Problem 13 - Chemistry: The Central Science in SI Units, Global Edition

Question

The wavenumber \(\bar{\lambda}\) is the number of waves that exist over a specified distance, very often \(1 \mathrm{~cm}\). The wavenumber can easily be calculated by taking the reciprocal of the wavelength. Give typical wavenumbers for (a) X-rays \((\lambda=1 \mathrm{nm})\) (b) visible light \((\lambda=500 \mathrm{nm})\) (c) microwaves \((\lambda=1 \mathrm{~mm})\).

Step-by-Step Solution

Answer

The typical wavenumbers for each type of wave are: (a) X-rays: \(\bar{\lambda}_{\mathrm{X-rays}} = 10^7 \mathrm{~cm}^{-1}\) (b) visible light: \(\bar{\lambda}_{\mathrm{visible}} = 2 \times 10^4 \mathrm{~cm}^{-1}\) (c) microwaves: \(\bar{\lambda}_{\mathrm{microwaves}} = 10 \mathrm{~cm}^{-1}\)

Step 1: (Step 1: Convert Wavelengths to Centimeters)

Before we find the wavenumbers, we need to make sure all wavelengths are in the same unit (centimeters). Since we are given the wavelengths in nanometers and millimeters, we can easily convert these values to centimeters: - For X-rays: \(1 \mathrm{nm} = 10^{-7}\mathrm{~cm}\) - For visible light: \(500 \mathrm{nm} = 5 \times 10^{-5}\mathrm{~cm}\) - For microwaves: \(1 \mathrm{~mm} = 0.1\mathrm{~cm}\)

Step 2: (Step 2: Calculate the Wavenumber for X-Rays)

Now, we can find the wavenumber of X-rays by taking the reciprocal of the wavelength (in centimeters): $$ \bar{\lambda}_{\mathrm{X-rays}} = \frac{1}{10^{-7}\mathrm{~cm}} = 10^7 \mathrm{~cm}^{-1} $$

Step 3: (Step 3: Calculate the Wavenumber for Visible Light)

Similarly, we can find the wavenumber of visible light by taking the reciprocal of the wavelength (in centimeters): $$ \bar{\lambda}_{\mathrm{visible}} = \frac{1}{5 \times 10^{-5}\mathrm{~cm}} = 2 \times 10^4 \mathrm{~cm}^{-1} $$

Step 4: (Step 4: Calculate the Wavenumber for Microwaves)

Finally, we can find the wavenumber of microwaves by taking the reciprocal of the wavelength (in centimeters): $$ \bar{\lambda}_{\mathrm{microwaves}} = \frac{1}{0.1\mathrm{~cm}} = 10 \mathrm{~cm}^{-1} $$

Step 5: (Step 5: State the Results)

We have found the typical wavenumbers for each type of wave: (a) X-rays: \(\bar{\lambda}_{\mathrm{X-rays}} = 10^7 \mathrm{~cm}^{-1}\) (b) visible light: \(\bar{\lambda}_{\mathrm{visible}} = 2 \times 10^4 \mathrm{~cm}^{-1}\) (c) microwaves: \(\bar{\lambda}_{\mathrm{microwaves}} = 10 \mathrm{~cm}^{-1}\)

Key Concepts

Wavelength Conversion

Understanding how to convert wavelengths into different units is a crucial skill in physics and various scientific fields. Wavelengths can be presented in units such as nanometers, millimeters, or meters, but they are often converted to centimeters for specific calculations.To convert:

• Nanometers to centimeters, use the factor: \( 1 ext{ nm} = 10^{-7} ext{ cm} \).
• Millimeters to centimeters, use the factor: \( 1 ext{ mm} = 0.1 ext{ cm} \).

For instance, a wavelength given as \( 1 ext{ nm} \) is equivalent to \( 10^{-7} ext{ cm} \) after conversion. And if you have \( 500 ext{ nm} \), it converts to \( 5 \times 10^{-5} ext{ cm} \). This unit consistency in centimeters is essential before performing further calculations, such as finding the wavenumber.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, organized by their wavelengths and frequencies. Each type of radiation on the spectrum, from radio waves to gamma rays, has unique properties and applications. - **Radio Waves:** Long wavelengths, suitable for wireless communications. - **Microwaves:** Shorter than radio waves, used for radar and heating food. - **Visible Light:** What the human eye perceives, with wavelengths from about 400 nm (violet) to 700 nm (red). - **X-rays:** Even shorter wavelengths, used for medical imaging. - **Gamma Rays:** The shortest wavelengths, with applications in cancer therapy. Understanding where each type of radiation falls on the electromagnetic spectrum helps in fields such as telecommunications, medicine, and astronomy. Each region has its typical wavenumber ranges, derived from their specific wavelengths.

Reciprocal Calculation

Reciprocal calculation is a mathematical process frequently used in physics to find relationships between physical quantities. Calculating the reciprocal essentially means taking one divided by the number.The concept of reciprocal is integral when discussing wavenumbers in physics:

• Wavenumber, \( \bar{\lambda} \), is found using the formula \( \bar{\lambda} = \frac{1}{\lambda} \), where \( \lambda \) is the wavelength in centimeters.
• This measurement indicates how many waves fit into a given length, typically expressed in \( \text{cm}^{-1} \).

For example, given a wavelength of \( 0.1 \text{ cm} \) for microwaves, the wavenumber is \( \frac{1}{0.1 \text{ cm}} = 10 \text{ cm}^{-1} \). The reciprocal relationship is vital for various scientific calculations, including analyzing the type of electromagnetic radiation based on its wavelength.