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Physics I. Microscopic Model of Gas. Prof. WAN, Xin xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/. The Naïve Approach, Again. N particles r i (t), v i (t); interaction V(r i -r j ). Elementary Probability Theory.

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Microscopic model of gas

Physics I

Microscopic Model of Gas

Prof. WAN, Xin



The na ve approach again
The Naïve Approach, Again

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

Elementary probability theory
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 theory1
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
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, 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
Applying the Ideal Gas Law

Boltzmann’s constant


  • 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
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
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?

Vibrational mode
Vibrational Mode

Solution 1:

Vibration with the reduced mass.

Translational mode
Translational Mode

Solution 1:


Two harmonic oscillators3
Two Harmonic Oscillators

In mathematics language, we solved an eigenvalue problem.

The two eigenvectors are orthogonal to each other. Independent!

Mode counting 1d
Mode Counting – 1D

  • 1D: N-atom linear molecule

    • Translation: 1

    • Vibration: N – 1

A straightforward generalization of the two-atom problem.

From 1d to 2d a trivial example



From 1D to 2D: A Trivial Example




Mode counting 2d
Mode Counting – 2D

  • 2D: N-atom (planer, nonlinear) molecule

    • Translation: 2

    • Rotation: 1

    • Vibration: 2N – 3

Mode counting 3d
Mode Counting – 3D

  • 3D: N-atom (nonlinear) molecule

    • Translation: 3

    • Rotation: 3

    • Vibration: 3N – 6

Vibrational modes of co 2
Vibrational Modes of CO2

  • N = 3, linear

    • Translation: 3

    • Rotation: 2

    • Vibration: 3N – 3 – 2 = 4

Vibrational modes of h 2 o
Vibrational Modes of H2O

  • N = 3, planer

    • Translation: 3

    • Rotation: 3

    • Vibration: 3N – 3 – 3 = 3

Contribution to specific heat
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 h 2
Specific Heat of H2

Quantum mechanics is needed to explain this.

Specific heat of solids
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

root mean square speed

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

Distribution of speed
Distribution of Speed




rotating drum

to pump

Speed selection
Speed Selection

  • Can you design an equipment to select gas molecules with a chosen speed?


to pump

Maxwell distribution1











Maxwell Distribution

number of moleculesv [v1, v2]

Maxwell distribution2
Maxwell Distribution

Total number of molecules

Characteristic speed
Characteristic Speed

Most probable speed

Characteristic speed1
Characteristic Speed

Root mean sqaure speed

Characteristic speed2
Characteristic Speed

Average speed

Boltzmann distribution
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
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
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
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
BEC of Dysprosium

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

Mean free path
Mean Free Path

Average distance between two collisions

Mean free path1
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 path2
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
Q&A: Collision Frequency

  • Consider air at room temperature.

    • How far does a typical molecule (with a diameter of 210-10m) move before it collides with another molecule?

Q a collision frequency1
Q&A: Collision Frequency

  • Consider air at room temperature.

    • How far does a typical molecule (with a diameter of 210-10m) move before it collides with another molecule?

Q a collision frequency2
Q&A: Collision Frequency

  • Consider air at room temperature.

    • Average molecular separation:

Q a collision frequency3
Q&A: Collision Frequency

  • Consider air at room temperature.

    • On average, how frequently does one molecule collidewith another?

Expect ~ 500 m/s

Expect ~ 2109 /s

Try yourself!

Transport viscous flow
Transport: Viscous Flow

Fluid flows layer by layer with varying v.

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



F, v


Cylindrical pipe nonviscous
Cylindrical Pipe, Nonviscous




(volumetric flow rate)

Cylindrical pipe viscous
Cylindrical Pipe, Viscous






(Poiseuille Law)


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