Ideal monatomic gas is enclosed in cylinder of radius a and...

Ideal monatomic gas is enclosed in cylinder of radius $a$ and length $L$. The cylinder rotates with angular velocity $\omega$ about its symmetry axis and the ideal gas is in equilibrium at temperature $T$ in the coordinate system rotating with the cylinder. Assume that the gas atoms have mass $m$, have no internal degrees of freedom, and obey classical statistics. (a) What is the Hamiltonian in the rotating coordinates system? (b) What is the partition function for the system? (c) What is the average particle number density as a function of $r$ ?

Ideal monatomic gas is enclosed in cylinder of radius $a$ and length $L$. The cylinder rotates with angular velocity $\omega$ about its symmetry axis and the ideal gas is in equilibrium at temperature $T$ in the coordinate system rotating with the cylinder. Assume that the gas atoms have mass $m$, have no internal degrees of freedom, and obey classical statistics.
(a) What is the Hamiltonian in the rotating coordinates system?
(b) What is the partition function for the system?
(c) What is the average particle number density as a function of $r$ ?

Key Concepts

Rotation-Induced Modifications in Equilibrium Systems

Rotating reference frames introduce non-inertial effects such as centrifugal and Coriolis forces which need to be included in the thermodynamic description. When the system is analyzed in the equilibrium state in the rotating frame, these effects modify the energy of the particles and, consequently, the statistical distributions. This adjustment is essential for correctly determining macroscopic properties such as the partition function, internal energy, and spatial density distributions.

Centrifugal Potential and Boltzmann Distribution

In a rotating system, the centrifugal potential emerges due to the outward effective force in the rotating frame. This potential modifies the Boltzmann factor, leading to a position-dependent probability density. In equilibrium, the average particle number density is determined by this modified Boltzmann factor, causing the density to vary with the radial coordinate. This concept is critical for understanding how rotation can lead to non-uniform density profiles in systems at thermal equilibrium.

Partition Function in Classical Statistical Mechanics

The partition function is a central concept in statistical mechanics that encapsulates all thermodynamic information of the system. For a classical ideal gas, the partition function is computed by integrating the Boltzmann factor exp(-H/(k_BT)) over the entire phase space. Here, the modified Hamiltonian in the rotating frame, which includes additional potential energy due to rotation, is used, leading to integrals that typically factor into contributions from kinetic energy and effective potential energy.

Hamiltonian in Rotating Coordinates

When dealing with a system in a rotating frame, the Hamiltonian must account for inertial forces. This includes not only the usual kinetic energy of the particles, but also additional terms such as the centrifugal potential and, depending on the treatment, the Coriolis contributions. In the case of an ideal gas in a rotating cylinder, the effective potential includes the term arising from the centrifugal effect, modifying the energy landscape in the rotating coordinate system.

Transcript

00:01 So we have a 10 centimeter diameter, 20 centimeter long cylinder.

00:06 It contains two times 10 to the 22nd molecules of or atoms of argon.

00:14 So argon is just monotomic at a temperature of 50 degrees celsius.

00:20 So and the first question is, is what is the number done to the gas? well, that means we just need to divide the number of molecules or atoms we have.

00:28 Since it's argon, we could just call it and say they're atoms.

00:31 Divided by the volume, and the volume, you know, is just the volume of a cylinder.

00:37 And we wind up with 1 .27 times 10 to the 25th atoms per cubic meter.

00:45 Then they ask for the root mean square speed, so that the, that is a square root of three times boltsman constant times temperature divided by the molecular, the weight of one of one argon atom.

01:03 Temperature we can convert into, we need to convert into calvin.

01:07 And the weight of one argon atom, so we know that a mole of argon weighs 40 kilograms, or 40 grams.

01:17 And so we divide that by avogadro's number, and we get 6 .64 times 10 to the minus 26 kilograms.

01:30 So we have that.

01:32 We can plug that in here.

01:33 We can plug this into here.

01:35 And boltzman's constant.

01:36 And tanking through this calculation, we get 449 meters per second, which again, seems reasonable given the information we have in the chapter.

01:48 So then we want to know what is vmax, the rms value of vx max, the rms value of the x component of velocity.

01:55 Well, that's just the rms value of the speed divided by square to three.

02:04 And that winds up being 259 meters per second.

02:12 Now, we're asked for the rate of atoms colliding with one end of the cylinder.

02:22 Well, we can see here that we have that that is the one half times the number density, times the area that we're looking at where they're hitting, and times the velocity in the x direction.

02:36 And we're assuming that the velocity along the length of the cylinder is x.

02:42 Again, we call it whatever we want.

02:45 Again, x, y, or z, if we would have then needed vz, v, sub z, which would have been the same value.

02:56 So x, y, and z are all the same...

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