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The Naïve Approach, Again

N particles ri(t), vi(t); interaction V(ri-rj)

Elementary Probability Theory

Assume the speeds of 10 particles are 0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0 m/s

When we have many particles, we may denote pa the probability of finding their velocities in the interval [va, va+1].

Elementary Probability Theory

Now, the averages become

In the continuous version, we may denote p(v)dv the probability of finding particles’ velocities in the interval [v, v+dv].

Assumptions of the Ideal Gas Model

- Large number of molecules and large average separation (molecular volume is negligible).
- The molecules obey Newton’s laws, but as a whole they move randomly with a time-independent distribution of speeds.
- The molecules undergo elastic collisions with each other and with the walls of the container.
- The forces between molecules are short-range, hence negligible except during a collision.
- That is, all of the gas molecules are identical.

Pressure, the Microscopic View

- Pressure that a gas exerts on the walls of its container is a consequence of the collisions of the gas molecules with the walls.

half of molecules moving right

r = N / V

Applying the Ideal Gas Law

Boltzmann’s constant

Temperature

- Temperature is a measure of internal energy (kB is the conversion factor). It measures the average energy per degree of freedom per molecule/atom.
- Equipartition theorem: can be generalized to rotational and vibrational degrees of freedom.

Heat Capacity at Constant V

- We can detect the microscopic degrees of freedom by measuring heat capacity at constant volume.
- Internal Energy U = NfkBT/2
- Heat capacity
- Molar specific heat cV = (f/2)R

degrees of freedom

Specific Heat at Constant V

- Monoatomic gases has a ratio 3/2. Remember?
- Why do diatomic gases have the ratio 5/2?
- What about polyatomic gases?

Two Harmonic Oscillators

Assume

Two Harmonic Oscillators

Assume

Two Harmonic Oscillators

In mathematics language, we solved an eigenvalue problem.

The two eigenvectors are orthogonal to each other. Independent!

Mode Counting – 1D

- 1D: N-atom linear molecule
- Translation: 1
- Vibration: N – 1

A straightforward generalization of the two-atom problem.

Mode Counting – 2D

- 2D: N-atom (planer, nonlinear) molecule
- Translation: 2
- Rotation: 1
- Vibration: 2N – 3

Mode Counting – 3D

- 3D: N-atom (nonlinear) molecule
- Translation: 3
- Rotation: 3
- Vibration: 3N – 6

Vibrational Modes of CO2

- N = 3, linear
- Translation: 3
- Rotation: 2
- Vibration: 3N – 3 – 2 = 4

Vibrational Modes of H2O

- N = 3, planer
- Translation: 3
- Rotation: 3
- Vibration: 3N – 3 – 3 = 3

Contribution to Specific Heat

Equipartition theorem: The mean value of each independent quadratic term in the energy is equal to kBT/2.

Specific Heat of H2

Quantum mechanics is needed to explain this.

Specific Heat of Solids

DuLong – Petit law

spatial dimension

vibration energy

Molar specific heat

Again, quantum mechanics is needed.

Root Mean Square Speed

root mean square speed

Estimate the root mean square speed of water molecules at room temperature.

Maxwell Distribution

Total number of molecules

Characteristic Speed

Most probable speed

Characteristic Speed

Root mean sqaure speed

Characteristic Speed

Average speed

Boltzmann Distribution

potential energy

- Continuing from fluid statics
- The probability of finding the molecules in a particular energy state varies exponentially as the negative of the energy divided by kBT.

Boltzmann distribution law

Laser Cooling

Figure: A CCD image of a cold cloud of rubidium atoms which have been laser cooled by the red laser beams to temperatures of a millionth of a Kelvin. The white fluorescent cloud forms at the intersection of the beams.

Bose-Einstein Condensation

Velocity-distribution data for a gas of rubidium atoms, confirming the discovery of a new phase of matter, the Bose–Einstein condensate.

Left: just before the appearance of a Bose–Einstein condensate. Center: just after the appearance of the condensate. Right: after further evaporation, leaving a sample of nearly pure condensate.

Earlier BEC Research

- BEC in ultracold atomic gases was first realized in 1995 with 87Rb, 23Na, and 7Li. This pioneering work was honored with the Nobel prize 2001 in physics, awarded to Eric Cornell, Carl Wieman, and Wolfgang Ketterle.

For an updated list, check http://ucan.physics.utoronto.ca/

BEC of Dysprosium

Strongly dipolar BEC of dysprosium, Mingwu Lu et al., PRL 107, 190401 (2011)

Mean Free Path

Average distance between two collisions

Mean Free Path

During time interval t, a molecule sweeps a cylinder of diameter 2d and length vt.

Volume of the cylinder

Average number of collisions

Mean free path

Mean Free Path

During time interval t, a molecule sweeps a cylinder of diameter 2d and length vt.

Average number of collisions

Relative motion

Mean free path

Q&A: Collision Frequency

- Consider air at room temperature.
- How far does a typical molecule (with a diameter of 210-10m) move before it collides with another molecule?

Q&A: Collision Frequency

- Consider air at room temperature.
- How far does a typical molecule (with a diameter of 210-10m) move before it collides with another molecule?

Q&A: Collision Frequency

- Consider air at room temperature.
- Average molecular separation:

Q&A: Collision Frequency

- Consider air at room temperature.
- On average, how frequently does one molecule collidewith another?

Expect ~ 500 m/s

Expect ~ 2109 /s

Try yourself!

Transport: Viscous Flow

Fluid flows layer by layer with varying v.

F = h A dv/dy h: coefficient of viscosity

A

y

F, v

A

Homework

CHAP. 22 Exercises 7, 8, 10, 21, 24 (P513)

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