OutlineTutorial

Review of the thermal movement of ideal gas moleculesThe distribution of the velocities of the gas molecules

Velocity distribution of ideal gas molecules

Jie Yan

August 25, 2006

Jie Yan Velocity distribution of ideal gas molecules

OutlineTutorial

Review of the thermal movement of ideal gas moleculesThe distribution of the velocities of the gas molecules

Contents

I Review of thermal movement of ideal gas molecules.

I Distribution of the velocity of a molecule in ideal gas.

I Boltzmann distribution.

Jie Yan Velocity distribution of ideal gas molecules

OutlineTutorial

Review of the thermal movement of ideal gas moleculesThe distribution of the velocities of the gas molecules

Contents

I Review of thermal movement of ideal gas molecules.

I Distribution of the velocity of a molecule in ideal gas.

I Boltzmann distribution.

Jie Yan Velocity distribution of ideal gas molecules

OutlineTutorial

Review of the thermal movement of ideal gas moleculesThe distribution of the velocities of the gas molecules

Contents

I Review of thermal movement of ideal gas molecules.

I Distribution of the velocity of a molecule in ideal gas.

I Boltzmann distribution.

Jie Yan Velocity distribution of ideal gas molecules

OutlineTutorial

Review of the thermal movement of ideal gas moleculesThe distribution of the velocities of the gas molecules

Time slots

I Tutorial starts from week 3 (28AUG-1SEP).

I A class tutorial every week, on Friday 9 am - 10 am.

I A small group tutorial every even week (4,6,8,10), on Mon(12.00-12.50pm) or on Wed (5.00-5.50pm). The small grouptutorial classroom is S13-04-02.

I Group A(30%): week 4, 8,12, Mon; group B(70%): week6,10, 12, Wed.

Jie Yan Velocity distribution of ideal gas molecules

OutlineTutorial

Review of the thermal movement of ideal gas moleculesThe distribution of the velocities of the gas molecules

Time slots

I Tutorial starts from week 3 (28AUG-1SEP).

I A class tutorial every week, on Friday 9 am - 10 am.

I A small group tutorial every even week (4,6,8,10), on Mon(12.00-12.50pm) or on Wed (5.00-5.50pm). The small grouptutorial classroom is S13-04-02.

I Group A(30%): week 4, 8,12, Mon; group B(70%): week6,10, 12, Wed.

Jie Yan Velocity distribution of ideal gas molecules

OutlineTutorial

Review of the thermal movement of ideal gas moleculesThe distribution of the velocities of the gas molecules

Time slots

I Tutorial starts from week 3 (28AUG-1SEP).

I A class tutorial every week, on Friday 9 am - 10 am.

I A small group tutorial every even week (4,6,8,10), on Mon(12.00-12.50pm) or on Wed (5.00-5.50pm). The small grouptutorial classroom is S13-04-02.

I Group A(30%): week 4, 8,12, Mon; group B(70%): week6,10, 12, Wed.

Jie Yan Velocity distribution of ideal gas molecules

OutlineTutorial

Review of the thermal movement of ideal gas moleculesThe distribution of the velocities of the gas molecules

Time slots

I Tutorial starts from week 3 (28AUG-1SEP).

I A class tutorial every week, on Friday 9 am - 10 am.

I A small group tutorial every even week (4,6,8,10), on Mon(12.00-12.50pm) or on Wed (5.00-5.50pm). The small grouptutorial classroom is S13-04-02.

I Group A(30%): week 4, 8,12, Mon; group B(70%): week6,10, 12, Wed.

Jie Yan Velocity distribution of ideal gas molecules

OutlineTutorial

Review of the thermal movement of ideal gas moleculesThe distribution of the velocities of the gas molecules

Review: random walk leads to Gaussian distribution

Jie Yan Velocity distribution of ideal gas molecules

OutlineTutorial

Review of the thermal movement of ideal gas moleculesThe distribution of the velocities of the gas molecules

Temperature is a measure of the molecule movements

I Ideal gas law: PV = NkBT .

I Pressure is determined by gas molecule move:PV = m < v2

x > N = m<v2>N3 .

I We thus have: The average kinetic energy εk = 12m < v2 >

of a molecule in an ideal gas is 32kBT . Each degree of

freedom contribute 12kBT .

Jie Yan Velocity distribution of ideal gas molecules

OutlineTutorial

Review of the thermal movement of ideal gas moleculesThe distribution of the velocities of the gas molecules

Speed of gas molecules on earth surface at roomtemperature

At the earth surface, P = 105Pa, the mole density isc = 1

24M = 124

molL (1mol = 6.02× 1023; 1L = 10−3m3).

I Let N = 1mol , please compute NkBT , PV , and NkBTPV . Show

that the ideal gas is a reasonable approximation to the gas onearth.

I Gas on earth mostly consists nitrogen. One nitrogen moleculehas a mass m ≈ 4.7× 10−26kg . At room temperatureT = 300K , please compute that its kinetic energy εk , andshow that the average of the magnitude of its speed√

< v2 > ≈ 500ms .

Jie Yan Velocity distribution of ideal gas molecules

OutlineTutorial

Review of the thermal movement of ideal gas moleculesThe distribution of the velocities of the gas molecules

Why gas molecules does not fall onto the ground in earth?

I The change in the potential: U(z) = mgz , assuming it is anitrogen molecule. So Zmax ≈ εk

mg . Please compute Zmax .

I Please show that in a room, the gravitation does not affectthe gas molecule distribution.

I At what mass the gravitation can affect the distribution of themolecules in a room on earth? At what mass the gravity cantrap a molecule within 1micron = 10−6m?

Jie Yan Velocity distribution of ideal gas molecules

OutlineTutorial

Review of the thermal movement of ideal gas moleculesThe distribution of the velocities of the gas molecules

Interaction strength between molecules

I General rule of estimating the interaction strength: fd ≈ kBT ,where f is the force, and d is the characteristic interactiondistance. So f ≈ kBT

d ≈ pN for d ≈ nm.

I Many proteins can bind to and unbind from DNA. Thebinding and unbinding equilibrium is driven by electrostaticinteractions and thermal fluctuation. In usual physiologicalsalt condition (≈ 150 mM NaCl), the interaction distance is≈ 1.7nm. If the binding and unbinding are both frequent,please show that the interaction strength must be aroundseveral Pico Newtons.

Jie Yan Velocity distribution of ideal gas molecules

OutlineTutorial

Review of the thermal movement of ideal gas moleculesThe distribution of the velocities of the gas molecules

Wild guess of the velocity distribution of the gas molecules

I Range of velocity: −∞ < vx < ∞.

I Symmetry in direction: ρ(vx) = ρ(−vx), so < vx >= 0.

I Since 12m < v2

x >= 12kBT , the higher the T , the bigger the

variance σ2vx

=< v2x >.

I Orientational symmetry: ρ(vx) = ρ(vy ) = ρ(vz).

I At small T , ρ(vx) quickly goes to zero for |vx | > 0.

I What function is a good candidate?

Jie Yan Velocity distribution of ideal gas molecules

OutlineTutorial

Review of the thermal movement of ideal gas moleculesThe distribution of the velocities of the gas molecules

Random walk in the velocity space

I We know a random walk in coordinate space leads to aGaussian distribution.

I Gas molecules are changing their their accelerations randomly.This can be thought as a random walk in the velocity space.

I Not strictly, we can guess the velocity distribution is aGaussian.

I We already knew < v2x >= kBT

m ,and < vx >= 0.

I Please show that: ρ(vx) =√

m2πkBT e

−mv2x

2kBT .

Jie Yan Velocity distribution of ideal gas molecules

OutlineTutorial

Review of the thermal movement of ideal gas moleculesThe distribution of the velocities of the gas molecules

More on ρ(vx)

Jie Yan Velocity distribution of ideal gas molecules

OutlineTutorial

Review of the thermal movement of ideal gas moleculesThe distribution of the velocities of the gas molecules

Maxwell distribution of the velocity of individual molecules

I We have learnt that: ρ(vx) =√

m2πkBT e

−mv2x

2kBT .

I But molecules are moving in 3-d. Reasonable guess: movingin each direction is independent. Soρ(~v) = ρ(vx , vy , vz) = ρ(vx) ∗ ρ(vy ) ∗ ρ(vz).

I Please show ρ(~v) = ( m2πkBT )3/2e

−mv2

2kBT , where

v2 = v2x + v2

y + v2z .

I Find < ~v >, u =√

< v2 >.

Jie Yan Velocity distribution of ideal gas molecules

OutlineTutorial

Review of the thermal movement of ideal gas moleculesThe distribution of the velocities of the gas molecules

More on ρ(~v)

Jie Yan Velocity distribution of ideal gas molecules

OutlineTutorial

Review of the thermal movement of ideal gas moleculesThe distribution of the velocities of the gas molecules

Maxwell distribution of the scalar velocity of individualmolecules

I What is the distribution of the magnitude of the velocity ρ(u),where u =

√~v?

I It can be shown that ρ(u) = 4π( m2πkBT )3/2u2e

−mu2

2kBT .

I Prove ρ(u) is normalized.

I Show that the most probable umax =√

2kBTm

(Hint: dduρ(u) = 0).

Jie Yan Velocity distribution of ideal gas molecules

OutlineTutorial

Review of the thermal movement of ideal gas moleculesThe distribution of the velocities of the gas molecules

More on ρ(u)

Jie Yan Velocity distribution of ideal gas molecules

OutlineTutorial

Review of the thermal movement of ideal gas moleculesThe distribution of the velocities of the gas molecules

Maxwell distribution of the velocity-N molecule

I We have N molecules in the tank. The velocity distribution of

each molecule follows ρ(~vi ) = ( m2πkBT )3/2e

−mv2i

2kBT .

I The movement of the molecules are independent, so the jointprob of the system is ρ(~v1, ~v2, · · · , ~vN) =

ρ(~v1)ρ(~v2) · · · ρ(~vN) = ( m2πkBT )3N/2e

−m(v21 +v2

2 +···+v2N )

2kBT .

I The above distribution only applies to ideal gas (nointeractions among the gas molecules).

Jie Yan Velocity distribution of ideal gas molecules