Problem 42 - Introductory Chemistry Atoms First

Question

Suppose a radio wave has a wavelength of \(10 \mathrm{~m}\). What is the energy of this radiation (in joules)?

Step-by-Step Solution

Answer

The energy of the radio wave with a wavelength of 10 meters is approximately 1.99 x 10^-26 joules.

Step 1: 1. Calculate the Frequency

We are given the wavelength (λ) and need to find the frequency (f). We can use the speed of light (c) formula to calculate the frequency: c = λf Here, c is the speed of light which is approximately 3 x 10^8 meters per second (m/s), and λ is the wavelength which is 10 meters. Rearranging the formula, we can solve for frequency: f = c/λ

Step 2: 2. Substitute the values and find the Frequency (f)

Now that we have the frequency formula (f = c/λ), we can substitute the values of the speed of light (c) and the wavelength (λ). f = (3 x 10^8 m/s) / (10 m)

Step 3: 3. Calculate the Frequency (f)

Simplify the expression to find the frequency of the radio wave: f = 3 x 10^7 s^-1 The frequency (f) of the radio wave is 3 x 10^7 s^-1.

Step 4: 4. Calculate the Energy of a wave (E)

Now we have the frequency (f). We can use the second formula to calculate the energy of this radiation. E = hf Planck's constant (h) is approximately 6.63 x 10^-34 Js.

Step 5: 5. Substitute the values and find the Energy (E)

Now that we have the energy formula (E = hf), we can substitute the values of Planck's constant (h) and the frequency (f). E = (6.63 x 10^-34 Js) * (3 x 10^7 s^-1)

Step 6: 6. Calculate the Energy of the radio wave (E)

Simplify the expression to find the energy of the radio wave: E = 1.99 x 10^-26 J The energy of the radio wave with a wavelength of 10 meters is approximately 1.99 x 10^-26 joules.

Key Concepts

Understanding Radio Waves

Radio waves are a type of electromagnetic radiation. They are widely used in communications technology, such as radios, TVs, and cell phones. An interesting fact about radio waves is that they have longer wavelengths than other types of electromagnetic waves. This is why they can travel long distances and through various materials, making them perfect for broadcasting.

• Wavelengths of radio waves range from 1 millimeter to 100 kilometers.
• The frequency range of radio waves is from 3 Hz to 300 GHz.

Understanding these properties helps us calculate and understand how radio waves work, especially when calculating their energy or frequency.

Calculating Frequency

When we talk about the frequency of a wave, we refer to how many cycles a wave completes in one second. This is measured in Hertz (Hz). For radio waves, calculating the frequency is vital to determine their energy and functionality in various applications. To calculate frequency when you know the wavelength, you can use the formula:\[ f = \frac{c}{\lambda} \]where:

• \( f \) is the frequency in Hertz,
• \( c \) is the speed of light (approximately \( 3 \, \times \, 10^8 \) meters per second),
• \( \lambda \) is the wavelength in meters.

For a radio wave with a wavelength of 10 meters, the frequency would be 3 x 10^7 s\(^{-1}\).

Planck's Constant Explained

Planck's constant is a fundamental constant in physics that relates the energy of a photon to its frequency. This constant is key when you calculate the energy of electromagnetic waves using the formula:\[ E = hf \]Within this context:

• \( E \) is the energy of the photon,
• \( h \) is Planck's constant (approximately \( 6.63 \, \times \, 10^{-34} \) Joule seconds),
• \( f \) is the frequency of the wave.

Planck's constant is a small yet mighty number that sets the scale for the quantum realm. Without it, we couldn't bridge the gap between energy and frequency in light and radio waves.

The Role of Wavelength

Wavelength is the spatial period of a wave — the distance over which the wave's shape repeats. In radio waves, this can vary greatly and affects both the wave's frequency and energy.To find the relationship between wavelength and other properties of waves, we use the equation:\[ c = \lambda f \]This equation shows how wavelength inversely affects frequency and vice versa, assuming the speed of light is constant.For radio waves with a given wavelength, calculating their energy becomes fairly straightforward using the above principles. Lower frequencies (and thus longer wavelengths) mean less energy, which aligns with what we've calculated for radio waves with a wavelength of 10 meters to have an energy value of \( 1.99 \, \times \, 10^{-26} \) Joules.