Kinetic Model of Gases
Section 1.9, 1.11
Assumptions
z A gas consists of molecules in ceaseless random motion
z The size of the molecules is
negligible in the sense that their diameters are much smaller than the average distance traveled
between collisions
z The molecules do not interact, except during collisions
Pressure of a gas
V P nMc
3
=
2z M molecular weight; V volume
z c root-mean-square speed (rms speed)
2 / 2 1 2
3 2
2 2
1
...
+ + + +
= N
s s
s
c s
NSpeed of gases
z r.m.s. speed
z mean speed
2 / 2 1 2
3 2
2 2
1
...
+ + + +
= N
s s
s
c s
N
+ + + +
= N
s s
s
c s
1 2 3...
Nc c
c 0.921
3
8 1/2 ≈
=
π
Average speed of gas molecules
2
3
1 nMc pV
nRT pV
=
=
nMc
2= nRT
3 1
2 /
3 1
=
M c RT
•Effect of Molecular Weight
•Temperature Effect
Kinetic Energy of Molecules
z Ek = 3/2 RT
Ek kinetic energy; T temperature; R gas constant
z The average kinetic energy per molecule
T k
B2
3
k
=
ε
kB Boltzmann constant = R/6.02×1023Partial Pressure
zDalton’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;
χJ molar fraction of component J.
Diffusion & Effusion
z Diffusion
Molecule of different substances mingle with each other.
z Effusion
Escape of a gas through a small hole.
Diffusion & Effusion
z Rates of diffusion and effusion of gases increase with increasing temperature.
z For effusion the rate decreases with increasing molar mass.
Diffusion & Effusion
z 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.
Rate of effusion 1
∝ M
Diffusion & Effusion
z The rate at which hydrogen and carbon dioxide effuse under the same conditions of pressure and temperature are in the ratio
2
2
1/ 2 1/ 2
2 2
Rate of effusion of H 44.01
4.672
Rate of effusion of CO 2.016
CO H
M M
= = =
Diffusion & Effusion
z 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
Effusion as a separation technique
Use porous membranes to separate light gases from heavy ones
z average speed of gas molecules depends on the masses of their molecules
z heavy molecules in a mixture move slower on average than light ones
z gases made of light molecules diffuse through pores in membranes faster than heavy molecules
Differences from dialysis
z membrane is permeable, not semipermeable: all gas molecules in the mixture can pass through it
z size of molecules isn't usually important: pores in membrane are much larger than gas molecules
z ...molecular velocity (and so, molecular mass) is the basis for separation, not size
Examples
z separating helium from oxygen
z separating uranium isotopes as volatile UF6
Molecular Collisions
z C = 平均自由路徑 / 飛行時間 = λ / [1/ z] = λ z
λ = RΤ / [21/2NAσ p]
Z = [21/2NAσ c p] / RT
λ: 平均自由路徑;
Z : 碰狀頻率 collision frequency
σ : 碰狀截面積; σ = πd2 p : 壓力; T: 溫度;
NA: 亞佛加厥常數
Molecule Collisions
z λ ∝ 1/p 平均路徑隨壓力減少而增加
z λ ∝ 1/σ 分子之碰狀截面積越大, 平均自由路徑隨之減短
z z ∝ p 碰撞頻率隨壓力增加而增大
z z ∝ c ∝ 1/ Μ1/2 分子量越大的分子其碰撞頻率會低於分 子量小的分子
Maxwell distribution of speeds
The Maxwell distribution of speeds f = F (s) ∆S
F (s) = 4π[m / 2πkBT](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;
∆s : 速率的範圍 interval of speed KB : Boltzmann Constant
Maxwell distribution of speeds
f = F (s) ∆S
F (s) = 4π[m/ 2πkBT](3/2) s2 e-(ms2/2kBT) f ∝ ∆S
當設定的速率範圍增大時, 所含蓋的分子比例也隨之增加
Maxwell distribution of speeds
f = F (s) ∆S
F (s) = 4π[m / 2πkBT](3/2) s2 e-(ms2/2kBT)
z 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.
Maxwell distribution of speeds
f = F (s) ∆S
F (s) = 4π[m / 2πkBT](3/2) s2 e-(ms2/2kBT)
e -x (x =ms2/2kBT)
z 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.
z 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.
z 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.
Maxwell distribution of speeds
f = F (s) ∆S
F (s) = 4π[m / 2πkBT](3/2) s2 e-(ms2/2kBT) 4π[m / 2πkBT](3/2)
使分子比例的呈現在0與1之間
Maxwell distribution of speeds
Distribution of Translational Energy
z Maxwell-Boltzmann Distribution Law
z ε kinetic energy (=1/2 mu2) f = F (ε) ∆ε
F (ε) = 2π / (πkBT)(3/2) ε1/2 e-(ε/kBT)
