Problem 81 - Chemistry The Molecular Nature of Matter

Question

Ozone protects the earth's inhabitants from the harmful effects of ultraviolet light arriving from the sun. This shielding is a maximum for UV light having a wavelength of \(295 \mathrm{nm}\). What is the frequency in hertz of this light?

Step-by-Step Solution

Answer

The frequency of the UV light with a wavelength of 295 nm is approximately 1.02 × 10^15 Hz.

Step 1: Identify the constants needed for the calculation

To find the frequency of the light, we will use the equation that relates the speed of light (c), frequency (ν), and wavelength (λ), which is c = λν. The speed of light, c, is a constant approximately equal to 3.00 × 10^8 m/s.

Step 2: Convert the wavelength into meters

The wavelength given is in nanometers (nm). To use the equation, we need to convert it to meters (the SI unit for length). Since 1 nm = 10^-9 m, convert 295 nm to meters: 295 nm × (1 meter/10^9 nm) = 295 × 10^-9 meters.

Step 3: Rearrange the equation to solve for frequency

We need to solve for frequency (ν), so we rearrange the equation c = λν to ν = c / λ.

Step 4: Substitute the values into the equation and solve for frequency

Now substitute the values for the speed of light (c) and the wavelength (λ) in meters into the equation ν = c / λ and solve for frequency (ν). Nu =(3.00 × 10^8 m/s) / (295 × 10^-9 m) = 1.02 × 10^15 Hz.

Key Concepts

The Speed of Light

When delving into the realm of physics and, more specifically, optics, the speed of light forms a cornerstone of many fundamental concepts. It is one of the constants in the universe, representing the fastest speed at which energy, information, or matter can travel. In a vacuum, light travels at an approximate speed of 3.00 x 10^8 meters per second (m/s). This constant isn’t just a number; it is central to equations that govern the behavior of light across the cosmos and in scientific experiments. The speed of light in materials other than a vacuum is slower and depends on the medium's refractive index.

Understanding this universal constant is crucial. For example, when light travels from the sun to the Earth, it takes about 8 minutes and 20 seconds, showcasing the vast distances in space that the light can cover at this incredible speed. Moreover, it illustrates why the speed of light is instrumental in calculations involving light's frequency and wavelength, as seen in exercises related to ultraviolet light frequency calculation.

Wavelength to Frequency Conversion

Moving on to the practical application of the speed of light in calculations, we turn our attention to the relationship between wavelength and frequency. The equation connecting these two properties of light is given by the equation \( c = \lambdau \), where \( c \) is the speed of light, \( \lambda \) represents the wavelength of light, and \( u \) denotes the frequency. To find one when you have the other, you can manipulate the equation to \( u = c / \lambda \). This is invaluable for tasks like determining the frequency of ultraviolet light which protects Earth's inhabitants by absorbing the sun's harmful rays.

In understanding this relationship, it's important to know that wavelength refers to the distance between successive crests of a wave, typically measured in meters, while frequency refers to the number of waves that pass a given point per second, measured in hertz (Hz). The frequency is inversely proportional to the wavelength; as the wavelength increases, the frequency decreases and vice versa. This inverse relationship helps scientists and engineers calculate one property if they know the other, and it's essential for fields like telecommunications and photonics.

Nanometers to Meters Conversion

In scientific measurements, especially in the field of optics, the need often arises to convert from one unit of measure to another. When dealing with light, such as ultraviolet (UV) light, wavelengths are often provided in nanometers (nm). A nanometer, which is one billionth of a meter (1 nm = 10^-9 m), is a convenient unit for the scale of light wavelengths.

To work with the commonly used scientific equations, it’s typically necessary to convert these measurements into meters, which are the standard unit of length in the International System of Units (SI). The conversion from nanometers to meters is straightforward but vital for accuracy in calculations. For example, when given the wavelength of UV light in nanometers, as in the exercise at hand, one must convert that number to meters by multiplying by 10^-9 to use the relationship between the speed of light, wavelength, and frequency. This step ensures that the values are in compatible units, facilitating the correct application of the formula and thereby yielding an accurate frequency, which is a critical step in understanding the properties and behavior of various types of light.