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KINETIC THEORY

KINETIC THEORY. Consider an ideal gas with molecules that are a point mass. KINETIC THEORY. Consider an ideal gas with molecules that are a point mass. The gas is confined in a cubic volume. KINETIC THEORY.

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KINETIC THEORY

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  1. KINETIC THEORY • Consider an ideal gas with molecules that are a point mass.

  2. KINETIC THEORY • Consider an ideal gas with molecules that are a point mass. The gas is confined in a cubic volume

  3. KINETIC THEORY • Consider an ideal gas with molecules that are a point mass. The gas is confined in a cubic volume and the density of molecules is rare so that there are very few molecular-molecular collisions

  4. KINETIC THEORY • Consider an ideal gas with molecules that are a point mass. The gas is confined in a cubic volume and the density of molecules is rare so that there are very few molecular-molecular collisions and there are no intermolecular forces.

  5. KINETIC THEORY • Consider an ideal gas with molecules that are a point mass. The gas is confined in a cubic volume and the density of molecules is rare so that there are very few molecular-molecular collisions and there are no intermolecular forces. When the molecules strike the inside of the container there are elastic collisions.

  6. KINETIC THEORY • The microscopic movement of molecules is used to describe macroscopic parameters.

  7. KINETIC THEORY • The microscopic movement of molecules is used to describe macroscopic parameters. • One uses variables such as N (number of particles), v (velocity) to produce pressure.

  8. KINETIC THEORY • The microscopic movement of molecules is used to describe macroscopic parameters. • One uses variables such as N (number of particles), v (velocity) to produce pressure. • See the java applet which simulates this • http://www.phy.ntnu.edu.tw/java/idealGas/idealGas.html

  9. KINETIC THEORY • The microscopic movement of molecules is used to describe macroscopic parameters. • One uses variables such as N (number of particles), v (velocity) to produce pressure. • See the java applet which simulates this • http://www.phy.ntnu.edu.tw/java/idealGas/idealGas.html • http://www.physics.org/Results/search.asp?q=Tell+me+about+kinetic+theory&uu=0

  10. KINETIC THEORY • When the molecules are in thermal equilibrium, then the average velocity in each direction is the same. • < vx > = < vy > = < vz >

  11. KINETIC THEORY • When the molecules are in thermal equilibrium, then the average velocity in each direction is the same. • < vx > = < vy > = < vz > • The average velocity of all the molecules is < v > = 0 (since they are confined)

  12. KINETIC THEORY • When the molecules are in thermal equilibrium, then the average velocity in each direction is the same. • < vx > = < vy > = < vz > • The average velocity of all the molecules is < v > = 0 (since they are confined) • Note the average speed is < v > ≠ 0 nor is the < v2 > ≠ 0 vRMS = √ (< v2 >

  13. KINETIC THEORY • When the molecules are in thermal equilibrium, then the average velocity in each direction is the same. • < vx > = < vy > = < vz > • The average velocity of all the molecules is < v > = 0 (since they are confined) • Note the average speed is < v > ≠ 0 nor is the < v2 > ≠ 0 vRMS = √ (< v2 > and < v2 > ≠ (< v >)2

  14. KINETIC THEORY • Let the internal energy be • U = N<K> where <K> is the average energy of one molecule.

  15. KINETIC THEORY • Let the internal energy be • U = N<K> where <K> is the average energy of one molecule. So U = N (½ m < v2 > )

  16. KINETIC THEORY • Let the internal energy be • U = N<K> where <K> is the average energy of one molecule. So U = N (½ m < v2 > ) but < v2 > = < vx2 + vy2 + vz2 >

  17. KINETIC THEORY • Let the internal energy be • U = N<K> where <K> is the average energy of one molecule. So U = N (½ m < v2 > ) but < v2 > = < vx2 + vy2 + vz2 > and because U = U(T)

  18. KINETIC THEORY • Let the internal energy be • U = N<K> where <K> is the average energy of one molecule. So U = N (½ m < v2 > ) but < v2 > = < vx2 + vy2 + vz2 > and because U = U(T) then < v2 > = < vx2 > + < vy2 > + < vz2 >

  19. KINETIC THEORY • Since the gas is not moving then < vx2 > = < vy2 > = < vz2 >

  20. KINETIC THEORY • Since the gas is not moving then < vx2 > = < vy2 > = < vz2 > therefore < v2 > = 3 < vx2 >

  21. KINETIC THEORY • Since the gas is not moving then < vx2 > = < vy2 > = < vz2 > therefore < v2 > = 3 < vx2 > and < vx2 > = < v2 > /3

  22. KINETIC THEORY • Since the gas is not moving then < vx2 > = < vy2 > = < vz2 > therefore < v2 > = 3 < vx2 > and < vx2 > = < v2 > /3 = 2/3 (U/(mN))

  23. KINETIC THEORY • Since the gas is not moving then < vx2 > = < vy2 > = < vz2 > therefore < v2 > = 3 < vx2 > and < vx2 > = < v2 > /3 = 2/3 (U/(mN)) Let us now consider all the molecules in an element of cylindrical volume in the direction x dV = A vx dt

  24. KINETIC THEORY • In that volume let us consider collisions with those molecules having a velocity vx .

  25. KINETIC THEORY • In that volume let us consider collisions with those molecules having a velocity vx . • The collisions will produce a change in momentum Δ PMOL = mvxf – mvxi = -2mvx .

  26. KINETIC THEORY • In that volume let us consider collisions with those molecules having a velocity vx . • The collisions will produce a change in momentum Δ PMOL = mvxf – mvxi = -2mvx . • The wall receives the reaction momentum • Δ Pwall(x) = 2mvx

  27. KINETIC THEORY • In that volume let us consider collisions with those molecules having a velocity vx . • The collisions will produce a change in momentum Δ PMOL = mvxf – mvxi = -2mvx . • The wall receives the reaction momentum • Δ Pwall(x) = 2mvx • Since we are considering all the molecules with an x component velocity; vx is <vx> .

  28. KINETIC THEORY • In order to obtain the total momentum transferred to the wall in a time dt, one must know the number of collisions

  29. KINETIC THEORY • In order to obtain the total momentum transferred to the wall in a time dt, one must know the number of collisions NCOLL = (N /V) dV = ½ (N/V) A vx dt where ½ N/V is the number density of molecules, in the element of volume dV.

  30. KINETIC THEORY • In order to obtain the total momentum transferred to the wall in a time dt, one must know the number of collisions NCOLL = (N /V) dV = ½ (N/V) A vx dt where ½ N/V is the number density of molecules, in the element of volume dV. The total x momentum transferred is dPx = NCOLL ΔPwall(x)

  31. KINETIC THEORY • dPx = ½ (N/V) (A vx dt) (2mvx )

  32. KINETIC THEORY • dPx = ½ (N/V) (A vx dt) (2mvx ) thus in a time dt, the change in momentum is dPx/dt = Fx = (N/V) m(vx)2 A

  33. KINETIC THEORY • dPx = ½ (N/V) (A vx dt) (2mvx ) thus in a time dt, the change in momentum is dPx/dt = Fx = (N/V) m(vx)2 A and the pressure is P = Fx/ A = (N/V) m(vx)2

  34. KINETIC THEORY • dPx = ½ (N/V) (A vx dt) (2mvx ) thus in a time dt, the change in momentum is dPx/dt = Fx = (N/V) m(vx)2 A and the pressure is P = Fx/ A = (N/V) m(vx)2 Since (vx)2 is an average over all molecules P = (N/V) m<vx2>

  35. KINETIC THEORY • Since < vx2 > = 2/3 (U/(mN))

  36. KINETIC THEORY • Since < vx2 > = 2/3 (U/(mN)) then P = (N/V) m 2/3 (U/(mN) = 2/3 (U/V)

  37. KINETIC THEORY • Since < vx2 > = 2/3 (U/(mN)) then P = (N/V) m 2/3 (U/(mN) = 2/3 (U/V) or PV = 2/3 U

  38. KINETIC THEORY • Since < vx2 > = 2/3 (U/(mN)) then P = (N/V) m 2/3 (U/(mN) = 2/3 (U/V) or PV = 2/3 U Using the Ideal Gas Law PV = NkT

  39. KINETIC THEORY • Since < vx2 > = 2/3 (U/(mN)) then P = (N/V) m 2/3 (U/(mN) = 2/3 (U/V) or PV = 2/3 U Using the Ideal Gas Law PV = NkT and U =N<K>

  40. KINETIC THEORY • Since < vx2 > = 2/3 (U/(mN)) then P = (N/V) m 2/3 (U/(mN) = 2/3 (U/V) or PV = 2/3 U Using the Ideal Gas Law PV = NkT and U =N<K> then kT = 2/3 U/N

  41. KINETIC THEORY • Since < vx2 > = 2/3 (U/(mN)) then P = (N/V) m 2/3 (U/(mN) = 2/3 (U/V) or PV = 2/3 U Using the Ideal Gas Law PV = NkT and U =N<K> then kT = 2/3 U/N = 2/3 <K> or 3/2 kT = U/N = <K>

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