Kinetic Model of Gases Section 1.9, 1.11
Jan 03, 2016
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
Pressure of a gas
M molecular weight; V volume c root-mean-square speed (rms speed)
V
nMcP
3
2
2/1223
22
21 ...
N
ssssc N
Average speed of gas molecules
2
3
1nMcpV
nRTpV
2/13
M
RTc
nRTnMc 2
3
1
•Effect of Molecular Weight
•Temperature Effect
Kinetic Energy of Molecules
Ek = 3/2 RT
Ek kinetic energy; T temperature; R gas constant
The average kinetic energy per molecule
TkB2
3k kB Boltzmann constant = R/6.021023
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; J molar fraction of component J.
Diffusion & Effusion
DiffusionMolecule of different substances mingle with each other.
EffusionEscape of a gas through a small hole.
Diffusion & Effusion
Rates of diffusion and effusion of gases increase with increasing temperature.
For effusion the rate decreases with increasing molar mass.
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.
1Rate of effusion
M
Diffusion & Effusion
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 44.014.672
Rate of effusion of CO 2.016CO
H
M
M
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.htmlhttp://www.uic.com.au/uicchem.htmhttp://www.uilondon.org/index.htm
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
Molecular Collisions
C = 平均自由路徑 / 飛行時間 = / [1/ z] = z
RNAp
ZNAc
pRT
: 平均自由路徑 ;
Z : 碰狀頻率 collision frequency
: 碰狀截面積 ; = d2
p : 壓力 ; T: 溫度 ;
NA: 亞佛加厥常數
Maxwell distribution of speeds
The Maxwell distribution of speeds
f = F (s) S
F (s) = 4[m / 2kBT](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
當設定的速率範圍增大時 , 所含蓋的分子比例也隨之增加
f = F (s) S
F (s) = 4[m/ 2kBT](3/2) s2 e-(ms2/2kBT)
f S
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) S
F (s) = 4[m / 2kBT](3/2) s2 e-(ms2/2kBT)
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) S
F (s) = 4[m / 2kBT](3/2) s2 e-(ms2/2kBT)
Maxwell distribution of speeds
4[m / 2kBT](3/2)
使分子比例的呈現在 0 與 1 之間
f = F (s) S
F (s) = 4[m / 2kBT](3/2) s2 e-(ms2/2kBT)