Question

The most intense line in the cerium spectrum is at 418.7 nm. (a) Determine the frequency of the radiation producing this line. (b) In what part of the electromagnetic spectrum does this line occur? (c) Is it visible to the eye? If so, what color is it? If not, is this line at higher or lower energy than visible light?

Short answer

The frequency of the radiation producing the observed line in the cerium spectrum is approximately \(7.16 \times 10^{14} Hz\). It occurs in the ultraviolet (UV) part of the electromagnetic spectrum, and it is not visible to the human eye, as it has higher energy than visible light.

Step-by-step solution

Finding the frequency

The formula linking wavelength (\( \lambda \)) and frequency (\( v \)) of light is \( c = v \lambda \), where \( c \) is the speed of light (\(3.00 \times 10^8 m/s\)). The given wavelength has to be converted into meters by multiplying with \( 10^{-9} \). The frequency is then calculated by dividing the speed of light \( c \) by the converted wavelength (\( v = \frac{c}{\lambda} \))

Identify the spectrum

The electromagnetic spectrum ranges from radio waves to gamma rays. Given the wavelength for the cerium line, we can reference the wavelength ranges for each category of the electromagnetic spectrum to find that the cerium line falls within the ultraviolet (UV) category.

Determine the visibility and color

The visible part of the electromagnetic spectrum ranges from approximately 400nm to 700nm. While the cerium line is near this range, at 418.7nm, it technically falls within the ultraviolet (UV) category and therefore is not visible to the human eye. Its energy is higher than that of visible light, as shorter wavelengths correspond to higher energy.

Key concepts

Wavelength and Frequency Relationship

The relationship between wavelength and frequency is a fundamental concept in understanding electromagnetic waves. Wavelength (\( \lambda \)) refers to the distance between successive peaks of a wave, while frequency (\( v \)) is the number of waves that pass a point per second.

These two properties are inversely related. This means that as the wavelength of a wave increases, its frequency decreases, and vice versa. The equation that describes this relationship isequation\[ c = v \lambda \]where

• \( c \) is the speed of light in a vacuum, which is approximately \( 3.00 \times 10^8 \text{m/s} \).
• \( v \) is the frequency of the wave.
• \( \lambda \) is the wavelength of the wave.

For example, to find the frequency of light at a given wavelength, one would rearrange the formula to read \( v = \frac{c}{\lambda} \).

This formula allows us to understand how energy is transferred through electromagnetic radiation.

Ultraviolet Radiation

Ultraviolet (UV) radiation is a type of electromagnetic radiation with a wavelength shorter than that of visible light but longer than X-rays.

In terms of wavelength, UV radiation typically ranges from about 10 nm to 400 nm. Because of its shorter wavelength compared to visible light, UV radiation carries more energy.

This high energy allows UV radiation to have powerful effects. It can cause chemical reactions and has practical applications such as sterilization and fluorescence.

• UV radiation is divided into three main types:
  • UVA (320–400 nm)
  • UVB (290–320 nm)
  • UVC (100–290 nm)
• The type and amount of UV radiation that reaches the Earth are influenced by the atmosphere, especially the ozone layer.

Though invisible to the human eye, UV radiation plays significant roles in various environmental and industrial processes.

Visible Light Spectrum

The visible light spectrum is the portion of the electromagnetic spectrum that can be detected by the human eye. This spectrum ranges approximately from 400 nm to 700 nm in terms of wavelength.

Within this 300 nm range, we perceive different colors. Red light, which is at the higher end of the visible spectrum (around 700 nm), has the longest wavelength.

• As the wavelength decreases, the colors transition from red to violet.
• Violet light has the shortest wavelength at around 400 nm, close to the beginning of the ultraviolet range.

The visible spectrum is important not just because it comprises all the colors we can see, but also because many biological processes are influenced by visible light.

In essence, the section of the spectrum we refer to as visible light is crucial for our perception of the world, aesthetics in art, and even for plant photosynthesis.