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Chapter 28: Problem 59

From the kinetic molecular theory of an ideal gas (Chapter 15), we know that the average kinetic energy of an atom is \(\frac{3}{2} k T .\) What is the wavelength of a photon that has this energy for a temperature of \(27^{\circ} \mathrm{C} ?\)

Short Answer

Expert verified
The wavelength of the photon is approximately \(3.20 \times 10^{-5}\) m.

Step by step solution

01

Convert Temperature to Kelvin

First, we need to convert the given temperature from Celsius to Kelvin because the kinetic molecular theory uses Kelvin. The formula for this conversion is: \ \( T(K) = T(掳C) + 273.15 \) \ Plugging in the given temperature: \ \( T(K) = 27 + 273.15 = 300.15 \, \text{K} \)
02

Calculate Average Kinetic Energy

The average kinetic energy of an atom in an ideal gas is given by the formula: \ \( E = \frac{3}{2} k T \) \ where \( k \) is the Boltzmann constant, approximately \( 1.38 \times 10^{-23} \, \text{J/K} \). \ Substituting the temperature: \ \( E = \frac{3}{2} \times 1.38 \times 10^{-23} \times 300.15 \approx 6.21 \times 10^{-21} \, \text{J} \)
03

Determine Wavelength from Energy

The energy of a photon is related to its wavelength by the equation: \ \( E = \frac{hc}{\lambda} \) \ where \( h \) is Planck's constant, approximately \( 6.626 \times 10^{-34} \, \text{J s} \), and \( c \) is the speed of light, approximately \( 3 \times 10^{8} \, \text{m/s} \). \ Rearranging the equation to solve for the wavelength: \ \( \lambda = \frac{hc}{E} \) \ Substituting the known values: \ \( \lambda = \frac{6.626 \times 10^{-34} \times 3 \times 10^{8}}{6.21 \times 10^{-21}} \approx 3.20 \times 10^{-5} \, \text{m} \)

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Key Concepts

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

Average Kinetic Energy
The average kinetic energy is a measure of how much energy particles in a substance, such as an atom or molecule in a gas, have due to their motion. In the context of kinetic molecular theory, it's usually defined for ideal gases. The formula for average kinetic energy is given by:
\[ E = \frac{3}{2} k T \]where:
  • \( E \) is the average kinetic energy.
  • \( k \) is the Boltzmann constant, approximately \( 1.38 \times 10^{-23} \text{J/K}. \)
  • \( T \) is the absolute temperature of the gas in Kelvin.
Temperature directly affects the average kinetic energy. As the temperature increases, the average kinetic energy of the particles also increases. This relationship makes it a core concept in understanding how temperature relates to particle movement in gases.
Ideal Gas
An ideal gas is a theoretical gas composed of many randomly moving point particles that interact only through elastic collisions. This is a simplified model used in physics and chemistry for studying real gases under many conditions. The assumptions of an ideal gas include:
  • Particles have no volume; they are point particles.
  • No intermolecular forces between particles; they do not attract or repel each other.
  • Collisions are perfectly elastic; no energy is lost during collisions.
  • Average kinetic energy of gas particles depends only on the temperature.
These assumptions lead us to the Ideal Gas Law: \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is moles of gas, \( R \) is the gas constant, and \( T \) is temperature in Kelvin. Understanding ideal gases helps explain real gas behaviors under a variety of conditions.
Photon Wavelength
Photon wavelength is linked to the energy of a photon, which is a fundamental particle of light. The relationship between the energy \( E \) of a photon and its wavelength \( \lambda \) can be described using the equation:
\[ E = \frac{hc}{\lambda} \]where:
  • \( h \) is Planck's constant, \( 6.626 \times 10^{-34} \text{J s}. \)
  • \( c \) is the speed of light, \( 3 \times 10^{8} \text{m/s}. \)
  • \( \lambda \) is the wavelength of the photon.
When energy or temperature changes, the wavelength of the photon adjusts accordingly. Shorter wavelengths correspond to higher energy, while longer wavelengths correspond to lower energy. This concept is key in understanding the electromagnetic spectrum and the behavior of light.
Temperature Conversion
Temperature conversion is essential in scientific calculations because different formulas require temperatures in specific units. The Kelvin scale is a primary unit for temperature in scientific contexts. Converting Celsius to Kelvin is common, and the formula is:
\[ T(K) = T(掳C) + 273.15 \]This conversion is crucial for using the kinetic molecular theory, which requires temperature in Kelvin. Converting between temperature scales ensures accurate computations and comparisons in scientific analysis, especially when discussing gas behaviors and thermodynamics.
Planck's Constant
Planck's constant, denoted as \( h \), is a fundamental constant in quantum mechanics. It links the energy of photons to their frequency, playing a crucial role in the field of quantum mechanics. The value of Planck's constant is approximately:
\[ h = 6.626 \times 10^{-34} \text{J s} \]Significance of Planck's constant includes:
  • Determining the energy levels of photons: \( E = hf \)
  • Formulating the Schr枚dinger equation and other quantum phenomena.
  • Implying that energy levels in quantum systems are quantized.
Understanding Planck's constant is essential for exploring atomic and subatomic processes, laying the groundwork for modern physics and chemistry.

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Most popular questions from this chapter

When ultraviolet light with a wavelength of \(254 \mathrm{nm}\) falls upon a clean metal surface, the stopping potential necessary to terminate the emission of photoelectrons is \(0.181 \mathrm{~V}\). (a) What is the photoelectric threshold wavelength for this metal? (b) What is the work function for the metal?

An incident X-ray photon is scattered from a free electron that is initially at rest. The photon is scattered straight back at an angle of \(180^{\circ}\) from its initial direction. The wavelength of the scattered photon is \(0.0830 \mathrm{nm}\). (a) What is the wavelength of the incident photon? (b) What is the magnitude of the momentum of the electron after the collision? (c) What is the kinetic energy of the electron after the collision?

(a) What is the de Broglie wavelength of an electron accelerated through \(800 \mathrm{~V} ?\) (b) What is the de Broglie wavelength of a proton accelerated through the same potential difference?

Why is it easier to use helium ions rather than neutral helium atoms in an atomic microscope? A. Helium atoms are not electrically charged, and only electrically charged particles have wave properties. B. Helium atoms form molecules, which are too large to have wave properties. C. Neutral helium atoms are more difficult to focus with electric and magnetic fields. D. The much larger mass of a helium atom compared to a helium ion makes it more difficult to accelerate.

A photon with wavelength of \(0.1100 \mathrm{nm}\) collides with a free electron that is initially at rest. After the collision, the photon's wavelength is \(0.1132 \mathrm{nm} .\) (a) What is the kinetic energy of the electron after the collision? What is its speed? (b) If the electron is suddenly stopped (for example, in a solid target), all of its kinetic energy is used to create a photon. What is the wavelength of this photon?
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