Emo] = %3D (10 with the same constant of proportionality. This is the principle of equipartitio of energy. (The constant kB is called Boltzman's constant.) Exercise 2. Using Equation (6), show that Equation (10) implies that m (11 B = 4k BT Exercise 3. Find the most likely speed of a monatomic gas in terms of the atomic mass, m and the temperature, T. Check that your answer has the correct units. Figure 1 shows the Maxwell distribution for four monatomic gases at the same temperature: Helium, Neon, Argon, and Xenon. Helium atoms have the smallest mass; xenon atoms have the largest mass. By Exercise 3, the heavier the atom, the smaller the most likely speed. Remarks: Physicists often quote the root mean square velocity which is
Emo] = %3D (10 with the same constant of proportionality. This is the principle of equipartitio of energy. (The constant kB is called Boltzman's constant.) Exercise 2. Using Equation (6), show that Equation (10) implies that m (11 B = 4k BT Exercise 3. Find the most likely speed of a monatomic gas in terms of the atomic mass, m and the temperature, T. Check that your answer has the correct units. Figure 1 shows the Maxwell distribution for four monatomic gases at the same temperature: Helium, Neon, Argon, and Xenon. Helium atoms have the smallest mass; xenon atoms have the largest mass. By Exercise 3, the heavier the atom, the smaller the most likely speed. Remarks: Physicists often quote the root mean square velocity which is
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Exercise 2
![Hence
E,mu*) = 3kpT
%3D
(10)
with the same constant of proportionality. This is the principle of equipartition
of energy. (The constant kB is called Boltzman's constant.)
Exercise 2. Using Equation (6), show that Equation (10) implies that
(11)
4kBT
Exercise 3. Find the most likely speed of a monatomic gas in terms of the
atomic mass, m and the temperature, T. Check that your answer has the correct
units.
Figure 1 shows the Maxwell distribution for four monatomic gases at the
same temperature: Helium, Neon, Argon, and Xenon. Helium atoms have the
smallest mass; xenon atoms have the largest mass. By Exercise 3, the heavier
the atom, the smaller the most likely speed.
Remarks: Physicists often quote the root mean square velocity which is
defined as E[V²].
Oet mean square velocity, or E[V]? Why?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F718b1378-40e4-4c32-83bc-211fc46d7de4%2F83e8deec-ee17-4d0a-a76e-837dd2821199%2F92r400w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Hence
E,mu*) = 3kpT
%3D
(10)
with the same constant of proportionality. This is the principle of equipartition
of energy. (The constant kB is called Boltzman's constant.)
Exercise 2. Using Equation (6), show that Equation (10) implies that
(11)
4kBT
Exercise 3. Find the most likely speed of a monatomic gas in terms of the
atomic mass, m and the temperature, T. Check that your answer has the correct
units.
Figure 1 shows the Maxwell distribution for four monatomic gases at the
same temperature: Helium, Neon, Argon, and Xenon. Helium atoms have the
smallest mass; xenon atoms have the largest mass. By Exercise 3, the heavier
the atom, the smaller the most likely speed.
Remarks: Physicists often quote the root mean square velocity which is
defined as E[V²].
Oet mean square velocity, or E[V]? Why?
Transcribed Image Text:h:(v.) = Ae Bu , i = 1, 2,3
(5)
Playing the usual game with volume differentials, imagining a spherical shell
with radius v and thickness dv, the differential volume is 4Tv2dv.
Maxwell concluded that the density for molecular speeds had the form
f(v) = 4Tv²A°e-Bv²
(6)
2
The Value of the Constants A and B
Recall our earlier definition:
Definition 1. The Gamma function, I: (0, 00) R, is
r(a) =
r-le-dx
(7)
Also recall that we proved the following properties of the Gamma function:
Theorem 2.1. Properties of the Gamma function:
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