(a) Calculate and compare the energy of a photon with a wavelength of 3.0 mm to that of a photon with a wavelength of 0.3 nm.

Name: Chemistry: The Central Science
Author: Brown
ISBN: 9780134414232

Verified step by step guidance

Step 1: Understand the relationship between the energy of a photon and its wavelength using the formula: \( E = \frac{hc}{\lambda} \), where \( E \) is the energy of the photon, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ J s}) \), \( c \) is the speed of light \( (3.00 \times 10^8 \text{ m/s}) \), and \( \lambda \) is the wavelength of the photon.

Step 2: Convert the given wavelengths into meters to ensure consistency in units. For the first photon, convert 3.0 mm to meters (1 mm = 1 \(\times\) 10^{-3} m). For the second photon, convert 0.3 nm to meters (1 nm = 1 \(\times\) 10^{-9} m).

Step 3: Calculate the energy of the first photon using its wavelength in meters. Substitute the values of \( h \), \( c \), and the converted wavelength into the energy formula.

Step 4: Calculate the energy of the second photon using its wavelength in meters. Again, substitute the values of \( h \), \( c \), and the converted wavelength into the energy formula.

Step 5: Compare the energies of the two photons. Discuss how the difference in wavelength affects the energy, noting that shorter wavelengths correspond to higher energy photons.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Photon Energy

The energy of a photon is directly related to its wavelength and can be calculated using the equation E = hc/λ, where E is energy, h is Planck's constant (6.626 x 10^-34 J·s), c is the speed of light (3.00 x 10^8 m/s), and λ is the wavelength in meters. Shorter wavelengths correspond to higher energy photons, while longer wavelengths correspond to lower energy.

Wavelength and Frequency Relationship

Wavelength and frequency are inversely related through the equation c = λν, where c is the speed of light, λ is the wavelength, and ν (nu) is the frequency. As the wavelength increases, the frequency decreases, which in turn affects the energy of the photon. Understanding this relationship is crucial for comparing the energies of photons with different wavelengths.

Units of Measurement

In calculations involving photon energy, it is essential to use consistent units. Wavelengths may be given in millimeters (mm) or nanometers (nm), which must be converted to meters (m) for calculations. Recognizing and converting these units correctly ensures accurate energy calculations and comparisons between different photons.

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