22-2 A molecular view of pressure(压强) How to relate pressure to Fig 22-2 microscopic quantities P We will take the ideal gas as our system Consider N molecules of an ideal gas X confined within a cubical box of edge length L, as in Fig 22-2
22-2 A molecular view of pressure(压强) We will take the ideal gas as our system. Consider N molecules of an ideal gas confined within a cubical box of edge length L, as in Fig 22-2. L L y x z m L A1 → v A2 How to relate pressure to Fig 22-2 microscopic quantities? P ~ ρ, v ...?
The average impulsive force exerted by the molecule on a is 2 2L (22-4 The total force on a, by all the gas molecules is the sum of the quantities mvx /L for all the molecules Then the pressure on Alis (Fx1+F32+…) (22-5) 1mv,+m1v,2+ V;+1 L
The average impulsive force exerted by the molecule on is A1 L mv v L mv F x x x x 2 2 2 = = (22-4) The total force on by all the gas molecules is the sum of the quantities for all the molecules. Then the pressure on A1 is (22-5) A1 mvx / L 2 ( ) 1 ( ...) 1 2 2 2 3 1 2 2 2 1 2 2 1 2 = + + + + = = + + x x x x x x v v L m L m v m v L F F L P
P=a 2 If N is the total number of molecules in container the total mass is nm the density is p=Nm/E P- Ly 22-6) N The quantity in parenthesis is average value of for all the molecules in the container P=p( av 22-7)
If N is the total number of molecules in container, the total mass is Nm. the density is . (22-6) The quantity in parenthesis is average value of for all the molecules in the container. (22-7) 3 = Nm/ L 2 x v x av P (v ) 2 = [ ] 2 2 3 N v N v L mN P i xi i xi = = ( ) 2 2 2 3 1 = + + x x v v L m P v v N i x av xi ( ) ( )/ 2 2 =
P=p(vx2)(227) For any molecules v2=v +v2+vand 1 so Eq 22-7)becomes 2 o(v av (22-8) 3 1. the result is true even when we consider collisions between molecules 2. the result is correct even with consideration of the collisions between molecules and other walls in the box 3. The result is correct for boxes with any kinds of shape
For any molecules, , and so Eq(22-7) becomes (22-8) 1.The result is true even when we consider collisions between molecules. 2. The result is correct even with consideration of the collisions between molecules and other walls in the box. 3. The result is correct for boxes with any kinds of shape. 2 2 2 2 x y z v = v + v + v x a v y a v z a v a v v v v (v ) 3 1 ( ) ( ) ( ) 2 2 2 2 = = = P v av ( ) 3 1 2 = x av P (v ) 2 = (22-7)
4.The“oot-mean- square”(均方根) speed of the molecules: 3P Vp (22-9) In eq(22-8, 9), we relate a macroscopic quantity( the pressure p)to an average value of a microscopic quantity, that is to (va,or v
(22-9) In Eq(22-8,9), we relate a macroscopic quantity ( the pressure P) to an average value of a microscopic quantity, that is to or . P v v rms a v 3 ( ) 2 = = rms v av (v ) 2 4. The “root-mean-square” (均方根)speed of the molecules：