associated with the given sodium D-line is approximately \(3.377 \times 10^{-19}\) joules.

Step-by-step solution

Convert given wavelength to meters

The provided wavelength is in angstroms (Å). We need to convert it to meters to work with SI units: 1 Å = \(1 \times 10^{-10}\) m Given wavelength λ = 5890 Å = \(5890 \times 10^{-10}\) m

Calculate frequency

Using the speed of light formula, we solve for frequency: \(c = \lambda \cdot f\) Rearrange the formula for frequency (f): \(f = \frac{c}{\lambda}\) Plugging in the values: - Speed of light (c) = \(3.0 \times 10^8\) m/s - Wavelength (λ) = \(5890 \times 10^{-10}\) m \(f = \frac{3.0 \times 10^8}{5890 \times 10^{-10}}\) Frequency (f) = \(5.093 \times 10^{14}\) Hz

Calculate energy difference

Now that we have the frequency, we can calculate the energy difference using the energy formula: \(E = h \cdot f\) Plugging in the values: - Planck's constant (h) = \(6.626 \times 10^{-34}\) J·s - Frequency (f) = \(5.093 \times 10^{14}\) Hz \(E = (6.626 \times 10^{-34}) (5.093 \times 10^{14})\) Energy (E) = \(3.377 \times 10^{-19}\) J The energy emitted or absorbed by an electron during the electronic transition associated with the given sodium D-line is approximately \(3.377 \times 10^{-19}\) joules.

Key concepts

Spectrometer Calibration

Calibration of a spectrometer is crucial to ensure its measurements are accurate and reliable. When using sodium D-lines, these lines are distinct and occur naturally at specific known wavelengths. This is why they serve as ideal reference points for calibration.
The sodium D-line at 5890 Å is particularly useful. By calibrating a spectrometer with this specific line, the device can maintain a high level of precision in identifying other spectral lines.

• Provides benchmark for accurate measurement
• Ensures data reliability in experiments
• Makes identification of unknown lines more precise

Precision in calibration ultimately affects the outcomes of any experiment relying on spectrometer readings, ensuring that scientists and researchers can trust the data they collect.

Wavelength Conversion

Converting wavelength measurements from angstroms to meters is necessary for working within the International System of Units (SI). Angstroms are often used for their convenience at the atomic scale, but meters are the standard.
To convert the sodium D-line from angstroms (Å) to meters (m), you remember that \(1 \, \text{Å} = 1 \times 10^{-10} \, \text{m}\). Thus, 5890 Å becomes \(5890 \times 10^{-10}\ m\).

• Helps use consistent units in calculations
• Ensures compatibility with other measurement units
• Aligns with standard scientific practices

This conversion makes it easier to apply physical constants like the speed of light \(c = 3.0 \times 10^8\, \text{m/s}\).

Frequency Calculation

Frequency represents how often a wave passes a specific point in one second and is calculated using the speed of light and the wavelength. This involves the formula:\[f = \frac{c}{\lambda}\]
Here, the speed of light \(c\) is \(3.0 \times 10^8\, \text{m/s}\) and \(\lambda\) is \(5890 \times 10^{-10}\, \text{m}\). Solving gives a frequency \(f\) of \(5.093 \times 10^{14}\, \text{Hz}\).

• Indicates light's cycles per second
• Crucial for further energy calculations
• Related directly to energy in quantum mechanics

Converting between wavelength and frequency is vital in many areas of science, such as quantum mechanics and optics.

Energy Transition

In atomic physics, energy transitions indicate an electron moving between energy levels, emitting or absorbing energy. For the sodium D-line, the energy difference can be calculated with:\[E = h \cdot f\]
This uses Planck's constant \(h = 6.626 \times 10^{-34}\, \text{J·s}\), and the frequency \(f = 5.093 \times 10^{14}\, \text{Hz}\), giving energy \(E = 3.377 \times 10^{-19}\, \text{J}\).

• Explains phenomena like light emission and absorption
• Shows quantum level energy interactions
• Key in understanding spectroscopic data

Energy transitions provide the foundation for interpreting spectral lines in terms of underlying atomic processes.

Planck's Constant

Planck's constant is a fundamental physical constant essential for describing the particles at the quantum level. It bridges the concepts of wave and particle nature of light and relates to energy (\(E\)) and frequency (\(f\)) through\[E = h \cdot f\].
It is valued at \(6.626 \times 10^{-34}\, \text{J·s}\), illustrating how small energy quantities are involved at the atomic level.

• Key to quantum mechanics
• Used in calculations of photon energy
• Integral in wave-particle duality discussion

Ultimately, Planck's constant helps provide a deeper understanding of how energy operates within the quantum realm, reinforcing the discrete nature of electronic transitions.