There is a larger fraction of molecules moving at 1000 m/s when the gas temperature is 1000 k than when the gas temperature is 300 K. Does this make sense? Explain. b. There is a smaller fraction of molecules moving at 50 m/s when the gas temperature is 1000 k than when the gas temperature is 300 K. Does this make sense? Explain
There is a larger fraction of molecules moving at 1000 m/s when the gas temperature is 1000 k than when the gas temperature is 300 K. Does this make sense? Explain. b. There is a smaller fraction of molecules moving at 50 m/s when the gas temperature is 1000 k than when the gas temperature is 300 K. Does this make sense? Explain
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
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a. There is a larger fraction of molecules moving at 1000 m/s when the gas temperature is 1000 k than when the gas temperature is 300 K. Does this make sense? Explain.
b. There is a smaller fraction of molecules moving at 50 m/s when the gas temperature is 1000 k than when the gas temperature is 300 K. Does this make sense? Explain.
![**Model 5: The Maxwell Distribution of Molecular Speeds**
_Not all gas particles within a collection of particles (at constant temperature) have the same velocity—some particles are moving slowly, others are moving rapidly. The problem of how to determine the most probable distribution of speeds was solved by J. Clerk Maxwell. Starting with the following assumptions:_
- **The probability of a molecular state depends only on the energy of the molecular state.**
- **The same probability distribution applies for all kinds of molecules.**
Maxwell's equation for the distribution of speeds is the following:
\[ \text{fraction of molecules per unit speed interval} = 4\pi v^2 \left( \frac{m}{2\pi k T} \right)^{3/2} e^{-mv^2 / 2kT} \]
where \( k \) (Boltzmann's constant) = \( 1.38066 \times 10^{-23} \) JK\(^{-1} \)
**Table 1. The Speeds of Gaseous N\(_2\) Molecules.**
| Speed of Molecule (m/s) | Fraction of Molecules per Unit Speed Interval at 300 K (s/m) | Fraction of Molecules per Unit Speed Interval at 1000 K (s/m) |
|-------------------------|---------------------------------------------------------------|----------------------------------------------------------------|
| 0 | 0 | 0 |
| 25 | 1.87 x 10\(^{-5}\) | 3.08 x 10\(^{-6}\) |
| 50 | 7.40 x 10\(^{-5}\) | 1.23 x 10\(^{-5}\) |
| 100 | 2.84 x 10\(^{-4}\) | 4.85 x 10\(^{-5}\) |
| 200 | 9.60 x 10\(^{-4}\) | 1.85 x 10\(^{-4}\) |
| 300 | 1.63 x 10\(^{-3}\) | 3.82 x 10\(^{-4}\) |
| 400 | 1.96 x 10\(^{-3}\) | 6.03 x 10\(^{-4}\) |
| 500 | 1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c0fb0ea-422c-4fd5-95c1-f570f6356a90%2Fcf2a6e38-6933-4c84-ac70-a542f17b26aa%2F629212_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Model 5: The Maxwell Distribution of Molecular Speeds**
_Not all gas particles within a collection of particles (at constant temperature) have the same velocity—some particles are moving slowly, others are moving rapidly. The problem of how to determine the most probable distribution of speeds was solved by J. Clerk Maxwell. Starting with the following assumptions:_
- **The probability of a molecular state depends only on the energy of the molecular state.**
- **The same probability distribution applies for all kinds of molecules.**
Maxwell's equation for the distribution of speeds is the following:
\[ \text{fraction of molecules per unit speed interval} = 4\pi v^2 \left( \frac{m}{2\pi k T} \right)^{3/2} e^{-mv^2 / 2kT} \]
where \( k \) (Boltzmann's constant) = \( 1.38066 \times 10^{-23} \) JK\(^{-1} \)
**Table 1. The Speeds of Gaseous N\(_2\) Molecules.**
| Speed of Molecule (m/s) | Fraction of Molecules per Unit Speed Interval at 300 K (s/m) | Fraction of Molecules per Unit Speed Interval at 1000 K (s/m) |
|-------------------------|---------------------------------------------------------------|----------------------------------------------------------------|
| 0 | 0 | 0 |
| 25 | 1.87 x 10\(^{-5}\) | 3.08 x 10\(^{-6}\) |
| 50 | 7.40 x 10\(^{-5}\) | 1.23 x 10\(^{-5}\) |
| 100 | 2.84 x 10\(^{-4}\) | 4.85 x 10\(^{-5}\) |
| 200 | 9.60 x 10\(^{-4}\) | 1.85 x 10\(^{-4}\) |
| 300 | 1.63 x 10\(^{-3}\) | 3.82 x 10\(^{-4}\) |
| 400 | 1.96 x 10\(^{-3}\) | 6.03 x 10\(^{-4}\) |
| 500 | 1
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