Problem 2 - Chemistry Matter and Change

Question

X rays can penetrate body tissues and are widely used to diagnose and treat disorders of internal body structures. What is the frequency of an \(\mathrm{X}\) ray with a wavelength of \(1.15 \times 10^{-10} \mathrm{m}\) ?

Step-by-Step Solution

Answer

The frequency of the X-ray with a wavelength of \(1.15 \times 10^{-10}\) meters is approximately \(2.61 \times 10^{18}\) Hertz.

Step 1: Write down the given information

The wavelength of the X-ray is given as \(1.15 \times 10^{-10}\) meters. We also know that the speed of light (c) is approximately \(3 \times 10^8\) meters per second.

Step 2: State the relationship between frequency, wavelength, and the speed of light

The relationship between frequency (f), wavelength (λ), and the speed of light (c) is given by: \[c = f \times \lambda\]

Step 3: Solve for the frequency

Rearranging the formula to solve for the frequency (f), we get: \[f = \frac{c}{\lambda}\] Now, we can plug in the given values for the speed of light (c) and the wavelength (λ) to find the frequency.

Step 4: Calculate the frequency

Using the values for c and λ, we can calculate the frequency: \[f = \frac{3 \times 10^8 \ \text{m/s}}{1.15 \times 10^{-10} \ \text{m}}\]

Step 5: Simplify the result

Divide the values and simplify the result to find the frequency of the X-ray: \[f = (3 \times 10^8 \ \text{m/s}) \div (1.15 \times 10^{-10} \ \text{m}) = 2.61 \times 10^{18} \ \text{Hz}\] The frequency of the X-ray is approximately \(2.61 \times 10^{18}\) Hertz.

Key Concepts

X-rays

X-rays are a form of electromagnetic radiation with exceptionally high energy and short wavelength. They fall in the spectrum between ultraviolet light and gamma rays. X-rays possess wavelengths ranging from about 0.01 to 10 nanometers. This allows them to penetrate various materials, including soft tissues in the human body.
This property makes X-rays particularly useful in medical imaging, such as in X-ray photography or computed tomography (CT) scans.

• X-rays are generated by high-energy collisions, often involving electrons and atoms.
• They can reveal the internal composition of objects, which is why they are prominently used in diagnostic medical applications.

Despite their benefits, excessive exposure to X-rays can be harmful due to the ionizing nature of this radiation. It is crucial to control the use and dosage during medical examinations to safeguard health.

frequency calculation

Calculating the frequency of electromagnetic waves, such as X-rays, is crucial in understanding their properties. The frequency ( f ) can be determined using the relationship between the speed of light ( c ) and the wavelength ( λ ).
The formula is: \[ c = f \times \lambda \] Rearranging it, we find: \[ f = \frac{c}{\lambda} \]
In practical calculations, it is essential to utilize consistent units to obtain an accurate frequency. As seen in our original problem, once you know the wavelength of the X-ray, you can use this formula to calculate its frequency.
Understanding frequency is important because it helps identify how much energy the wave carries. Higher frequencies indicate higher energy, which is characteristic of X-rays and why they are capable of penetrating matter.

wavelength

The wavelength of electromagnetic radiation is a measure of the distance between consecutive peaks or troughs in the wave. In the context of X-rays, as seen in the original exercise, the wavelength provided was \(1.15 \times 10^{-10} \text{ m}\).
This is extremely short, which corresponds to the high energy levels of X-rays. Wavelength is inversely proportional to frequency. This means shorter wavelengths, like those of X-rays, result in higher frequencies.

• Understanding wavelength helps determine the energy and penetrating power of electromagnetic waves.
• The shorter the wavelength, the higher the energy and the greater the ability to penetrate materials.

In scientific studies and practical applications, knowing the wavelength of radiation is essential for predicting how the wave will interact with different substances.

speed of light

The speed of light is a fundamental constant in physics, symbolized by \( c \). It denotes the speed at which electromagnetic waves propagate through a vacuum, approximately \(3 \times 10^8 \text{ m/s}\).
This constant is critical when calculating the frequency of electromagnetic waves, including X-rays. In most solutions involving light and other electromagnetic waves, the speed of light bridges the relationship between frequency and wavelength.

• The speed of light is the cornerstone for various physics equations, highlighting the universality and efficiency of electromagnetic wave propagation.
• It is also essential in relativity theory, impacting our understanding of space, time, and energy.

Properly measuring and understanding the speed of light is crucial in both theoretical frameworks and experimental practices in physics. This value serves not only as an integral part of equations but also as a lead into more advanced concepts, illustrating the nature of energy and wave interactions.