Question

Careful spectral analysis shows that the familiar yellow light of sodium lamps (such as street lamps) is made up of photons of two wavelengths, \(589.0 \mathrm{nm}\) and \(589.6 \mathrm{nm}\). What is the difference in energy (in joules) between photons with these wavelengths?

Short answer

The energy difference is approximately \(3.39 \times 10^{-26}\) joules.

Step-by-step solution

Understand the Photon Energy Formula

The energy of a photon, given its wavelength, can be calculated using the formula \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant \((6.626 \times 10^{-34} \text{ J s})\), \( c \) is the speed of light \((3 \times 10^8 \text{ m/s})\), and \( \lambda \) is the wavelength of the photon in meters.

Convert Wavelengths to Meters

Convert the given wavelengths from nanometers to meters by multiplying by \(10^{-9}\): \(589.0 \text{ nm} = 589.0 \times 10^{-9} \text{ m}\) and \(589.6 \text{ nm} = 589.6 \times 10^{-9} \text{ m}\).

Calculate Energy of First Wavelength

Substitute the first wavelength \(589.0 \times 10^{-9} \text{ m}\) into the photon energy formula: \(E_1 = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{589.0 \times 10^{-9}}\). Calculate \(E_1\).

Calculate Energy of Second Wavelength

Substitute the second wavelength \(589.6 \times 10^{-9} \text{ m}\) into the photon energy formula: \(E_2 = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{589.6 \times 10^{-9}}\). Calculate \(E_2\).

Calculate the Difference in Energy

Subtract the energy of the photon with the second wavelength from the energy of the photon with the first wavelength: \(\Delta E = E_1 - E_2\). Calculate the result.

Key concepts

Planck's Constant

The concept of Planck's constant is instrumental in understanding quantum mechanics. It acts as a bridge between the energy of a photon and its frequency. This constant, denoted by \( h \), has a fixed value of \( 6.626 \times 10^{-34} \text{ J s} \).
It represents the smallest action or quantum of electromagnetic action in quantum mechanical systems. In simpler terms, it helps define the scales at which quantum effects become significant.

The significance of Planck's constant is evident in the photon energy formula \( E = \frac{hc}{\lambda} \), where it links directly to energy, wavelength, and the speed of light.
Any change in wavelength, as observed in spectral analysis, influences the photon's energy because of this constant factor. Understanding Planck's constant allows students to appreciate how quantized energy levels are calculated.

Speed of Light

The speed of light, symbolized as \( c \), is a fundamental constant in physics with a value of \(3 \times 10^8 \mathrm{m/s}\). It represents the speed at which light travels in a vacuum.
This constant is crucial because it is used in the formula to calculate the energy of a photon, \( E = \frac{hc}{\lambda} \). By multiplying Planck's constant and the speed of light, we find the photon energy at specific wavelengths.

Understanding how speed affects calculations is crucial, especially in equations dealing with light and electromagnetic radiation. Whether in natural phenomena or scientific applications, the speed of light is key in defining how light and radiation move through space.
Its inclusion in computations, like those in spectral analysis, shows its universal importance in physics.

Wavelength Conversion

Converting wavelength units is a necessary step in calculations involving light and photon energy. Wavelength can be given in various units, such as nanometers or meters, but calculations often require it to be in meters.
To convert nanometers to meters, multiply the value by \( 10^{-9} \). For example:

• 589.0 nm converts to \(589.0 \times 10^{-9} \) m
• 589.6 nm converts to \(589.6 \times 10^{-9} \) m

Understanding unit conversion ensures that calculations using the photon energy formula are accurate.
It's a basic skill that aids in accurate scientific analysis, ensuring each element in a formula uses compatible units.

Energy Difference

Knowing how to calculate the difference in energy can help determine the impact of varying photonic wavelengths. We use the photon energy formula for each wavelength to find their energies, then subtract the results.
This tells us how much more or less energy a photon holds at different wavelengths. For example, if \( E_1 \) is the energy of the first wavelength and \( E_2 \) is the energy of the second, then the energy difference \( \Delta E \) is found by computing \( E_1 - E_2 \).

The achieved energy difference holds practical importance in various fields, including telecommunication and astronomy, where precise wavelength changes can result in notable signal shifts or stellar observations.

Spectral Analysis

Spectral analysis is the method of studying how light interacts with matter, often used to identify chemical compositions or study astronomical objects.
By dissecting light into its component wavelengths, scientists can infer various properties of the light source.

This technique relies heavily on understanding photon energy as different wavelengths correspond to different energy levels. The detailed measurement of these wavelengths enables researchers to interpret the energy emitted or absorbed under different conditions.
Being familiar with spectral analysis allows students to connect theory with practical observation, deepening their understanding of how light contributes to our knowledge of the universe.