Problem 59
Question
At what temperature does the average translational \(\mathrm{KE}\) of a molecule in a gas becomes equal to the \(\mathrm{K} . \mathrm{E}\). of an electron accelerated through a \(\mathrm{PD}\) of \(3 \mathrm{~V} ?\) (a) \(232 \mathrm{~K}\) (b) \(2320 \mathrm{~K}\) (c) \(23,200 \mathrm{~K}\) (d) \(2,32,000 \mathrm{~K}\)
Step-by-Step Solution
Verified Answer
The temperature at which the average translational KE of a gas molecule equals the KE of an electron accelerated through 3 V is approximately 23,200 K.
1Step 1: Understand the concept of kinetic energy of a gas molecule
The average kinetic energy (KE) of a molecule in an ideal gas can be expressed using the formula: \( KE = \frac{3}{2}kT \), where \( k \) is the Boltzmann constant (\(1.38 \times 10^{-23} \text{J/K}\)) and \( T \) is the temperature in Kelvin.
2Step 2: Calculate the kinetic energy of the electron
The kinetic energy (KE) of an electron accelerated through a potential difference (PD) of 3 V can be calculated using the equation: \( KE = eV \), where \( e \) is the elementary charge (\(1.6 \times 10^{-19} \text{C}\)) and \( V \) is the potential difference in volts. \( KE = 1.6 \times 10^{-19} \text{C} \times 3 \text{V} = 4.8 \times 10^{-19} \text{J} \).
3Step 3: Equate the kinetic energies and solve for temperature
To find the temperature at which the average KE of the gas molecule equals the KE of the electron, set the two expressions equal to each other: \( \frac{3}{2}kT = 4.8 \times 10^{-19} \text{J} \). Now solve for \( T \): \( T = \frac{2 \times 4.8 \times 10^{-19} \text{J}}{3 \times 1.38 \times 10^{-23} \text{J/K}} \).
4Step 4: Calculate the temperature
Perform the calculation to find \( T \): \( T = \frac{9.6 \times 10^{-19} \text{J}}{4.14 \times 10^{-23} \text{J/K}} \approx 23,188 \text{K} \). This value is closest to option (c) 23,200 K.
Key Concepts
Boltzmann ConstantElectron Kinetic EnergyTemperature and Kinetic Energy
Boltzmann Constant
To fully understand how energy within a gas is related to temperature, one must be acquainted with the Boltzmann constant. This fundamental physical constant is denoted by the symbol \( k \) and is one of the cornerstones in the field of statistical mechanics.
It acts as a bridge between macroscopic and microscopic physics, connecting observable quantities such as temperature with microscopic details including the energy of atoms and molecules. The value of the Boltzmann constant is approximately \(1.38 \times 10^{-23} \) joules per kelvin (J/K). In essence, it quantifies the amount of energy, in joules, for each degree of temperature when expressed in Kelvin.
In gas kinetic theory, the Boltzmann constant is frequently utilized to express the average kinetic energy of particles in a gas at a given temperature. By understanding this constant, students can grasp the foundations of energy distribution in thermodynamics and how temperature scales with the energy of microscopic particles.
It acts as a bridge between macroscopic and microscopic physics, connecting observable quantities such as temperature with microscopic details including the energy of atoms and molecules. The value of the Boltzmann constant is approximately \(1.38 \times 10^{-23} \) joules per kelvin (J/K). In essence, it quantifies the amount of energy, in joules, for each degree of temperature when expressed in Kelvin.
In gas kinetic theory, the Boltzmann constant is frequently utilized to express the average kinetic energy of particles in a gas at a given temperature. By understanding this constant, students can grasp the foundations of energy distribution in thermodynamics and how temperature scales with the energy of microscopic particles.
Electron Kinetic Energy
When considering microscopic particles such as electrons, their kinetic energy becomes a crucial aspect to examine. In physics, kinetic energy is the energy possessed by an object due to its motion. For electrons, this energy can be markedly increased by accelerating them through an electric potential difference, measured in volts (V).
When an electron is subjected to a potential difference, it gains energy equal to the charge of the electron times the potential difference it is accelerated through, expressed as \( KE = eV \). Here, \( e \) stands for the elementary charge of the electron, equal to about \(1.6 \times 10^{-19} \) coulombs (C), and \( V \) is the potential difference in volts.
This concept is particularly useful in fields like electronics and particle physics, where understanding the energy transfer to electrons is vital for designing circuits or for the acceleration of particles in a collider, respectively.
When an electron is subjected to a potential difference, it gains energy equal to the charge of the electron times the potential difference it is accelerated through, expressed as \( KE = eV \). Here, \( e \) stands for the elementary charge of the electron, equal to about \(1.6 \times 10^{-19} \) coulombs (C), and \( V \) is the potential difference in volts.
This concept is particularly useful in fields like electronics and particle physics, where understanding the energy transfer to electrons is vital for designing circuits or for the acceleration of particles in a collider, respectively.
Temperature and Kinetic Energy
The relationship between temperature and kinetic energy in gases is an essential topic in thermodynamics and statistical mechanics. The average kinetic energy (KE) of a gas molecule is directly proportional to the temperature of the gas. For an ideal gas, this relationship is defined by the equation \( KE = \frac{3}{2}kT \), where \( T \) is the absolute temperature measured in Kelvin, and \( k \) represents the Boltzmann constant.
Temperature serves as a measure of the average kinetic energy of the particles within a substance. As the temperature increases, the molecules in the gas move faster, and their average kinetic energy rises. This correlation implies that by measuring the temperature, we also gain knowledge about the internal kinetic energy of the gas particles.
The practical application of this can be seen in everyday scenarios, from the behavior of air molecules that affect weather patterns to the controlled environment of a lab where gas behavior is vital for chemical reactions. Hence, the relation between temperature and kinetic energy gives us a powerful tool to predict and understand the behavior of gases under various conditions.
Temperature serves as a measure of the average kinetic energy of the particles within a substance. As the temperature increases, the molecules in the gas move faster, and their average kinetic energy rises. This correlation implies that by measuring the temperature, we also gain knowledge about the internal kinetic energy of the gas particles.
The practical application of this can be seen in everyday scenarios, from the behavior of air molecules that affect weather patterns to the controlled environment of a lab where gas behavior is vital for chemical reactions. Hence, the relation between temperature and kinetic energy gives us a powerful tool to predict and understand the behavior of gases under various conditions.
Other exercises in this chapter
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