Question

Microwave radiation has a wavelength on the order of \(1.0 \mathrm{~cm}\). Calculate the frequency and the energy of a single photon of this radiation. Calculate the energy of an Avogadro's number of photons (called an einstein) of this radiation.

Short answer

The frequency of the microwave radiation is \(3.0 * 10^{10} Hz\), the energy of a single photon is \(1.99 * 10^{-23} J\), and the energy of an Avogadro's number of photons (einstein) is \(12.0 J\).

Step-by-step solution

Convert the wavelength to meters

We are given the wavelength (λ) in centimeters, and we need to convert it to meters for calculations. 1.0 cm = \(1.0 * 10^{-2} m\)

Calculate the frequency

Now, we can calculate the frequency (ν) using the provided wavelength (λ) and the speed of light (c) with the equation ν = \(c / λ\). Frequency (ν) = \(\frac{3.0 * 10^8 m/s}{1.0 * 10^{-2} m}\) ν = \(3.0 * 10^{10} Hz\)

Calculate the energy of a single photon

With the frequency obtained in the previous step, we can now calculate the energy (E) of a single photon of microwave radiation using the equation E = \(h * ν\). Energy (E) = \((6.63 * 10^{-34} Js) * (3.0 * 10^{10} Hz)\) E = \(1.99 * 10^{-23} J\)

Calculate the energy of an Avogadro's number of photons

Now, we will calculate the total energy of an Avogadro's number of photons (Einstein) using the energy (E) of a single photon and Avogadro's number (N). Energy of an Avogadro's number of photons = \(E * N\) = \((1.99 * 10^{-23} J) * (6.022 * 10^{23})\) Total energy = \(12.0 J\) Therefore, the frequency of the microwave radiation is \(3.0 * 10^{10} Hz\), the energy of a single photon is \(1.99 * 10^{-23} J\), and the energy of an Avogadro's number of photons (einstein) is \(12.0 J\).

Key concepts

Wavelength to Frequency Conversion

Understanding the relationship between wavelength and frequency is crucial when dealing with various forms of electromagnetic radiation, such as microwaves. The wavelength of any wave is inversely proportional to its frequency, which means that as the wavelength increases, the frequency decreases and vice versa. This relationship is encapsulated in the formula:
\( u = \frac{c}{\tau} \)
Here, \( u \) represents the frequency, \( c \) is the speed of light in a vacuum (approximately \( 3.0 \times 10^8 \) meters per second), and \( \tau \) represents the wavelength. When we convert the given wavelength from centimeters to meters, we can then calculate the frequency directly by dividing the speed of light by the wavelength. This conversion is essential for further calculations involving the energy of photons, as we'll see in the application of Planck's constant.

Let’s recall the problem scenario. It asks for the frequency of a photon with a wavelength of 1.0 cm. With the conversion to meters and the formula provided, we find that the frequency is \( 3.0 \times 10^{10} \) Hz. This step is an essential foundation for calculating the photon's energy in the next section.

Planck's Constant Application

When delving into photon energy calculations, one cannot bypass the quintessential role of Planck's constant (denoted as \( h \)). This fundamental constant of nature measures the quantization of energy levels in particles at the microscopic scale. Its value is \( 6.63 \times 10^{-34} \) joule-seconds (Js). Planck's constant is at the heart of the equation that relates the energy (E) of a photon to its frequency (\( u \)):
\( E = h \times u \)
Using this equation, we can calculate the energy of a photon if we know its frequency, which we determined from the wavelength conversion. Thus, by applying Planck's constant and the frequency we obtained in the previous step, we can compute how much energy a single photon carries. It's this relationship that bridges the gap between the concepts of wavelength, frequency, and energy in the realm of quantum mechanics.

In our microwave example, the energy of a photon with the frequency of \( 3.0 \times 10^{10} \) Hz is calculated as \( 1.99 \times 10^{-23} \) joules. This energy represents the quantum of action needed for a photon of microwave radiation to interact with other particles.

Avogadro's Number in Energy Calculations

Avogadro's number, symbolized as \( N \), signifies the amount of entities, be it atoms, ions, or in our case, photons, in one mole, which is approximately \( 6.022 \times 10^{23} \). When we talk about an Einstein in the context of photons, we refer to Avogadro's number of these light particles. More specifically, we use Einstein's concept to quantify the energy contained in one mole of photons. This is where Avogadro's number plays an integral role in bridging particle physics with chemical quantities.

To find the energy in one mole of photons, we can multiply the energy of a single photon by Avogadro's number. The result is a macroscopic energy amount that can be measured in joules. This process is pervasive in photosynthesis and other photochemical reactions, where the calculation of energies associated with large numbers of photons becomes relevant.

In wrapping up our microwave radiation example, we multiply the energy of a single photon, \( 1.99 \times 10^{-23} \) joules, by Avogadro's number, yielding an einstein of microwave radiation with an energy total of 12 joules. This quantity demonstrates the vast energy carried by a mole of seemingly minuscule photons.