Problem 39 - Chemistry An Atoms First Approach

Question

The laser in an audio CD player uses light with a wavelength of \(7.80 \times 10^{2} \mathrm{nm} .\) Calculate the frequency of this light.

Step-by-Step Solution

Answer

The frequency of the light used in the audio CD player is approximately \(3.846 × 10^{14} \mathrm{Hz}\).

Step 1: Convert the wavelength to meters

First, we need to convert the given wavelength from nanometers (nm) to meters (m). We know that 1 nm = \(10^{-9}\) m. So, we can write the conversion as: \(λ = 7.80 × 10^2 \mathrm{nm} × \frac{1 \mathrm{m}}{10^9 \mathrm{nm}}\)

Step 2: Calculate the frequency

Now, we can plug in our given values into the formula, \(c = λ × f\), and rearrange to solve for the frequency, f: \(f = \frac{c}{λ}\) Given that c ≈ \(3 × 10^8 \mathrm{m/s}\) and λ = 7.80 × 10^2 nm, then: \(f = \frac{3 × 10^8 \mathrm{m/s}}{7.80 × 10^2 \mathrm{nm} × \frac{1 \mathrm{m}}{10^9 \mathrm{nm}}}\)

Step 3: Simplify and find the frequency

Now, let's simplify the equation and solve for the frequency: \(f = \frac{3 × 10^8 \mathrm{m/s}}{7.80 × 10^2 \mathrm{nm} × 10^{-9} \mathrm{m/nm}}\) \(f = \frac{3 × 10^8 \mathrm{m/s}}{7.80 × 10^{-7} \mathrm{m}}\) \(f ≈ 3.846 × 10^{14} \mathrm{Hz}\) So, the frequency of the light used in the audio CD player is approximately \(3.846 × 10^{14} \mathrm{Hz}\).

Key Concepts

Wavelength Conversion

Understanding how to convert wavelengths is crucial when dealing with light and its properties. Wavelength, typically denoted as \( \text{λ} \), is the distance between two consecutive peaks of a wave. In the context of the exercise we discussed, the conversion from nanometers to meters is essential to comply with the standard units used in the formula for calculating frequency. One nanometer (nm) is equal to \( 10^{-9} \) meters. To convert \( 780 \) nm to meters, we multiply by \( 10^{-9} \), yielding \( 780 \times 10^{-9} \) meters, or \( 7.80 \times 10^{-7} \) meters when expressed in scientific notation.

It is pivotal to remember that correct unit conversion simplifies the calculations significantly and prevents errors in the final result.

Speed of Light

The speed of light, symbolized as \( c \), is a constant that represents how fast light travels in a vacuum. It is approximately \( 3 \times 10^8 \) meters per second. This value is fundamental to several areas of physics, particularly optics and electromagnetic theory. In our exercise, the speed of light is used in conjunction with the wavelength to calculate the frequency of light. Since the speed of light is constant, it acts as a pivotal reference point which, when paired with the wavelength, can determine the light's frequency.

Consequently, the speed of light is not just a speed limit for information in the universe, but a cornerstone from which we can derive other properties of electromagnetic waves, such as frequency.

Frequency Formula

Frequency, denoted by \( f \), is the number of waves that pass a point in space during any time interval, typically one second. The unit of frequency is Hertz (Hz), which is equivalent to one wave per second. To find the frequency of light, we use the formula: \[ f = \frac{c}{\text{λ}} \] where \( c \) is the speed of light and \( \text{λ} \) is the wavelength. By rearranging the formula, we solve for \( f \) to describe the relationship directly.

For a CD player's laser light with a known wavelength, the frequency can be calculated to determine the light's color and energy. The energy, in turn, is instrumental in understanding the light's capability to read data off the CD, manifesting the practical importance of the frequency formula in technology.

Scientific Notation

Scientific notation is a method of writing very large or very small numbers in a compact form. It's expressed as the product of a number between 1 and 10 and a power of 10. For example, \( 3.846 \times 10^{14} \) Hz is the scientific notation of the frequency calculated in the exercise. This notation is essential in physics and other scientific fields because it simplifies the representation and calculation of the extreme values often encountered.

In our light frequency calculation example, we use scientific notation to handle the very small distance measured in nanometers and the very large frequency value measured in Hertz. It makes it easier to see the scale of the values and also simplifies multiplication and division by allowing manipulation of exponents.