(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.

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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(a) A green laser pointer emits light with a wavelength of 532 nm. What is the frequency of this light?

Textbook Question

(b) What is the energy of one of these photons?

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Textbook Question

(c) The laser pointer emits light because electrons in the material are excited (by a battery) from their ground state to an upper excited state. When the electrons return to the ground state, they lose the excess energy in the form of 532-nm photons. What is the energy gap between the ground state and excited state in the laser material?

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Textbook Question

An AM radio station broadcasts at 1000 kHz and its FM partner broadcasts at 100 MHz. Calculate and compare the energy of the photons emitted by these two radio stations.

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Textbook Question

One type of sunburn occurs on exposure to UV light of wavelength in the vicinity of 325 nm. (a) What is the energy of a photon of this wavelength?

Textbook Question

One type of sunburn occurs on exposure to UV light of wavelength in the vicinity of 325 nm. (b) What is the energy of a mole of these photons?