Velocity distribution of ideal gas molecules

Transcription

1 August 25, 2006

2 Contents Review of thermal movement of ideal gas molecules.

3 Contents Review of thermal movement of ideal gas molecules. Distribution of the velocity of a molecule in ideal gas.

4 Contents Review of thermal movement of ideal gas molecules. Distribution of the velocity of a molecule in ideal gas. Boltzmann distribution.

5 Time slots starts from week 3 (28AUG-1SEP).

6 Time slots starts from week 3 (28AUG-1SEP). A class tutorial every week, on Friday 9 am - 10 am.

7 Time slots starts from week 3 (28AUG-1SEP). A class tutorial every week, on Friday 9 am - 10 am. A small group tutorial every even week (4,6,8,10), on Mon ( pm) or on Wed ( pm). The small group tutorial classroom is S

8 Time slots starts from week 3 (28AUG-1SEP). A class tutorial every week, on Friday 9 am - 10 am. A small group tutorial every even week (4,6,8,10), on Mon ( pm) or on Wed ( pm). The small group tutorial classroom is S Group A(30%): week 4, 8,12, Mon; group B(70%): week 6,10, 12, Wed.

9 Review: random walk leads to Gaussian distribution

10 Temperature is a measure of the molecule movements Ideal gas law: PV = Nk B T. Pressure is determined by gas molecule move: PV = m < v 2 x > N = m<v 2 >N 3. We thus have: The average kinetic energy ε k = 1 2 m < v 2 > of a molecule in an ideal gas is 3 2 k BT. Each degree of freedom contribute 1 2 k BT.

11 Speed of gas molecules on earth surface at room temperature At the earth surface, P = 10 5 Pa, the mole density is c = 1 24 M = 1 mol 24 L (1mol = ; 1L = 10 3 m 3 ). Let N = 1mol, please compute Nk B T, PV, and Nk BT PV. Show that the ideal gas is a reasonable approximation to the gas on earth. Gas on earth mostly consists nitrogen. One nitrogen molecule has a mass m kg. At room temperature T = 300K, please compute that its kinetic energy ε k, and show that the average of the magnitude of its speed < v 2 > 500 m s.

12 Why gas molecules does not fall onto the ground in earth? The change in the potential: U(z) = mgz, assuming it is a nitrogen molecule. So Z max ε k mg. Please compute Z max. Please show that in a room, the gravitation does not affect the gas molecule distribution. At what mass the gravitation can affect the distribution of the molecules in a room on earth? At what mass the gravity can trap a molecule within 1micron = 10 6 m?

13 Interaction strength between molecules General rule of estimating the interaction strength: fd k B T, where f is the force, and d is the characteristic interaction distance. So f k BT d pn for d nm. Many proteins can bind to and unbind from DNA. The binding and unbinding equilibrium is driven by electrostatic interactions and thermal fluctuation. In usual physiological salt 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 around several Pico Newtons.

14 Wild guess of the velocity distribution of the gas molecules Range of velocity: < v x <. Symmetry in direction: ρ(v x ) = ρ( v x ), so < v x >= 0. Since 1 2 m < v 2 x >= 1 2 k BT, the higher the T, the bigger the variance σ 2 v x =< v 2 x >. Orientational symmetry: ρ(v x ) = ρ(v y ) = ρ(v z ). At small T, ρ(v x ) quickly goes to zero for v x > 0. What function is a good candidate?

15 Random walk in the velocity space We know a random walk in coordinate space leads to a Gaussian distribution. Gas molecules are changing their their accelerations randomly. This can be thought as a random walk in the velocity space. Not strictly, we can guess the velocity distribution is a Gaussian. We already knew < v 2 x >= k BT m,and < v x >= 0. Please show that: ρ(v x ) = m mv 2 2πk B T e x 2k B T.

16 More on ρ(v x )

17 Maxwell distribution of the velocity of individual molecules We have learnt that: ρ(v x ) = m mv 2 2πk B T e x 2k B T. But molecules are moving in 3-d. Reasonable guess: moving in each direction is independent. So ρ( v) = ρ(v x, v y, v z ) = ρ(v x ) ρ(v y ) ρ(v z ). Please show ρ( v) = ( v 2 = vx 2 + vy 2 + vz 2. Find < v >, u = < v 2 >. m 2πk B T )3/2 e mv 2 2k B T, where

18 More on ρ( v)

19 Maxwell distribution of the scalar velocity of individual molecules What is the distribution of the magnitude of the velocity ρ(u), where u = v? 2 2k B T. m It can be shown that ρ(u) = 4π( 2πk B T )3/2 u 2 e mu Prove ρ(u) is normalized. Show that the most probable u max = (Hint: d du ρ(u) = 0). 2k B T m

20 More on ρ(u)

21 Maxwell distribution of the velocity-n molecule We have N molecules in the tank. The velocity distribution of each molecule follows ρ( v i ) = ( mv i 2 m 2πk B T )3/2 2k e B T. The movement of the molecules are independent, so the joint prob of the system is ρ( v 1, v 2,, v N ) = ρ( v 1 )ρ( v 2 ) ρ( v N ) = ( m(v 1 2 m +v v2 N ) 2πk B T )3N/2 2k e B T. The above distribution only applies to ideal gas (no interactions among the gas molecules).