Problem 32 - Chemistry: The Central Science in SI Units, Global Edition

Question

A stellar object is emitting radiation at \(3.0 \mathrm{~mm} .\) (a) What type of electromagnetic spectrum is this radiation? (b) If a detector is capturing \(3.0 \times 10^{8}\) photons per second at this wavelength, what is the total energy of the photons detected in 1 day?

Step-by-Step Solution

Answer

(a) The radiation with a wavelength of \(3.0 \, \mathrm{mm}\) belongs to the microwave region of the electromagnetic spectrum. (b) The total energy of the photons detected in 1 day is approximately \(5.653 \times 10^{-15} \, \mathrm{J}\).

Step 1: Identify the type of electromagnetic spectrum

To identify the type of electromagnetic spectrum, we can refer to a table or a chart listing the different types and their respective wavelength ranges. Let's check the wavelength of the emitted radiation, which is \(3.0 \, \mathrm{mm}\), and compare it to the ranges of various types of electromagnetic spectrum.

Step 2: Calculate the energy of a single photon

In order to calculate the total energy of the detected photons, we need to find the energy of a single photon first. We can use the formula for the energy of a photon: \(E = \dfrac{hc}{\lambda}\), where \(E\) is the energy of a photon, \(h \approx 6.626 \times 10^{-34} \, \mathrm{Js}\) is the Planck's constant, \(c \approx 3.0 \times 10^8 \, \mathrm{m/s}\) is the speed of light, and \(\lambda = 3.0 \, \mathrm{mm} = 3.0 \times 10^{-3}\, \mathrm{m}\) is the wavelength of the radiation.

Step 3: Calculate the total energy of the photons detected in 1 day

Now that we have the energy of a single photon, we can calculate the total energy of the photons detected in 1 day. To do this, we need to multiply the energy of a single photon by the number of photons captured per second and by the total number of seconds in 1 day (86400 seconds): \(E_{total} = E \times N \times t\), where \(E_{total}\) is the total energy of the photons detected in 1 day, \(N = 3.0 \times 10^8 \, \mathrm{photons/s}\) is the number of photons captured per second, and \(t = 86400 \, \mathrm{s}\) is the number of seconds in 1 day. Now, let's use the given values and calculate the energy of a single photon and the total energy of photons detected in 1 day.

Key Concepts

Photon Energy Calculation

To find the energy of a photon, you can use the relationship between its energy, wavelength, and some fundamental constants. This is essential if you want to understand how much energy is being transferred in the form of radiation. The formula we use is \( E = \dfrac{hc}{\lambda} \).
This equation tells us that the energy \(E\) of a photon is inversely proportional to its wavelength \(\lambda\). This means shorter wavelengths carry more energy.

• \( h \) is Planck's constant, approximately \( 6.626 \times 10^{-34} \, \mathrm{Js} \).
• \( c \) is the speed of light in a vacuum, about \( 3.0 \times 10^8 \, \mathrm{m/s} \).
• \( \lambda \) is given in meters.

For example, if you have a photon with a wavelength of \(3.0 \, \mathrm{mm} = 3.0 \times 10^{-3} \, \mathrm{m}\), you insert these values into the formula to find its energy.

Planck's Constant

Planck's constant \(h\) is a crucial element in quantum physics and plays a pivotal role in photon energy calculation. Named after Max Planck, it's one of the fundamental constants of physics. Planck's constant represents the proportionality constant between the energy of a photon and its frequency. This constant bridges the wave and particle nature of light, showing how they relate through energy. Its approximate value is \(6.626 \times 10^{-34} \, \mathrm{Js}\).

• It helps in understanding how energy is quantized, meaning energy levels are discrete, not continuous.
• Used extensively in physics, especially in the study of quantum phenomena.
Knowing Planck's constant allows us to calculate photon energy using the energy-frequency relation \(E = hu\), where \(u\) is frequency. This highlights the quantized nature of energy transactions at microscopic scales.

Speed of Light

The speed of light \(c\) is a fundamental constant that is vitally important in physics. It denotes how fast electromagnetic waves, including light, travel in a vacuum. When working with the electromagnetic spectrum, it's crucial to understand that the speed of light is approximately \(3.0 \times 10^8 \, \mathrm{m/s}\). This constant is pivotal in calculations involving photons and waves.

• Acts as a limit for how fast information can travel in the universe.
• Is used to convert between the wavelength and frequency of electromagnetic waves through the relationship \(c = \lambda u\), where \(u\) is frequency.

In photon energy calculations, \(c\) helps determine the energy carried by photons, as seen in formulas like \(E = \dfrac{hc}{\lambda}\). Understanding this constant allows us to describe the interactions of light and matter accurately.