Kinetic Model of Gases

1. Kinetic Model of Gases Section 1.9, 1.11

2. Assumptions • A gas consists of molecules in ceaseless random motion • The size of the molecules is negligible in the sense that their diameters are much smaller than the average distance traveled between collisions • The molecules do not interact, except during collisions

3. Pressure of a gas • M molecular weight; V volume • c root-mean-square speed (rms speed)

4. Speed of gases • r.m.s. speed • mean speed

5. Average speed of gas molecules • Effect of Molecular Weight • Temperature Effect

6. Kinetic Energy of Molecules • Ek = 3/2 RT Ekkinetic energy; T temperature; R gas constant • The average kinetic energy per molecule kB Boltzmann constant = R/6.021023

7. Partial Pressure • Dalton’s Law The total pressure observed for a mixture of gases is equal to the sum of the pressures that each individual component gas would exert Ptotal = P1+P2+P3+…+PJ PJ = xJ Ptotal Ptotal total pressure; PJ partial pressure of component J;cJ molar fraction of component J.

8. Diffusion & Effusion • Diffusion Molecule of different substances mingle with each other. • Effusion Escape of a gas through a small hole.

9. Diffusion & Effusion • Rates of diffusion and effusion of gases increase with increasing temperature. • For effusion the rate decreases with increasing molar mass.

10. Diffusion & Effusion • Graham’s Law At a given pressure and temperature, the rate of effusion of a gas is inversely proportional to the square root of its molar mass.

11. Diffusion & Effusion • The rate at which hydrogen and carbon dioxide effuse under the same conditions of pressure and temperature are in the ratio

12. Diffusion & Effusion • Separation of uranium-235 from uranium-238, in the form of volatile solids UF6 http://www.columbia.edu/itc/chemistry/chem-c1403/text_chapters/nukes.html http://www.uic.com.au/uicchem.htm http://www.uilondon.org/index.htm

13. Effusion as a separation technique Use porous membranes to separate light gases from heavy ones • average speed of gas molecules depends on the masses of their molecules • heavy molecules in a mixture move slower on average than light ones • gases made of light molecules diffuse through pores in membranes faster than heavy molecules Differences from dialysis • membrane is permeable, not semipermeable: all gas molecules in the mixture can pass through it • size of molecules isn't usually important: pores in membrane are much larger than gas molecules • ...molecular velocity (and so, molecular mass) is the basis for separation, not size Examples • separating helium from oxygen • separating uranium isotopes as volatile UF6

14. Molecular Collisions • C= 平均自由路徑 / 飛行時間 = l / [1/ z] =l z • : 平均自由路徑;  Z : 碰狀頻率 collision frequency • s : 碰狀截面積;  s = pd2 • p : 壓力; T: 溫度; • NA: 亞佛加厥常數 l=RT / [21/2NAsp] Z=[21/2NAsc p] / RT

15. Molecule Collisions • l 1/p平均路徑隨壓力減少而增加 • l 1/s 分子之碰狀截面積越大, 平均自由路徑隨之減短 • zp 碰撞頻率隨壓力增加而增大 • z c  分子量越大的分子其碰撞頻率會低於分 子量小的分子

16. Maxwell distribution of speeds The Maxwell distribution of speeds f = F (s) DS F (s) = 4p[m / 2pkBT](3/2) s2 e-(ms2/2kBT) f : 運動速率在某個範圍內的分子之比例 s: 分子運動速率 speed; • Ds : 速率的範圍 interval of speed • KB: Boltzmann Constant

17. Maxwell distribution of speeds f = F (s) DS F (s) = 4p[m/ 2pkBT](3/2) s2 e-(ms2/2kBT) fDS 當設定的速率範圍增大時, 所含蓋的分子比例也隨之增加

18. Maxwell distribution of speeds • s2 當速率值趨向極小質 s2 趨於0.這表示具有極低 運動速率分子所佔的比力是非常小的. f = F (s) DS F (s) = 4p[m / 2pkBT](3/2)s2 e-(ms2/2kBT)

19. Maxwell distribution of speeds e -x(x =ms2/2kBT) • 這是一個"衰減"函數.當速率 (s) 非常大的時候指數值就相當小.也就是說具有極高運動速率的氣體分子比例是非常小的. • 分子量(M)越大, 指數值就越小. 大分子具有高運動速率的比例較小. • 溫度(T)升高, 指數值越大. 溫度越高具有較快運動速率的分子比例也越大. f = F (s)DS F (s) = 4p[m / 2pkBT](3/2) s2e-(ms2/2kBT)

20. Maxwell distribution of speeds 4p[m / 2pkBT](3/2) 使分子比例的呈現在0與1之間 f = F (s)DS F (s) = 4p[m / 2pkBT](3/2) s2 e-(ms2/2kBT)

21. Maxwell distribution of speeds

22. Distribution of Translational Energy • Maxwell-Boltzmann Distribution Law • ekinetic energy (=1/2 mu2) f = F (e) De F (e) = 2p / (pkBT)(3/2)e1/2 e-(e/kBT)