Problem 23 - Chemistry: An Atoms-Focused Approach

Question

In 1895 German physicist Wilhclm Röntgen discovered X-rays. He also discovered that X-rays emitted by different metals have different wavelengths. Which X-rays have the higher frequency, those emitted by (a) Cu \((\lambda=0.154 \mathrm{nm})\) or (b) iron \((\lambda=194 \mathrm{pm}) ?\)

Step-by-Step Solution

Answer

Answer: Copper emits X-rays with a higher frequency compared to iron.

Step 1: Review the relationship between frequency, speed of light, and wavelength

Recall the equation relating the speed of light (\(c\)), wavelength (\(\lambda\)), and frequency (\(f\)): \(c = \lambda \cdot f\) We can rearrange the equation to solve for frequency: \(f = \frac{c}{\lambda}\)

Step 2: Substitute given wavelengths and the speed of light in the formula

For Cu (\(\lambda = 0.154 \, \mathrm{nm}\)) and iron (\(\lambda = 194 \, \mathrm{pm}\)), let's first convert the wavelengths to the same unit (meters) to make calculations easier. 1 nm = \(10^{-9}\) m, so \(\lambda_{Cu} = 0.154 \times 10^{-9} \, \mathrm{m}\) 1 pm = \(10^{-12}\) m, so \(\lambda_{Fe} = 194 \times 10^{-12} \, \mathrm{m}\) Now, let's substitute the values of \(\lambda_{Cu}\) and \(\lambda_{Fe}\) into the formula for frequency. The speed of light (c) is \(3 \times 10^8 \, \mathrm{m/s}\). $f_{Cu} = \frac{3 \times 10^8 \, \mathrm{m/s}}{0.154 \times 10^{-9} \, \mathrm{m}}$ \(f_{Fe} = \frac{3 \times 10^8 \, \mathrm{m/s}}{194 \times 10^{-12} \, \mathrm{m}}\)

Step 3: Calculate the frequencies

Now, we can calculate the frequencies of the X-rays emitted by Cu and iron. \(f_{Cu} = \frac{3 \times 10^8}{0.154 \times 10^{-9}} = 1.948 \times 10^{18} \, \mathrm{Hz}\) \(f_{Fe} = \frac{3 \times 10^8}{194 \times 10^{-12}} = 1.546 \times 10^{18} \, \mathrm{Hz}\)

Step 4: Compare the frequencies

Finally, let's compare the frequencies of the X-rays emitted by Cu and iron. Since \(f_{Cu} = 1.948 \times 10^{18} \, \mathrm{Hz}\) and \(f_{Fe} = 1.546 \times 10^{18} \, \mathrm{Hz}\), we can see that the X-rays emitted by Cu have a higher frequency than those emitted by iron.

Key Concepts

Frequency Calculation

Frequency is defined as the number of cycles that a wave completes in a unit of time. In physics, especially when dealing with waves such as X-rays, frequency (\(f\)) is a fundamental concept. To find frequency, we utilize the fundamental equation that links frequency to the speed of light (\(c\)) and wavelength (\(\lambda\)):\[ f = \frac{c}{\lambda} \]This equation tells us that frequency is inversely proportional to wavelength. Meaning, as the wavelength increases, frequency decreases and vice versa. This is key because high frequency corresponds to shorter wavelengths, which is a typical characteristic of X-rays.When evaluating which X-rays possess a higher frequency, it’s crucial to correctly substitute the wavelengths into the formula. Always ensure that measurements are in meters, since the speed of light is given in meters per second. This allows for consistent and accurate calculations.

Wavelength Conversion

Accurate wavelength conversion is necessary when dealing with scientific equations. Wavelength (\(\lambda\)) is often given in nanometers (nm) or picometers (pm), requiring conversion to meters for calculations involving the speed of light.Here's how you can convert:

• 1 nanometer (nm) is equal to \(10^{-9}\) meters.
• 1 picometer (pm) is equal to \(10^{-12}\) meters.

To convert, simply multiply the given wavelength by its respective scaling factor to turn it into meters:- For copper: \[ \lambda_{Cu} = 0.154 \times 10^{-9} \text{ m} \]- For iron: \[ \lambda_{Fe} = 194 \times 10^{-12} \text{ m} \]By using these conversions, you can seamlessly compute frequencies using the relationship between speed of light, frequency, and wavelength.

Speed of Light

The speed of light (\(c\)) is a constant in physics characterized by its symbol "c". Precisely, it is equal to \(3 \times 10^8\) meters per second in a vacuum. This value is essential when computing wave properties such as wavelength and frequency together. In the context of X-ray computations, understanding that the speed of light remains constant allows us to focus on how wavelength changes affect frequency.Remember these fundamental points:

• The speed of light is consistently used as \(3 \times 10^8\) m/s in our calculations.
• It serves as a critical link between frequency and wavelength in the formula \[ c = \lambda \cdot f \].

Knowing the speed of light ensures that we can accurately determine how fast electromagnetic waves, including X-rays, propagate irrespective of medium, making it a pillar in wave mechanics.