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Understanding Maxwell-Boltzmann Distribution and Ideal Gas Laws

This comprehensive overview covers the Maxwell-Boltzmann distribution and its application to translational degrees of freedom in ideal gases. We discuss key principles, including the ideal gas law (PV = NkT), Boyle's law, and Charles's law under standard conditions (T=300K, P=1 atm). The document explores macroscopic gas pressure, average kinetic energy, and how to measure gas speed distributions. Additionally, it investigates how various temperatures affect molecular speeds and positions in atmospheric pressure contexts.

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Understanding Maxwell-Boltzmann Distribution and Ideal Gas Laws

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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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