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Chapter 7: Problem 44

What is the energy of a photon corresponding to microwave radiation of frequency \(1.258 \times 10^{10} / \mathrm{s}\) ?

Short Answer

Expert verified
The energy is approximately \(8.336 \times 10^{-24} \text{ J}\).

Step by step solution

01

Identify the Formula

The energy of a photon is given by the formula \( E = h \cdot f \), where \( E \) is the energy, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ J}\cdot\text{s}) \), and \( f \) is the frequency of the radiation.
02

Substitute the Given Values

Use the given frequency \( f = 1.258 \times 10^{10} \text{ Hz} \) and substitute it into the formula \( E = h \cdot f \): \[ E = (6.626 \times 10^{-34} \text{ J}\cdot\text{s}) \times (1.258 \times 10^{10} \text{ Hz}) \]
03

Calculate the Energy

Perform the multiplication: \[ E = 6.626 \times 10^{-34} \times 1.258 \times 10^{10} \]Calculate the result to find the energy: \[ E \approx 8.336 \times 10^{-24} \text{ J} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Microwave Radiation
Microwave radiation is a type of electromagnetic radiation with wavelengths ranging from one meter to one millimeter, which puts it in the frequency range of 300 MHz to 300 GHz. It falls between infrared radiation and radio waves on the electromagnetic spectrum.
This form of radiation is commonly used in microwave ovens, as it efficiently heats water and food by causing water molecules to rotate and produce thermal energy.
  • Microwaves are also widely used in telecommunications, such as in satellite and mobile communications.
  • The study of microwave radiation involves understanding both its frequency and wavelength as they are inversely proportional.
Understanding microwave radiation is crucial when discussing photon energy because it affects the energy computations of the photons involved.
Planck's Constant
The Planck's constant (\( h \)), denoted as \( 6.626 \times 10^{-34} \text{ J}\cdot\text{s} \), is a fundamental physical constant that plays a key role in quantum mechanics. It relates the energy of a photon to its frequency according to the formula \( E = h \cdot f \).
This constant is named after Max Planck, a physicist who was pivotal in developing quantum theory.
  • Planck's constant is incredibly tiny, which accounts for the small energy quantity even at high frequencies, as seen in electromagnetic waves like microwaves.
  • Its introduction marked the beginning of quantum mechanics and provided a bridge between the microscopic and macroscopic worlds.
Understanding \( h \) allows us to calculate the photon energy precisely, which is vital for fields ranging from quantum physics to nanotechnology.
Frequency of Radiation
Frequency of radiation refers to the number of cycles a wave completes in one second, measured in Hertz (Hz). It is a key factor in determining the energy of a photon. In the given scenario, the microwave radiation has a frequency of \( 1.258 \times 10^{10} \text{ Hz} \).
Frequency directly influences the energy a photon carries, as portrayed in the relationship \( E = h \cdot f \). Here are some fundamental points to remember:
  • A higher frequency means higher energy; hence, frequencies found in microwaves are capable of transferring energy efficiently.
  • The frequency determines where the wave lies on the electromagnetic spectrum, affecting properties like penetration power and thermal energy generation.
Calculating photon energy often involves substituting the frequency into formulas, highlighting its pivotal role in energy computations.

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Most popular questions from this chapter

Explain why each of the following sets of quantum numbers would not be permissible for an electron, according to the rules for quantum numbers. a. \(n=1, l=0, m_{l}=0, m_{s}=+1\) b. \(n=1, l=3, m_{l}=+3, m_{s}=+\frac{1}{2}\) c. \(n=3, l=2, m_{l}=+3, m_{s}=-\frac{1}{2}\) d. \(n=0, l=1, m_{l}=0, m_{s}=+\frac{1}{2}\) e. \(n=2, l=1, m_{l}=-1, m_{s}=+\frac{3}{2}\)

Give a brief wave description of light. What are two characteristics of light waves?

At what speed must an electron travel to have a wavelength of \(10.0 \mathrm{pm} ?\)

An electron in a hydrogen atom in the level \(n=5\) undergoes a transition to level \(n=3 .\) What is the frequency of the emitted radiation?

In your own words, explain the photoelectric effect. How does the photon concept explain this effect?
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