Problem 86 - Chemistry The Central Science In Si Units

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

Approximately \(8.82 \times 10^{19}\) photons strike the DVD during 90 minutes.

Step 1: Understanding Power and Energy

First, convert the given power of the laser into watts: \(5.0 \text{ mW} = 5.0 \times 10^{-3} \text{ W}\). Power is the rate at which energy is emitted, \(P = E/t\), where \(E\) is energy and \(t\) is time.

Step 2: Convert Time to Seconds

The playing time of the DVD is 90 minutes. Convert it to seconds: \[90 \text{ minutes} \times 60 \text{ seconds/minute} = 5400 \text{ seconds}.\]

Step 3: Calculate Total Energy Emitted

Using \(P = E/t\), we can find the total energy emitted \(E\):\[E = P \times t = 5.0 \times 10^{-3} \text{ W} \times 5400 \text{ s} = 27 \text{ J}.\]

Step 4: Find Energy of a Single Photon

The energy of a single photon is given by \(E = \frac{hc}{\lambda}\), where \(h = 6.626 \times 10^{-34} \text{ Js}\) is Planck's constant, \(c = 3.00 \times 10^8 \text{ m/s}\) is the speed of light, and \(\lambda = 650 \text{ nm} = 650 \times 10^{-9} \text{ meters}\).Calculate:\[E = \frac{6.626 \times 10^{-34} \text{ Js} \times 3.00 \times 10^{8} \text{ m/s}}{650 \times 10^{-9} \text{ m}} = 3.06 \times 10^{-19} \text{ J}\]

Step 5: Calculate Number of Photons

To find the number of photons, divide the total energy by the energy of a single photon:\[\text{Number of photons} = \frac{27 \text{ J}}{3.06 \times 10^{-19} \text{ J/photon}} \approx 8.82 \times 10^{19} \text{ photons}.\]

Key Concepts

Power and Energy

Power is essentially a measure of how much energy is being transferred or converted per unit of time. It tells us how "quickly" energy is used or produced in a certain process. In physics, power is typically measured in watts (W), and one watt is equivalent to one joule per second (J/s). This means if a device has a power rating of 1 watt, it uses or produces 1 joule of energy every second.
Let's consider the example of the semiconductor laser in a DVD player. It has a power level of 5.0 mW (milliwatts). To simplify calculations, we first convert this power to watts:

• 5.0 mW = 5.0 x 10^-3 W

This laser, with its power, emits energy at the rate of 5.0 x 10^-3 joules every second. If you know how long the device is operating (in seconds), you can calculate the total energy emitted by multiplying power (5.0 x 10^-3 watts) by time in seconds. This forms the basic understanding behind the equation, \( P = \frac{E}{t} \), where \(E\) is energy, and \(t\) is time.

Planck's Constant

Planck's constant is a fundamental value in physics, represented by the symbol \(h\). It's a very small number, approximately equal to \(6.626 \, \times \, 10^{-34} \, \text{Js}\). This constant is a crucial part of quantum mechanics as it relates energy and frequency of a photon.
Max Planck, a physicist, introduced this constant at the dawn of the 20th century in his work on black body radiation, which helped in forming the quantum theory. Planck's constant allows us to calculate the energy of a single photon using the formula:\[E = \frac{hc}{\lambda}\]where:

• \(E\) is the energy of a photon,
• \(h\) is Planck's constant, \(6.626 \times 10^{-34} \text{ Js}\),
• \(c\) is the speed of light, approximately \(3.00 \times 10^8 \text{ m/s}\),
• \(\lambda\) is the wavelength of the light in meters.

This formula shows that as the wavelength of light decreases (becomes "bluer"), the energy of each photon increases.

Photon Energy Calculation

Photon energy calculations help us determine how much energy an individual photon of light carries based on its wavelength. Light behaves both as a wave and as a particle, these particles of light are photons. Each photon consists of energy that can be quantified using a relationship involving Planck's constant.
For example, in our DVD player problem, we have a wavelength of \(650\, \text{nm}\), which we convert to meters \(650 \, \text{nm} \, = \, 650 \times 10^{-9} \, \text{m}\). From there, we can use the formula to calculate photon energy:\[E = \frac{hc}{\lambda} \]

• Using \(h = 6.626 \, \times \, 10^{-34} \, \text{Js}\)
• \(c = 3.00 \, \times \, 10^8 \, \text{m/s}\)
• \(\lambda = 650 \, \text{nm}\)

We find each photon has an energy of approximately \(3.06 \times 10^{-19} \text{ J}\). This energy per photon is then used to calculate the total number of photons emitted over time by dividing the total energy output by the energy per individual photon.