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20 B Week I Chapter 9

• Maxwell-Boltzmann Distribution Translational Degrees of Freedom x, y and z • The Ideal Gas or Perfect Gas Model PV=NkT. @ Standard Condition: T=300 K and P=1 atm • Macroscopic Gas Pressure (P=Force/Unit Area): Boyle ’ s Law: PV=constant @ T=const.

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20 B Week I Chapter 9

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  1. • Maxwell-Boltzmann Distribution Translational Degrees of Freedom x, y and z • The Ideal Gas or Perfect Gas Model PV=NkT. @ Standard Condition: T=300 K and P=1 atm • Macroscopic Gas Pressure (P=Force/Unit Area): Boyle’s Law: PV=constant @ T=const. • Charles’ Law: V/T=const. @ P=const 20 B Week I Chapter 9

  2. The Maxwell- Boltzmann Speed (u=|v|) distribution? For an Ideal gas at a temperature T, the pressure (Force per molecule)x(number of molecules N colliding with a unit area) is: PV=NkT Where <KE>=<mu2/2>=(m/2) <u2> =(3/2) kT Which equi-partitions (1/2)kT in each translational Degrees of freedom: x, y and z! How is <u2> calculated?

  3. A molecular beam formed by heating material in effusion (Knudsen) cell at temperature T Fast slow Picks out atoms traveling at a certain speed u If the distribution of speeds are Plotted for different temperatures

  4. <u>=(8kBT/πm)1/2 average speed <u2>=(3kBT/m) average square speed urms= (3kBT/m)1/2 root mean square speed ump = (2kBT/m)1/2 most probable spped Fig. 9-14, p. 384

  5. f(u)du=fraction particles with speed between u and u + du Area under Curve is unity f(u) du u u + du speed

  6. Average speed=<u> = ( u1 + u2 + u3 +……….)/N <u> = ∑ui/N If N particles have the same speed u1 and u2 Then <u>= (u1 N1 + u2 N2 + …..)/N=∑ui Ni/N But N1/N  f(u1) as u0 <u>=∑uif(ui)u  <u>=∫uf(u)du as u 0 also <u2>=∫u2f(u)du We found f(u) by measuring the speed distribution or from:

  7. . Probability of having Kinetic Energy mu2/2 If the average energy (3/2)kT u2 ~ the number of possible velocities with speed u Fig. 9-15, p. 385

  8. Spheres of surface area ~ u2 the larger surface area allow more velocities uy u’ > u u’ u ux

  9. Therefore f(u) ~ probability or fraction of particles with speed between u and u + du <u>=∫uf(u)du=√8kT/πm <u2>=∫u2f(u)du= 3kT/m urms=√3kT/m root-mean-square speed ump = √2kT/m found by setting df(u)/du=0

  10. f(u)du=fraction particles with speed between u and u + du Area under Curve is unity f(u) du u u + du speed

  11. <u>=∫uf(u)du=√8kT/πm For example: on average how long does an atom/molecule to take to travel a distance L? <t>=L/<u> <u2>=∫u2f(u)du= 3kT/m urms=√3kT/m root-mean-square speed What is the average kinetic energy of an atom in a gas at temperature T <KE>=m<u2>/2=m(urms)2/2

  12. The Boltzmann factor ~ exp(-mu2/2kT) Is a very general concept and is really the probability of the particle having total energy E for an average energy ~kT For example: what is the Maxwell-Boltzmann distribution for atmospheric O2 molecule at altitude (h) on a day of temperature T ~u2 exp(-mgh/kT) exp(-mu2/2kT)=u2exp(-{mgh+mu2/2}/kT) where m is the mass of O2 and V(h) = mgh is the potential energy of O2 due to gravity. E= mgh+mu2/2 is the total energy of the molecule

  13. • Maxwell-Boltzmann Distribution Translational Degrees of Freedom x, y and z • The Ideal Gas or Perfect Gas Model PV=NkT. @ Standard Condition: T=300 K and P=1 atm • Macroscopic Gas Pressure (P=Force/Unit Area): Boyle’s Law: PV=constant @ T=const. • Charles’ Law: V/T=const. @ P=const 20 B Week I Chapter 9

  14. Measuring Pressure Hg Barometer mm

  15. Pressure= Force/unit area F=mg=Vg= hAg =mass/V Mass Density kg/m3 or kg/cm3 mm FHg Fair FHg = Fair Pair =Fair/A=FHg/A= hAg/A= hg: Pair= hg

  16. Experimental measurements of Pressure is possible using a Barometer! Lets do some experiments to find out how volume (V) changes with Pressure (P) for hot and cold gases.

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