Problem 29 - Chemistry The Central Science

Question

A diode laser emits at a wavelength of \(987 \mathrm{nm}\). (a) In what portion of the electromagnetic spectrum is this radiation found? (b) All of its output energy is absorbed in a detector that measures a total energy of \(0.52 \mathrm{~J}\) over a period of \(32 \mathrm{~s}\). How many photons per second are being emitted by the laser?

Step-by-Step Solution

Answer

The given radiation with a wavelength of 987 nm falls within the near-infrared (NIR) range of the electromagnetic spectrum. The laser emits approximately 8.09 × 10^{16} photons per second.

Step 1: Identify the portion of the electromagnetic spectrum

The given wavelength for the diode laser is 987 nm. We can use the wavelength to find the portion of the electromagnetic spectrum it belongs to. By referring to the electromagnetic spectrum ranges, we can identify that a 987 nm wavelength falls within the near-infrared (NIR) range.

Step 2: Calculate the energy of a single photon

To calculate the energy of a single photon, we can use the formula: Energy (E) = \( \cfrac{hc}{λ} \) where h = Planck's constant (6.63 × 10^{-34} Js), c = speed of light (3 × 10^8 m/s), λ = wavelength (987 nm or 987 × 10^{-9} m). Plugging the values into the equation, we get: E = \( \cfrac{6.63 × 10^{-34} Js × 3 × 10^8 m/s}{987 × 10^{-9} m} \) Calculating this value, we obtain: E ≈ 2.01 × 10^{-19} J

Step 3: Calculate the total number of photons

Now, we can find the total number of photons using the absorbed energy and the energy of a single photon. The absorbed energy is given as 0.52 J and the energy of a single photon is 2.01 × 10^{-19} J. Total number of photons (N) = \( \cfrac{Total~energy~absorbed}{Energy~of~a~single~photon} \) N = \( \cfrac{0.52 J}{2.01 × 10^{-19} J} \) Calculating this value, we obtain: N ≈ 2.59 × 10^{18}

Step 4: Calculate photons emitted per second

We can find the number of photons emitted per second by dividing the total number of photons by the duration of time (32 seconds). Photons emitted per second = \( \cfrac{Total~number~of~photons}{Time~duration} \) Photons emitted per second = \( \cfrac{2.59 × 10^{18}}{32 s} \) Calculating this value, we obtain: Photons emitted per second ≈ 8.09 × 10^{16} Hence, the laser emits approximately 8.09 × 10^{16} photons per second.

Key Concepts

Photon Energy

Photon energy is a fundamental concept when exploring the electromagnetic spectrum, as it represents the energy carried by a single photon. Photons are the building blocks of light, and their energy is directly proportional to its frequency. This means that as the frequency increases, so does the energy of the photon.

To calculate photon energy, the formula used is:

• Energy (E) = \( \cfrac{hc}{\lambda} \)

where:

• \( h \) is Planck's constant, approximately \( 6.63 \times 10^{-34} \) Js.
• \( c \) is the speed of light, about \( 3 \times 10^8 \) m/s.
• \( \lambda \) symbolizes the wavelength.

In our example, a diode laser emits light with a wavelength of \( 987 \) nm. To convert this to meters for our calculation, we use \( \lambda = 987 \times 10^{-9} \) m. Substituting these values into the formula, the energy per photon works out to \( 2.01 \times 10^{-19} \) Joules.

Understanding photon energy helps in calculating how much energy is interacting with devices such as detectors, which is crucial in fields from communications to physics research.

Wavelength Calculation

Wavelength calculation is key in categorizing light within the electromagnetic spectrum. The wavelength of a photon determines its type, ranging from radio waves to gamma rays. By knowing where a specific wavelength falls, we understand its properties and applications.

For a diode laser emitting at \( 987 \) nm, identifying this wavelength's place in the electromagnetic spectrum reveals it is within the near-infrared region. The electromagnetic spectrum includes, in order of increasing frequency:

• Radio waves
• Microwaves
• Infrared (of which near-infrared is a part)
• Visible light
• Ultraviolet
• X-rays
• Gamma rays

Near-infrared (NIR) is often utilized in remote controls, optical fiber communications, and medical diagnostics. Recognizing these uses helps us comprehend how technology applies different wavelengths to perform tasks from simple household to advanced medical equipment.

Diode Laser

A diode laser is a type of laser widely used across numerous applications due to its compact size and efficiency. Unlike other lasers, it uses a semiconductor as the gain medium, which produces a coherent light beam. It’s primarily used in communications, medicine, and CD/DVD players due to its suitable wavelength emissions.

The term 'diode' refers to the electrical component integral to the laser’s function. When electricity is applied, it emits light in the near-infrared, visible, or ultraviolet spectrum. For instance, our diode laser with a wavelength of \( 987 \) nm falls in the near-infrared spectrum.

How Photons Emerge from a Diode Laser

When electric current passes through the diode, energy is provided to electrons in the semiconductor. These electrons transition to higher energy levels. When they fall back to their original levels, they release energy in the form of photons, generating the laser light.

Diode lasers are valued for their efficiency, reliability, and the ability to produce a focused beam of light. These properties make them advantageous in applications from fiber-optic communications to surgical instruments.