Kinetic Model of Gases

Kinetic Model of Gases

Section 1.9, 1.11

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

Speed of gases

r.m.s. speed

mean speed

2/1223

22

21 ...

N

ssssc N

N

ssssc N...321

ccc 921.03

82/1

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.


Rates of diffusion and effusion of gases increase with increasing temperature.

For effusion the rate decreases with increasing molar mass.


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


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


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

Molecule Collisions

1/p 平均路徑隨壓力減少而增加

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

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


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


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)


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)


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)

Distribution of Translational Energy

Maxwell-Boltzmann Distribution Law

kinetic energymu

f = F ()

F () = 2 / (kBT)(3/2) 1/2 e-(/kBT)

Kinetic Model of Gases

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