Question

High-pressure sodium vapor lamps are used in street lighting. The two brightest lines in the sodium spectrum are at 589.00 and \(589.59 \mathrm{nm}\). What is the difference in energy per photon of the radiations corresponding to these two lines?

Short answer

The difference in energy per photon of the radiations corresponding to these two lines can be found by first calculating the energies associated with each line using Planck's equation and then subtracting to find the difference.

Step-by-step solution

Conversion of Wavelengths to Frequencies

The first step involves converting the wavelengths of the sodium lines given in nanometers (nm) to frequencies in Hz. The speed of light (\(c\)) in a vacuum and the formula \(c = \lambda f\) can be used for this conversion, where \(\lambda\) is the wavelength and \(f\) is the frequency. Note: The speed of light \(c\) is \(3.00 × 10^8 m/s\). Since the wavelengths are in nm, they should be converted to meters: \(589.00 nm = 589.00 × 10^-9 m\), \(589.59 nm = 589.59 × 10^-9 m\). Frequencies can then be calculated as \(f_1 = c / \lambda_1\) and \(f_2 = c / \lambda_2\)

Calculate the Energy of Each Photon

Energies corresponding to each frequency can then be calculated using Planck’s formula \(E = hf\), where \(h\) is Planck’s constant (\(6.626 × 10^{-34} J.s\)). Therefore, the energies for the two frequencies would be \(E_1 = h f_1\) and \(E_2 = h f_2\)

Find the Difference in Energy

Finally, subtract the smaller energy from the larger one to get the difference in energy per photon for the two lines. \( \Delta E = E_2 - E_1 \)

Key concepts

High-pressure sodium vapor lamps

High-pressure sodium vapor lamps are a common choice for street lighting and industrial facilities, owing to their high efficiency and long life span. But how do they work? At their core, these lamps use sodium and mercury under high pressure to emit light. Sodium atoms, when excited by an electric current, jump to higher energy levels. When they return to lower energy states, they release photons, which are particles of light. This process is what produces the characteristic orange-yellow glow.

These lamps are incredibly efficient because they convert a significant portion of electrical power into visible light. Due to their structure:

• They contain a small tube filled with sodium and other gases.
• Once electricity passes through this tube, it excites the sodium atoms.
• The merging of this excitation and relaxation of sodium atoms generates light, especially in the visible spectrum.

This makes them ideal for large-scale, low-energy lighting needs.

Sodium spectrum wavelengths

When we discuss sodium spectrum wavelengths, we're referring to the very specific colors of light that sodium atoms emit. These emissions fall primarily in the visible spectrum in forms of narrow lines called emission lines. The two lines at 589.00 nm and 589.59 nm are particularly notable due to their brightness, and are often referred to as the "sodium doublet."

These wavelengths result from specific electron transitions within the sodium atom. As these electrons jump between energy levels, they release energy in precise wavelengths that we can see as specific colors. Understanding these wavelengths helps us precisely calculate the energy levels of atoms, which is crucial in fields like quantum mechanics and spectroscopy.

The sodium doublet lines are important because:

• They are used to identify sodium in various chemical and astronomical observations.
• They act as a reference in calibrating spectroscopic instruments.

This dual-line emission is a testament to the unique properties of sodium and its place in the study of atomic spectra.

Photon energy calculations

Photon energy calculations involve the relationship between a photon's energy and its wavelength. These calculations are pivotal in understanding how much energy is associated with light of different wavelengths, including the ones emitted by sodium lamps. We use Planck’s equation for these calculations, which is expressed as:\[ E = hf \] Here, \( E \) denotes the energy of the photon, \( h \) signifies Planck's constant \( (6.626 \times 10^{-34} \, \text{J.s}) \), and \( f \) indicates the frequency of the light.

The frequency can be derived from the wavelength using the speed of light \( c \), where:\[ f = \frac{c}{\lambda} \] By calculating the frequency for each wavelength, we can then determine their corresponding photon energies. The final step in understanding the energy emitted by these photons is calculating the difference between them, which shows how minor variations in wavelength can encapsulate different energy levels:

• This is critical in fields such as quantum physics, where understanding energy levels within atoms is foundational.
• It also aids in the development of technology that is reliant on specific wavelengths or energies, such as lasers or LED lights.

This approach to calculating photon energy underpins much of our understanding of light and energy interactions in atom-level physics.