Hoffman D.M., Singh B., Thomas J.H. (Eds). Handbook of Vacuum Science and Technology

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14 Chapter 1.3: Molecular Processes and Kinetic Theory

1.3.3

KINETIC THEORY DERIVATION OF THE GAS LAWS

1.3.3.1

Boyle's Law

Let S be the gas density, thus

mn = S

where m is the molecular mass and n is molecular volume density. Hence, the

pressure is given by

P = -8 = ~mnv^ (7)

3 3

Thus

PV=^V (8)

mv^

where m is the total mass of the gas. This relationship shows that at constant tem-

perature, the pressure varies directly with density or conversely with volume,

which is Boyle's law.

1.3.3.2

Charles and the Ideal Gas Law

(9)

(10)

That is, the pressure is equal to two-thirds of the total kinetic energy per unit

volume. From the expression for v^^, we have

= — (11)

Thus

P = nkr (12)

The average energy

and therefore

of molecule is given by

P =

~~~2~

2nE^^

1.3.4

Pressure

where n is the number of molecules per unit volume. Substituting for n = N/V

we get

P = ^ (13)

— = Nk (14)

which is the ideal gas law.

Consider equal volumes of any two different gases at same values of P and T.

Since P and V are same for each gas, and ^ are the same at the same tempera-

ture,

it follows that n must be the same for each gas. Thus we have "equal vol-

umes of all gases at any given values of temperature and pressure contain equal

number of molecules." This is, of course, Avogadro's hypothesis, which was

stated around

1811.

Introducing Avogadro's number,

N/^

= 6.023 X 10^^/mole as

Np,

= M/m and n = NA/V^ (15)

where m is the molar mass and V^ is the molar volume.

PV^ = AT^Rr (16)

PV^ = RT (17)

where R is the universal gas constant, and T

temperature in °Kelvin.

1.3.4

PRESSURE

The following is a simplified derivation of the pressure exerted by a gas on the

walls of its container as a result of molecular impacts. Since the molecular im-

pacts are perfectly elastic, energy and momentum are conserved and each mole-

cule is thrown back from the wall with the same speed as its incident speed.

Consider a molecule of mass m that approaches the enclosure wall with a ve-

locity

and rebounds with the same velocity, thus

Change in momentum = 2mv

Chapter 1.3: Molecular Processes and Kinetic Theory

If n molecules strike a unit area of the wall in unit time, with average velocity v,

then

Total impulse exerted on unit area of wall in unit time = 2mnv

Pressure = rate of change of momentum per unit area

Thus

P = 2mnv

(18)

To estimate P, we need to calculate v, the frequency of molecular impact of the

unit area of the enclosure surface.

Let n = number of molecules in cube of unit volume. Each molecule can move

in six possible directions. The velocity of each molecule is v. On average, ^ of the

molecules will move toward each face of the cube. Thus, for a gas with volume

density n.

Thus

(n/6)v molecules will cross a unit area in unit time

(19)

Hence

P = 2mvn = 2mv

•

—

(20)

Thus

-{mnv-)

Pressure = P

•NmiCy

(21)

where P

N-

pressure in MKS units (newton m~^)

number of molecules per unit volume

m = mass of molecule (g)

^ = mean square velocity {cm^ls~^)

(C)

The average value of the kinetic energy of translation of molecules (jmC^) is

a direct measure of the absolute temperature of a gas, and therefore the average

kinetic energy of molecules of all gases is the same at the same temperature.

1.3.5 Molecular Mean Free Path 17

1.3.5

MOLECULAR MEAN FREE PATH

The average distance a molecule moves before colliding with another (collisions

with chamber walls being excluded) is called the mean free path (mfp) and is

given by

1 1 kT

—

i^nd,^)

y/lTTdiP

(22)

where d^ = molecular diameter (~

"^ cm)

n = number of molecules per unit volume

Example:

At 760 torr, mfp for air at 20°C = 6.4 X

"^ cm

"^ torr, mfp for air at 20°C = 49 m

Derivation: Consider a molecule of diameter d^, velocity v, moving in a gas of

density n. The molecule moves distance vdt in time dt so the molecule will collide

if there is another molecule of the same diameter within the volume:

8V=7rdo^vdt (23)

Since n = molecules/cm^, the volume associated with

molecule is l/n/cm^.

Thus when 8v becomes equal to l/n, a collision has occurred:

^ = TTdo^VT, (24)

where r^ is the average time between collisions.

The mean free path = distance traveled:

A = VT, = -^ (25)

If we take into account the molecular velocity distribution, we get a more cor-

rect relation:

A = r-^ ^ (26)

v27rndQ

Hence the mjp differs for various gases and is inversely proportional to the

pressure. The mean free path must not be confused with the average distance be-

I-

-S

«a

^1?

tx:

C/3

-3

°.

•—2

t>0

•^

2 6

c«

^ —

<«

C C

«>

"«

:&!

c kn cd bZlJ cQ

I—~

o o

«d

c-^

.18.

"-^

c«

l-l

•a

C/5

f<^

-§

"cd

*i2

1.3.6 Number of Impacts With the Chamber Wall 19

tween molecules, which remains very small even at high vacuum, whereas the

mean free path becomes very long due to the small size of molecules and the re-

sulting unlikelihood of a collision occurring.

i.3.6

NUMBER

IMPACTS WITH THE CHAMBER WALL

The impact frequency is a quantity of considerable interest in vacuum technology

is the number of molecules that strike a surface in unit

time.

From kinetic theory, it

can be shown that rate of bombardment of each unit surface area is

= ii-.i(2SY°

(27,

4 2

TTm

The volume density is given by

9.656 X

>«^ (28)

and

thus

= 1.45 X lO'*! j- ) (29)

r = 3.5 X 10^2 PI{MT)

^'^

molecules-Vcm-2 (30)

where F is in torr

M is in g

nsin°K

P is pressure in torr

M is molecular weight

At a pressure of 760 torr and 20°C, for nitrogen molecules there are 2.93 X 10^^

impacts s-Vcm"-^. At a pressure of 10"^ torr and 20°C, for nitrogen molecules

there are 3.87 X

10^"^

impacts s" Vcm"'^.

20 Chapter 1.3: Molecular Processes and Kinetic Theory

1.3.7

TIME TO FORM A MONOLAYER

The number of molecular impacts per second is a difficult quantity to compre-

hend. It can be expressed in a more useful way as the time required for the surface

to become covered by a single layer of molecules, as for instance in surface

physics studies where a clean surface is prepared under high vacuum.

At a pressure of 760 torr and 20°C, the time to form a monolayer of nitrogen

molecules is 3 X 10 ~^ seconds. At a pressure of 10"^ torr and 20°C, the time to

form a monolayer of nitrogen molecules is about 2000 seconds. These values

have been calculated assuming an accommodation coefficient of 1, that is, assum-

ing that an incident molecule sticks on first impact.

1.3.8

THERMAL TRANSPIRATION

For two chambers maintained at T^ and T2 °K and separated by a porous partition,

equilibrium chamber pressures are established such that

Thermal transpiration is a process that occurs at all pressures, the porous parti-

tion being necessary to ensure that at one part of the system the mean free path

of the gas is large compared with system dimensions, in this case the dimensions

of the pores of the porous material. In high-vacuum systems, this condition exists

without the presence of the porous partition, and thermal transpiration requires that

the pressure (Pi) registered by a gauge at room temperature (T^) be corrected to ob-

tain the pressure (P2) existing at points maintained at a different temperature (72).

Coefficient of viscosity: If

= coefficient of viscosity, then

77 oc

Anv (32)

where m

m = mass of molecule (g)

V = arithmetic average molecular velocity = 14,55 l(r/M)^^^ cm

s-i

Hence

It is surprising that the

is independent of pressure. This is verified by experi-

ment at normal pressures where molecular collisions are frequent, that is, where

1.3.10

Coefficient of Diffusion 21

the assumption of the kinetic theory holds good. At low pressures, less than 10"^

torr, when the mean free path is 5 cm or more the molecules can travel directly

from a moving surface to the walls of the vacuum chamber. Thus the momentum

exchange between a moving surface and the walls or the drag on the moving sur-

face depends on the number of molecules present (the pressure), and viscosity

gauges are constructed to measure pressures below 10"-^ torr.

1.3.9

COEFFICIENT OF THERMAL CONDUCTIVITY

Thermal conductivity is the transfer of thermal energy, whereas viscosity is the

transfer of mechanical energy through a gas. Kinetic theory gives an expression

for the coefficient of thermal conductivity Cj similar to that for viscosity. In fact,

Cj = vC, (33)

where Cy = the specific heat of the gas at constant volume and is constant over

wide ranges of pressures and temperature. With the same reasoning as applied to

viscosity, it is possible to construct a thermal conductivity gauge to measure be-

low 10"^ torr. Because of their simple construction, versatile control, and detec-

tion circuit techniques, thermal conductivity gauges are widely used to measure

pressures below 10 torn

1.3.10

COEFFICIENT OF DIFFUSION

The rate at which molecules of gas 1 diffuse into the molecules of gas 2 is given

by the following formula:

^ = D^ (34)

where D = coefficient of diffusion {cm^s~^) of gas 1 through gas 2 and depends

on the concentrating gradient dp^/dx. It can be shown that

T^'^ V(l/Mi + I/M2)

7 (di + d^f

D = constant X -^ \^ . ^— (^^^

By putting

= M2 = 28, d^ = d2 = 3.75 X 10"^ cm, a factor for the

self-

diffusion for nitrogen is obtained.

CHAPTER

1.4

Throughput, Pumping

Speed, Evacuation Rate,

Outgassing Rate,

and Leaic Rate

The process of evacuation is in effect the removal of mass from the vacuum

vessel: Thus the rate of mass removal (the mass flow) determines the rate at

which pressure falls. The relationship between the throughput, Q, and the corre-

sponding volumetric flow, 5, are related to the gas pressure as follows:

If the vessel initially contains

molecules of mass m/, the rate of change of

the total mass m is given by

~dt

Substituting for

from the equation

~di~

_ ^<

[Nm,)

state

{PV =

dt\

miPV\

kT )

In practice, the type of gas and the temperature

stant during the evacuation, so that

~dt^

ISBN 0-12-325065-7

$25.00

. '"'

d{PV)

(36)

NkT),

(37)

can be considered to be con-

(38)

All rights of reproduction in any form reserved.

Chapter 1.4: Throughput, Pumping

Speed,

and

Rates 23

Both the pressure and volume are easily measured quantities in vacuum ves-

sels,

so that it is convenient to define a gas flow rate, or throughput, Q, as

Q can then be measured in practical cases and is related to the mass flow rate by

dm rrii _ ^^^^

provided always that m and T are constant. Q is expressed in units of Pa m^s~'

and is often referred to as gas throughput.

The number of molecules per unit volume is proportional to the pressure and

the volumetric flow rate, and is usually defined as the product:

Q = SP (40)

This equation is one of the fundamental equations of vacuum physics.

Pumping speed: The speed of a pump is defined as the volumetric rate of flow

into the pump (Ls ~') where the volume is measured at the pressure at the pump

inlet (torr). To be correct, therefore, a speed should be quoted as 500 Ls~^ at

"^ torr. It is common practice, however, to refer only to Ls ~' (diffusion pumps

and sputter-ion pumps) or L min ~' (for mechanical pumps). Hence a particular

diffusion pump that has a speed of 500 Ls

~ ^

"^ torr will have the same

speed of 500 Ls

~ ^

10 "^

torr. In the first case the pump pumps 500 Ls

~ ^

"1000 "^ torr gas," and in the second case it pumps 500 Ls

~ ^

of "10 "^ torr gas."

The amount of gas pumped (the throughput) has fallen by a factor of 10^, but the

pumping speed or volumetric flow rate remains the same.

Leak rates: A leak rate is the throughput of a small hole from atmospheric

pressure to the vacuum space, and is conveniently measured in torr-Ls~^ An

older unit of leak rate is the liter micron s~K

Outgassing rates: An outgassing rate of a system is the rate at which gas is

desorbed from system walls to be pumped away by the vacuum pump, and is

expressed as a throughput. It is dependent on the surface area of the system

from which desorption is taking place. The outgassing rate of a surface is there-

fore expressed as throughput per unit area, or torr-Ls "' cm"-^. Typically

a clean, unbaked, stainless steel surface has an outgassing rate, after one to

two hours pumping, of 10"^ to 10"^ torr-Ls~^ cm~^. Outgassing rates are

particularly influenced by surface history such as exposure to moist atmo-

sphere, baking, etc.

Permeability: Construction materials, especially glasses and elastomers, are

permeable to the light constituents of the atmosphere. Permeability rates are ex-

pressed in the same terms as outgassing rates.