Question

Cobalt- \(60\left({ }^{60}{ }_{72} \mathrm{CO}\right)\) is an artificially produced radioisotope that is important in cancer therapy and in the manufacture and sterilization of certain plastics. If gamma radiation from cobalt- 60 has a wavelength of \(1.0 \times 10^{-8} \AA\), what is its frequency?

Short answer

The frequency of the gamma radiation from Cobalt-60 is approximately \(3 \times 10^{26} s^{-1}\) or \(3 \times 10^{26} Hz\).

Step-by-step solution

Convert Wavelength to Meters

Since the wavelength given is in Angstroms, we need to convert it to meters. We know that: 1 Angstrom (Å) = \(1 \times 10^{-10}\) meters Thus, the given wavelength in meters is: \(\lambda = 1.0 \times 10^{-8}\ Å \times \frac{1 \times 10^{-10} m}{1 Å} = 1.0 \times 10^{-18} m\)

Calculate Frequency Using the Speed of Light Formula

Now we will use the formula \(c = \lambda \nu\) to calculate the frequency \(\nu\): \(\nu = \frac{c}{\lambda}\) Plug in the values for the speed of light and the wavelength (in meters): \(\nu = \frac{3 \times 10^8 m/s}{1.0 \times 10^{-18} m} = 3 \times 10^{26} s^{-1}\) The frequency of the gamma radiation from Cobalt-60 is approximately \(3 \times 10^{26} s^{-1}\) or \(3 \times 10^{26} Hz\).

Key concepts

Cobalt-60

Cobalt-60 is a synthetic radioisotope of cobalt. It is represented as \( {}^{60}\text{Co} \). This isotope is not found naturally but is produced artificially. Its significance lies in its ability to emit gamma rays, which are a type of high-energy electromagnetic radiation. These gamma rays are very effective in medical treatments, especially in radiation therapy for cancer.
Cobalt-60 is preferred for such treatments because its gamma radiation can target and destroy cancerous cells while minimizing damage to healthy tissue. This makes it a valuable tool in medical therapies. Additionally, because of its sterilizing properties, it is also used in the production and sterilization of certain plastics.
Cobalt-60's applications highlight how radioisotopes can be beneficial in both medical and industrial settings. It allows for non-invasive treatment options in medicine and quality control in manufacturing.

wavelength to frequency conversion

The conversion from wavelength to frequency is essential to understanding electromagnetic waves. The wavelength (λ) is the distance between two corresponding points of a wave, while the frequency (ν) is the number of times the wave passes a particular point per second.
When converting wavelength to frequency, we use the relationship provided by the speed of light formula:

• The formula is given by \( c = \lambda u \), where \( c \) is the speed of light (\( 3 \times 10^8 m/s \)), \( \lambda \) is the wavelength, and \( u \) is the frequency.


• To find the frequency, you rearrange the formula to \( u = \frac{c}{\lambda} \).

This formula is used because electromagnetic waves, including gamma rays, travel at the speed of light. When the wavelength is known, we can easily calculate the frequency by dividing the speed of light by the wavelength, both of which must be in compatible units.

radioisotopes in medicine

Radioisotopes are isotopes that are radioactive; they have nuclei that are unstable and will decay over time. In medicine, these isotopes are used extensively for both diagnosis and treatment.
Cobalt-60, for example, is widely used in radiation therapy to treat cancer. Here's why radioisotopes are so useful:

• They emit radiation that can be used to image or treat parts of the human body.


• They can target specific areas within the body, such as tumors.


• Radioisotopes can provide precise treatment, minimizing damage to surrounding healthy tissues.

Apart from treatment, radioisotopes are also used in diagnostic imaging. Techniques such as PET scans utilize these radioactive substances to provide detailed images of the inside of the body, helping doctors make accurate diagnoses. This application of radioisotopes offers non-invasive methods that are crucial to modern medicine.

speed of light formula

The speed of light is a fundamental constant that plays a crucial role in the physics of electromagnetic waves. Denoted by the symbol \( c \), the speed of light is approximately \( 3 \times 10^8 m/s \). This velocity is the speed at which electromagnetic waves, including light and gamma rays, travel through a vacuum.
The relationship between speed, wavelength, and frequency is described by the speed of light formula: \( c = \lambda u \). This formula provides a direct link between the wavelength of a wave and its frequency.

• \( \lambda \) is the wavelength, measured in meters.


• \( u \) is the frequency, measured in hertz (Hz), or cycles per second.

This formula allows us to move between frequency and wavelength seamlessly. It shows how, for electromagnetic waves traveling at light speed, knowing one property (either wavelength or frequency) lets you calculate the other. This principle underlies much of the analysis done in wave-related physics and engineering.