Problem 86 - Chemistry: The Central Science in SI Units, Global Edition

Question

The watt is the derived SI unit of power, the measure of energy per unit time: \(1 \mathrm{~W}=1 \mathrm{~J} / \mathrm{s}\). A semiconductor laser in a DVD player has an output wavelength of \(650 \mathrm{nm}\) and a power level of \(5.0 \mathrm{~mW}\). How many photons strike the DVD surface during the playing of a DVD 90 minutes in length?

Step-by-Step Solution

Answer

The number of photons that strike the DVD surface during the playing of a 90-minute DVD is approximately \(8.8 \cdot 10^{19}\) photons.

Step 1: Calculate the energy per photon

To calculate the energy per photon, we will use the formula: \[E = h \cdot f\] Where E is the energy, h is the Planck's constant (approx. \(6.63 \cdot 10^{-34} Js\)), and f is the frequency of the light. Since we are given the wavelength \(\lambda\), we can calculate the frequency using the speed of light (approx. \(3\cdot 10^8 m/s\)): \[f = \frac{c}{\lambda}\] Now plug in the values to find the frequency and then the energy per photon: \[ f = \frac{3\cdot 10^8 m/s}{650 \cdot 10^{-9} m} \Rightarrow f \approx 4.62 \cdot 10^{14} s^{-1} \] \[ E = h \cdot f = (6.63 \cdot 10^{-34} Js) \cdot (4.62 \cdot 10^{14} s^{-1}) \Rightarrow E \approx 3.06 \cdot 10^{-19} J \]

Step 2: Calculate the total energy delivered by the laser

We have the power of the laser given as \(5.0 mW\). To find the total energy delivered by the laser, we will use the formula: \[E_\text{total} = P \cdot t\] Where \(E_\text{total}\) is the total energy, P is the power, and t is the time. First, we need to convert the power to watts and the time to seconds: \[ P = 5.0 mW = 5.0 \cdot 10^{-3} W \] \[ t = 90 \text{ minutes} = (90)(60) \text{ seconds} = 5400 \text{ seconds} \] Now, plug in the values to find the total energy delivered in the given time: \[ E_\text{total} = (5.0 \cdot 10^{-3} W) \cdot (5400 s) \Rightarrow E_\text{total} \approx 27 J \]

Step 3: Calculate the number of photons striking the DVD surface

Now we can find the number of photons by dividing the total energy by the energy per photon: \[ \text{Number of photons} = \frac{E_\text{total}}{E} = \frac{27 J}{3.06 \cdot 10^{-19} J} \] \[ \text{Number of photons} \approx 8.8 \cdot 10^{19} \] So, approximately \(8.8 \cdot 10^{19}\) photons strike the DVD surface during the playing of a 90-minute DVD.

Key Concepts

Energy per Photon

When we talk about energy per photon, we're discussing the amount of energy carried by a single particle of light, known as a photon. To determine this, we use Planck's equation:

• \(E = h \cdot f\)

Here, \(E\) represents energy, \(h\) is Planck’s constant \(\approx 6.63 \times 10^{-34} Js\), and \(f\) is the frequency of the light. Frequency is crucial to finding the energy since it tells us how many wave cycles pass a point per second. More cycles mean more energy per photon.
Given a certain wavelength \(\lambda\), the frequency \(f\) can be found using the speed of light \(c\):

• \(f = \frac{c}{\lambda}\)

This relationship helps convert the known wavelength to frequency value, allowing the calculation of energy per photon.

Semiconductor Laser

A semiconductor laser is a type of laser that uses semiconductor material as its gain medium, typically known for its role in DVD players and other optical devices. These lasers emit light through the process of "stimulated emission," a fundamental concept in laser physics.
When the semiconductor material is appropriately energized, it can emit photons in a coherent beam — that is, with the photons all lined up phase-wise, which is why lasers are so focused and precise.
For the particular laser in a DVD player, understanding the semiconductor laser's wavelength provides important data about its operation. Here, the wavelength of 650 nm lies in the visible portion of the electromagnetic spectrum.
Semiconductor lasers are efficient and compact, making them ideal for consumer electronics where size and power consumption are critical.

Wavelength and Frequency

Wavelength and frequency are two interlinked properties of light that describe its nature. Wavelength \(\lambda\) is the distance between successive crests of a wave, measured typically in nanometers for visible light.
Frequency \(f\) is the number of cycles of a wave passing a point per unit time, measured in hertz (Hz).
These two properties have an inverse relationship, described with the equation:

• \(f = \frac{c}{\lambda}\)

Here, \(c\) represents the speed of light, approximately \(3 \times 10^{8} m/s\).
A shorter wavelength means a higher frequency and vice versa, which influences the energy per photon. Understanding the balance between these two helps specify the laser's operational characteristics and its visible color when dealing with visible spectrum lasers.

Laser Power Calculation

The calculation of laser power, especially in a DVD player, involves understanding how much energy the laser emits over time. In our exercise, the laser has a power output of \(5.0\ mW\), which translates to \(5.0 \times 10^{-3} W\).
Power, in physics, is defined as energy per unit time:

• \(1 \text{ Watt} = 1 \text{ Joule per second}\)

To find the total energy used by the laser over the span of 90 minutes, the formula used is:

• \(E_{\text{total}} = P \cdot t\)

Converting the 90 minutes into seconds gives us 5400 seconds, and multiplying with the power in watts provides \(27 Joules\) of energy.
This total energy is then divided by the energy per photon to calculate the number of photons hitting the DVD surface, resulting in a vast number of photons, highlighting the incredible efficiency of laser technology in data reading.