Problem 82 - Chemistry The Molecular Nature of Matter

Question

The meter is defined as the length of the path light travels in a vacuum during the time interval of \(1 / 299,792,458\) of a second. It is recommended that a helium-neon laser is used for defining the meter. The light from the laser has a wavelength of \(632.99139822 \mathrm{nm}\). What is the frequency of this light, in hertz?

Step-by-Step Solution

Answer

The frequency of the helium-neon laser light is approximately 474 terahertz (THz).

Step 1: Understanding the Relationship between Speed, Wavelength, and Frequency

To find the frequency of the light, use the formula for the speed of light: speed (c) = wavelength (λ) × frequency (f). The speed of light in a vacuum is a constant, approximately equal to 299,792,458 meters per second.

Step 2: Convert Wavelength to Meters

Since the wavelength is given in nanometers (nm), we need to convert it to meters (m) for consistency with the speed of light's units. This is done by using the conversion factor that 1 nm equals 1 x 10^-9 meters.

Step 3: Calculate the Frequency

With the wavelength in meters, insert the values into the equation c = λf to solve for the frequency (f). Rearrange the equation to find f = c / λ, and then substitute the speed of light (c) and the converted wavelength (λ) into the equation to solve for f.

Key Concepts

Wavelength to Frequency Conversion

The relationship between wavelength and frequency is a fundamental concept in physics, particularly in the study of light. Here's the key thing to know: as the wavelength of light becomes shorter, its frequency becomes higher, and vice versa. This phenomenon occurs because the speed of light is constant in a vacuum.

To convert the wavelength of light to its frequency, you use the formula: \[ f = \frac{c}{\lambda} \]where \( f \) is the frequency in hertz (Hz), \( c \) is the speed of light in meters per second (m/s), and \( \lambda \) is the wavelength in meters (m).

Applying the Formula

Once you have the wavelength in the correct unit of meters, simply divide the speed of light by the wavelength to obtain the frequency. It's essential to ensure that the units for speed and wavelength match to avoid any errors in your calculation. If you start with nanometers (nm), you must convert them to meters by multiplying by \( 10^{-9} \) before using them in the formula.

Speed of Light in Vacuum

The speed of light in a vacuum is a cornerstone of the theory of relativity and holds tremendous significance in the realm of physics. It is denoted by the symbol \( c \) and is approximately \( 299,792,458 \) meters per second.

Understanding that this speed is a constant and unchanging value is critical. No matter how light's wavelength or frequency varies, the speed at which light travels through a vacuum remains the same. This is why we can use it as a fundamental unit of measurement, like defining a meter as the distance light covers in a specific fraction of a second.

Critical to Calculations

In any calculation involving light (such as the exercise in question), having the precise value for the speed of light allows for accurate wavelength to frequency conversions and vice versa. This figure is one of the defined constants in the International System of Units (SI), making it integral to scientific measurements and calculations worldwide.

Units Conversion

When working with physical quantities, the accurate conversion of units is pivotal to obtaining correct results. For the conversion from wavelength to frequency or any calculation involving differing units, they must be compatible.

Let's take the given exercise. The wavelength is initially provided in nanometers (nm), and we need it in meters (m) to use with the speed of light. Nanometers are a much smaller unit – there are one billion nanometers in a meter. So the conversion factor here is \( 1 nm = 10^{-9} m \). It is part of the metric system which is decimal-based and makes such conversions logical and consistent.

Correct Conversion Application

To convert the wavelength given in the exercise from nanometers to meters, you would multiply the numerical value by \( 10^{-9} \). These conversions are crucial to maintain the integrity of calculations across various domains of physics and are a standard aspect of ensuring that all units in a given problem are appropriately aligned for the intended computations.