Problem 63 - Chemistry: Principles and Reactions

Question

A carbon dioxide laser produces radiation of wavelength \(10.6\) micrometers (1 micrometer \(=10^{-6}\) meter). If the laser produces about one joule of energy per pulse, how many photons are produced per pulse?

Step-by-Step Solution

Answer

Answer: Approximately 5.34 × 10^{18} photons are produced per pulse.

Step 1: Calculate the energy of a single photon

First, let's calculate the energy of a single photon using the given wavelength and the formula E = h * c / λ. We have: - h (Planck's constant) = 6.626 × 10^{-34} J·s - c (speed of light) = 3.0 × 10^8 m/s - λ (wavelength) = 10.6 × 10^{-6} m (converting from micrometers to meters) Using these values, we can calculate the energy of a single photon: E = (6.626 × 10^{-34} J·s) * (3.0 × 10^8 m/s) / (10.6 × 10^{-6} m).

Step 2: Evaluate the energy of a single photon

Now, let's evaluate the energy of a single photon using the values from the previous step: E = (6.626 × 10^{-34} J·s) * (3.0 × 10^8 m/s) / (10.6 × 10^{-6} m) = 1.874 × 10^{-19} J The energy of a single photon is 1.874 × 10^{-19} Joules.

Step 3: Calculate the number of photons produced per pulse

We are given that the laser produces about one Joule of energy per pulse. To find the number of photons produced in each pulse, we can divide the total energy per pulse by the energy of a single photon: Number of photons = (Total energy per pulse) / (Energy of a single photon) = 1 J / (1.874 × 10^{-19} J).

Step 4: Evaluate the number of photons produced per pulse

Finally, let's evaluate the number of photons produced per pulse: Number of photons = 1 J / (1.874 × 10^{-19} J) = 5.34 × 10^{18} Approximately 5.34 × 10^{18} photons are produced per pulse by the carbon dioxide laser.

Key Concepts

Planck's Constant

Planck's constant is a fundamental quantity that plays a crucial role in quantum mechanics. It is denoted by the letter \(h\) and is used to describe the sizes of quanta, the smallest possible discrete unit of energy. The value of Planck's constant is approximately \(6.626 \times 10^{-34} \) Joule seconds (J·s). This constant helps us understand how energy and frequency relate on a quantum scale.

When we use Planck's constant in formulas, it allows us to calculate the energy of a photon, which is a particle of light, based on its frequency or wavelength. This relationship is given by the equation \(E = h \times f\), where \(E\) represents the energy, \(h\) is Planck's constant, and \(f\) is the frequency of the photon.

• The smaller the Planck's constant, the smaller the energy for a given frequency.
• It emphasizes the particle-like properties of light and establishes a bridge to quantum mechanics.

Understanding Planck's constant is essential for studying quantum physics and helps explain the behavior of particles at microscopic scales.

Speed of Light

The speed of light, often denoted by the symbol \(c\), is one of the fastest and most fundamental constants known in physics. It represents the speed at which light travels in a vacuum and is approximately \(3.0 \times 10^{8} \) meters per second (m/s). The speed of light is pivotal in understanding various concepts in physics, including relativity and quantum mechanics.

The speed of light is used in calculations involving the energy of photons since it connects the wavelength and frequency of light to its energy. This is illustrated in the equation \(E = \frac{h \times c}{\lambda}\), where \(E\) is the energy of the photon, \(h\) is Planck's constant, and \(\lambda\) is the wavelength.

• In a vacuum, the speed of light is constant and does not depend on the motion of the light source or observer.
• This constant acts as a cosmic speed limit, setting the maximum speed at which information and matter can travel.

Knowing the speed of light is crucial for numerous calculations in physics and helps us explore the universe's diverse phenomena.

Wavelength

Wavelength is a concept used to describe the distance between repeating units in a wave pattern, such as the crests of a wave. In the context of light, it refers to the distance between two consecutive points of similar phase in the wave, like two peaks. Wavelength is commonly represented by the Greek letter \(\lambda\) and is usually measured in meters (m).

Wavelength and frequency are inversely related; as one increases, the other decreases. This relationship is described by the equation \(c = \lambda \times f\), where \(c\) is the speed of light, \(\lambda\) is the wavelength, and \(f\) is the frequency.

• Longer wavelengths correspond to lower-energy waves, such as radio waves.
• Shorter wavelengths, like those of gamma rays, mean higher-energy waves.

Wavelength is a key parameter when calculating photon energy because it helps determine the energy associated with a particular photon. It allows us to apply the equation \(E = \frac{h \times c}{\lambda}\) to find the energy per photon based on its wavelength.

Energy per Photon

The energy per photon is a measure of the energy contained in a single particle of light. To calculate this, we use the formula \(E = \frac{h \times c}{\lambda}\), where \(E\) is the energy of the photon, \(h\) denotes Planck's constant, \(c\) stands for the speed of light, and \(\lambda\) is the wavelength of the light.

This equation shows that the energy of a photon is inversely proportional to its wavelength. Therefore, photons with shorter wavelengths carry more energy than those with longer wavelengths. For example, ultraviolet light photons have more energy than infrared photons.

• Energy per photon is crucial when analyzing the photon emissions from light sources, such as lasers.
• The calculated energy per photon helps determine the number of photons needed to produce a certain amount of energy.

Understanding the energy of a photon is vital in various applications, including spectroscopy, telecommunications, and the study of atomic structures. This concept also forms the basis for many technological advancements in fields like solar energy and quantum computing.