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PROBABILITY DISTRIBUTIONS. FINITE CONTINUOUS ∑ N g = N N v Δ v = N. PROBABILITY DISTRIBUTIONS. FINITE CONTINUOUS ∑ N g = N N v Δ v = N P g = N g /N ∫N v dv = N P v = N v /N. PROBABILITY DISTRIBUTIONS. FINITE CONTINUOUS - PowerPoint PPT Presentation

### Transcript of PROBABILITY DISTRIBUTIONS

• PROBABILITY DISTRIBUTIONS FINITE CONTINUOUS Ng = N Nv v = N

• PROBABILITY DISTRIBUTIONS FINITE CONTINUOUS Ng = N Nv v = N Pg = Ng /N Nv dv = N Pv = Nv /N

• PROBABILITY DISTRIBUTIONS FINITE CONTINUOUS Ng = N Nv v = N Pg = Ng /N Nv dv = N Normalized Pv = Nv /N Pg = 1 Pv dv = 1

• PROBABILITY DISTRIBUTIONS FINITE CONTINUOUS Ng = N Nv v = N Pg = Ng /N Nv dv = N Normalized Pv = Nv /N Pg = 1 Pv dv = 1 < g> = g Pg < v > = vPv dv

• PROBABILITY DISTRIBUTIONS FINITE CONTINUOUS Ng = N Nv v = N Pg = Ng /N Nv dv = N Normalized Pv = Nv /N Pg = 1 Pv dv = 1 < g> = g Pg < v > = vPv dv = g2 Pg < v2> = v2 Pv dv

• Velocity Distribution of GasesMaxwell Velocity Distribution for gases is N(v) dv = N 4v2 (m/2kT)3/2 e mv^2/2kT dv where N is the number of molecules of mass m and temperature T.

• Velocity Distribution of GasesMaxwell Velocity Distribution for gases is N(v) dv = N 4v2 (m/2kT)3/2 e mv^2/2kT dv where N is the number of molecules of mass m and temperature T. If one divides by N and changes the differential element dv to d3v = dvx dvy dvz ,

• Velocity Distribution of GasesMaxwell Velocity Distribution for gases is N(v) dv = N 4v2 (m/2kT)3/2 e mv^2/2kT dv where N is the number of molecules of mass m and temperature T. If one divides by N and changes the differential element dv to d3v = dvx dvy dvz , then the normalized probability function F(v) is: F(v) = (m/2kT)3/2 e mv^2/2kT

• Velocity Distribution of GasesThis velocity probability distribution has all the properties given before: F(v) d3v = 1

• Velocity Distribution of GasesThis velocity probability distribution has all the properties given before: F(v) d3v = 1 and the mean velocity and the mean of the square velocity are: = v F(v) d3v = v2 F(v) d3v

• Velocity Distribution of GasesThis velocity probability distribution has all the properties given before: F(v) d3v = 1 and the mean velocity and the mean of the square velocity are: = v F(v) d3v = v2 F(v) d3v (remember d3v means one must do a triple integration over dvx dvy dvz )

• Velocity Distribution of GasesThe results of this are: = (8kT/(m)) = 1.59 kT/m

• Velocity Distribution of GasesThe results of this are: = (8kT/(m)) = 1.59 kT/m = (3kT/m) = 1.73 kT/m

• Velocity Distribution of GasesThe results of this are: = (8kT/(m)) = 1.59 kT/m = (3kT/m) = 1.73 kT/mIf one sets the derivative of the probability function to zero (as was done for the Planck Distribution) one obtains the most probable value of v

• Velocity Distribution of GasesThe results of this are: = (8kT/(m)) = 1.59 kT/m = (3kT/m) = 1.73 kT/mIf one sets the derivative of the probability function to zero (as was done for the Planck Distribution) one obtains the most probable value of v vmost prob = (2kT/m) = 1.41kT/m

• Maxwell-Boltzmann DistributionMolecules with more complex shape have internal molecular energy. Boltzmann realized this and changed Maxwells Distribution to include all the internal energy. FM (v) FMB (v) FMB (v) = (1/Z) e E/kT where Z = the normalization factor

• Maxwell-Boltzmann DistributionMolecules with more complex shape have internal molecular energy.

• Maxwell-Boltzmann DistributionMolecules with more complex shape have internal molecular energy. Boltzmann realized this and changed Maxwells Distribution to include all the internal energy. FM (v) FMB (v)

• Maxwell-Boltzmann DistributionMolecules with more complex shape have internal molecular energy. Boltzmann realized this and changed Maxwells Distribution to include all the internal energy. FM (v) FMB (v) FMB (v) = (1/Z) e E/kT where Z = the normalization factor

• MOLECULAR INTERNAL ENERGYDiatomic, Triatomic and more complex molecules can rotate and vibrate about their symmetrical axes.

• MOLECULAR INTERNAL ENERGYDiatomic, Triatomic and more complex molecules can rotate and vibrate about their symmetrical axes. EINT = < E > = ETRANS + EROT + EVIBR

• MOLECULAR INTERNAL ENERGYDiatomic, Triatomic and more complex molecules can rotate and vibrate about their symmetrical axes. EINT = < E > = ETRANS + EROT + EVIBR ETRANS = < ETRANS > = m

• MOLECULAR INTERNAL ENERGYDiatomic, Triatomic and more complex molecules can rotate and vibrate about their symmetrical axes. EINT = < E > = ETRANS + EROT + EVIBR ETRANS = < ETRANS > = m EROT = Ix x2 + Iy y2 + Iz z2

• MOLECULAR INTERNAL ENERGYDiatomic, Triatomic and more complex molecules can rotate and vibrate about their symmetrical axes. EINT = < E > = ETRANS + EROT + EVIBR ETRANS = < ETRANS > = m EROT = Ix x2 + Iy y2 + Iz z2 Diatomic (2 axes) Triatomic (3 axes)

• MOLECULAR INTERNAL ENERGYDiatomic, Triatomic and more complex molecules can rotate and vibrate about their symmetrical axes. EINT = < E > = ETRANS + EROT + EVIBR ETRANS = < ETRANS > = m EROT = Ix x2 + Iy y2 + Iz z2 Diatomic (2 axes) Triatomic (3 axes) EVIBR = - k x2 VIBR (for each axis)

• INTERNAL MOLECULAR ENERGYFor a diatomic molecule then = 5/2 kT

• INTERNAL MOLECULAR ENERGYFor a diatomic molecule then = 5/2 kTOne of the basic principles used in the Kinetic Theory of Gases is that each degree of freedom has an average energyof kT.

• INTERNAL MOLECULAR ENERGYFor a diatomic molecule then = 5/2 kTOne of the basic principles used in the Kinetic Theory of Gases is that each degree of freedom has an average energyof kT. Or = (s/2) kT

• INTERNAL MOLECULAR ENERGYFor a diatomic molecule then = 5/2 kTOne of the basic principles used in the Kinetic Theory of Gases is that each degree of freedom has an average energyof kT. Or = (s/2) kT where s = the number of degrees of freedom

• INTERNAL MOLECULAR ENERGYFor a diatomic molecule then = 5/2 kTOne of the basic principles used in the Kinetic Theory of Gases is that each degree of freedom has an average energyof kT. Or = (s/2) kT where s = the number of degrees of freedomThis is called the EQUIPARTION THEOREM

• INTERNAL MOLECULAR ENERGYFor dilute gases which still obey the ideal gas law, the internal energy is:

• INTERNAL MOLECULAR ENERGYFor dilute gases which still obey the ideal gas law, the internal energy is: U = N = (s/2) NkT

• INTERNAL MOLECULAR ENERGYFor dilute gases which still obey the ideal gas law, the internal energy is: U = N = (s/2) NkT

Real gases undergo collisions and hence can transport matter called diffusion.

• INTERNAL MOLECULAR ENERGYFor dilute gases which still obey the ideal gas law, the internal energy is: U = N = (s/2) NkT

Real gases undergo collisions and hence can transport matter called diffusion. The average distance a molecule moves between collisions is

• COLLISIONS OF MOLECULESLet D be the diameter of a molecule.

• COLLISIONS OF MOLECULESLet D be the diameter of a molecule. The collision cross section is merely the cross-sectional area = D2 .

• COLLISIONS OF MOLECULESLet D be the diameter of a molecule. The collision cross section is merely the cross-sectional area = D2 . If there is a collision then the molecule traveles a distance = vt.

• COLLISIONS OF MOLECULESLet D be the diameter of a molecule. The collision cross section is merely the cross-sectional area = D2 . If there is a collision then the molecule traveles a distance = vt. If one averages this = vRMS where = mean collision time.

• COLLISIONS OF MOLECULESLet D be the diameter of a molecule. The collision cross section is merely the cross-sectional area = D2 . If there is a collision then the molecule traveles a distance = vt. If one averages this = vRMS where = mean collision time. During this time there are N collisions in a volume V.

• MOLECULAR COLLISIONSThe molecule sweeps out a volume which is V = AvRMS =

• MOLECULAR COLLISIONSThe molecule sweeps out a volume which is V = AvRMS = vRMS

• MOLECULAR COLLISIONSThe molecule sweeps out a volume which is V = AvRMS = vRMS Since there are number / volume (ndens ) molecules undergoing a collision then the average number of collisions per unit time is = 1/N

• MOLECULAR COLLISIONSThe molecule sweeps out a volume which is V = AvRMS = vRMS Since there are number / volume (ndens ) molecules undergoing a collision then the average number of collisions per unit time is = 1/N = 1/(nV/)

• MOLECULAR COLLISIONSThe molecule sweeps out a volume which is V = AvRMS = vRMS Since there are number / volume (ndens ) molecules undergoing a collision then the average number of collisions per unit time is = 1/N = 1/(nV/) = 1/n vRMS .

• MOLECULAR COLLISIONSThe molecule sweeps ou