Section 27.8 Kinetic Theory Bootcamp
Subsection 27.8.1 Pressure in an Ideal Gas
Problem 27.8.1. Root Mean Squared Speed of a Helium Molecule.
Follow the link: Checkpoint 27.1.3.
Problem 27.8.2. Pressure from a Stream of Particles Striking on a Wall.
Follow the link: Checkpoint 27.1.4.
Problem 27.8.3. Pressure from a Stream of Particles Striking on a Wall and Sticking to the Wall.
Follow the link: Checkpoint 27.1.5.
Subsection 27.8.2 Maxwell's Distribution
Problem 27.8.4. Average Speed and RMS Speed of a Nitrogen Molecule.
Follow the link: Checkpoint 27.2.2.
Problem 27.8.5. (Calculus) Mean of the Maxwell's Distribution.
Follow the link: Checkpoint 27.2.4.
Problem 27.8.6. (Calculus) Exploring Maxwell's Distribution of Energy of Molecules.
Follow the link: Checkpoint 27.2.5.
Subsection 27.8.3 Mean Free Path
Problem 27.8.7. Mean Free Path of an Ideal Gas Illustrated.
Follow the link: Example 27.3.3.
Problem 27.8.8. Mean Free Path, Mean Free Time, and Intermolecular Distance in Helium Gas.
Follow the link: Checkpoint 27.3.5.
Problem 27.8.9. Variation of Mean Free Path with Pressure, Temperature, Volume, Amount of Gas.
Follow the link: Checkpoint 27.3.6.
Problem 27.8.10. (Calculus) Exploring Distribution of Free Lengths.
Follow the link: Checkpoint 27.3.7.
Subsection 27.8.4 Effusion
Problem 27.8.11. (Calculus) Reduction of Pressure in a Container due to Escaping Molecules by Effusion.
Follow the link: Checkpoint 27.5.1.
Subsection 27.8.5 Water Vapor and Humidity
Problem 27.8.12. Partial Pressure of Water Vapor from Humidity.
Follow the link: Checkpoint 27.7.2.
Subsection 27.8.6 Miscellaneous
Problem 27.8.13. Mean Free Path from Distances Between Successive Collisions of a Molecule.
The mean free path is the average distance a molecule travels in-between collisions. In reality a molecule will travel different distances in-between collisions. Suppose a molecule of size 0.3 nm travels the following distances between collisions: 11 nm, 5 nm, 12 nm, 15 nm, 25 nm, 2 nm, 11 nm, 30 nm, 18 nm, 20 nm, 14 nm, 15 nm, 10 nm, 28 nm, 9 nm, 17 nm, 5 nm, 10 nm, 6 nm, 13 nm, 17 nm, 15 nm, 10 nm, 22 nm, 9 nm, 23 nm, 19nm, 4 nm, 15 nm, 12 nm, 10 nm, 16 nm, 11 nm.
(a) Find the mean free path from this data.
(b) What is the standard deviation of space between collisions from the mean free path?
(c) What is the most probable free length?
(d) What is the rms free length?
(e) Supposing, the temperature of the gas is 200 K and the pressure of the gas is 1 atm, what would be the diameter of molecules of the gas
Use a data processing sofware such as Mathematica or Excel or Google Sheets.
(a) - (d) See solution, (e) \(D = 0.66\text{ nm}\text{.}\)
I typed up the data in Mathematica and used their statistical functions. (a) 14 nm, (b) 6.7 nm, (c) Most common function gave two most-commonly occurring number, 10 nm and 15 nm. When I drew the histogram I found that 10 nm has more data points clustered around it than 15 nm points. The most probable length will be 10 nm. (d) 15.4 nm.
(e) Using the formula for the mean free path we get
Now, you can put in the numbers given.
This gives \(D = 0.66\text{ nm}\text{.}\)
Problem 27.8.14. Surface Temperature of a Star for Helium Atoms to Escape Gravitational Pull.
What should be the maximum temperature of a sun-sized star if Helium atoms cannot escape the gravitational pull of the star? Hint: Equate gravitational escape speed with the rms speed.
The escape speed \(v_e\) of a body of mass \(m\) from a body of mass \(M\) and radius \(R\) is given by equating the kinetic energy to the negative of the potential energy.
This gives
The root-mean square speed of a molecule when in the environment of temperature \(T\) is
Equating the two speeds will give the condition we seek in this problem.
The temperature will be
Now, we put in the following numbers.
These give the following for \(T\text{.}\)
This temperature is much larger than the surface temperature of the Sun.