CHAPTER 2

Gas Properties

In this chapter we discuss the properties of gases at atmospheric and reduced pressures. The properties developed here are based on the kinetic picture of a gas. Kinetic theory has its limitations, but with it we are able to describe particle motion, pressure, effusion, viscosity, diffusion, thermal conductivity, and thermal transpiration of ideal gases. We will use these ideas as the starting point for discussing gas flow, gauges, pumps and systems.

2.1 KINETIC PICTURE OF A GAS

The kinetic picture of a gas is based on several assumptions. (i) The volume of gas under consideration contains a large number of molecules. A cubic meter of gas at a pressure of lo5 Pa and a temperature of 22°C contains 2 . 4 8 ~ 1 0 ~ ~ molecules, whereas at a pressure of Pa, a very high vacuum, it contains 2 . 5 ~ 1 0 ' ~ molecules. Indeed, any volume and pressure normally used in the laboratory will contain a large number of molecules. (i i) Adjacent molecules are separated by distances that are large compared with their individual diameters. If we could stop all molecules instantaneously and place them on the coordinates of a grid, the average spacing between them would be about 3 . 4 ~ 1 0-9 m at atmospheric pressure (1 O5 Pa). The diameter of most molecules is of order 2 4 x lo-'' m and their separation distances are -6-15 times their diameter at atmospheric pressures. For extremely low pressures, say Pa, the separation distance is about 3x10" m. (iii) Molecules are in a constant state of motion. All directions of motion are equally likely and all velocities are possible, although not equally probable. (iv) Molecules exert no force on one another except when they collide. If this is true, then molecules will be uniformly distributed throughout the volume and travel in straight lines until they collide with a wall or with one another.

Using these assumptions, many interesting properties of ideal gases have been derived. Some elementary properties are reviewed here.

9

A User’s Guide to Vacuum Technology, 3rd Edition. John F. O’Hanlon Copyright 0 2003 John Wiley & Sons, Inc.

ISBN: 0-471-27052-0

10 GAS PROPERTIES

2.1.1 Velocity Distribution

As the individual molecules move about they collide with elastic collisions. Elastic collisions conserve energy, whereas the colliding particle's velocity is changed after each collision. We stated that all velocities are possible, but not with equal probability. The distribution of particle velocities calculated by Maxwell and Boltzmann is

312 dn - 2N m 2 -rnv2/(2kT) __- - dv z1l2 ( 2 k T )

rn is the particle mass and T is the Kelvin temperature. The relation between the Kelvin scale and the Celsius scale is T(K) = 273.16 + T C ) . In (2.1) N is the total number of particles, and k is Boltzmann's constant. Figure 2.1 illustrates (2.1) for nitrogen molecules (air) at three temperatures. It is a plot of the relative number of molecules between velocity v and v + dv. We see that there are no molecules with zero or infiite velocity, and that the peak or most probable velocity vp is a function of the average gas temperature. The particle velocity also depends on the molecular mass, the peak velocity can be expressed as vp = (2kT/m)". The arithmetic mean or average velocity v is useful when describing particle flow.

112 v'(s) b (2.2)

0 500 1000 1500 2000 Velocity (m/s)

Fig. 2.1 Relative velocity distribution of air at O"C, 25"C, and 400°C.

2.1 THE KINETIC PICTURE OF A GAS 11

Velocity (m/s)

Fig. 2.2 Relative velocity distribution of several gases at 25°C.

The average velocities of several gas and vapor molecules are given in Appendix B.2. The root of the mean square velocity, is v,, = (3kT/m)ln. The rms velocity is the square root of the average or mean of each velocity squared times the number of particles with that velocity. For Maxwell- Boltzmann statistics the average velocity is always 1.128 times as large as vp, while v,, = 1 . 2 2 5 ~ ~ In Fig. 2.1 we illustrated the temperature dependence of the velocity distribution. As the temperature is increased the peak is broadened and shifted to a higher velocity. We may also plot (2.1) for different gases having the same temperature. Figure 2.2 illustrates the velocity distribution for H2, He, H 2 0 , N2, CO2, and Xe. There are two concepts illustrated in Figs. 2.1 and 2.2. First the average velocity of a particle is proportional to (T/m)ln. An increase in temperature or decrease in mass causes an increase in a particle’s velocity and the frequency with which it collides with other particles or nearby walls. Second, not all the particles in a distribution have the same velocity. The Maxwell-Boltzmann distribution is quite broad-over 5% of the molecules travel at velocities greater than two times the average velocity.

2.1.2 Energy Distribution

Maxwell and Boltrmann also derived an energy distribution, which is based on the same assumptions as the velocity distribution. It is

12 GAS PROPERTIES

Fig. 2.3 Relative energy distribution of a gas at 25°C.

From this expression the average energy can be calculated as Eave = 3kT/2, and the most probable energy as Ep = kT/2. Notice that neither the energy distribution nor the average energy is a function of the molecular mass. Each is only a h c t i o n of temperature, as shown in Fig. 2.3. For example, all the gases depicted in Fig. 2.2 have the same energy distribution, because they are all have the same average temperature. See Fig. 2.3.

2.1.3 Mean Free Path

The fact molecules are randomly distributed and move with different velocities implies that each travels a different straight-line distance, known as a p e e path before suffering a collision. As illustrated in Fig. 2.4, not all free paths are the same length. The average, or mean, of the free paths h, is found from kinetic theory

b (2.4)

where do is the molecular diameter in meters, and n is the gas density in molecules per cubic meter. The mean free path is clearly gas density dependent. If the temperature is constant, it is also pressure dependent. See (2.12). For air at room temperature the mean free path is most easily remembered by one of the following expressions

0.005 or h(cm)=

p (Pa) P (Ton-) 0.67

h (cm) = ___ b(2.5)

2.1 THE KINETIC PICTURE OF A GAS 13

'"0 d"

Fig. 2.4 Individual molecular paths.

where h has units of cm, and P is the pressure in Pascal or Torr, respectively. Kinetic theory also describes the distributions of free paths.

(2.6) N = Nqe-xIL

Ilr is the number of molecules in the volume and N is the number of molecules that traverse a distance x before suffering a collision. Equation 2.6 states that 63% of the collisions occur in a distance 0 5 x I h, whereas about 37% of the collisions occur in range XI x I 5h. Only about 0.6% of the particles travel distances greater than 5h without suffering a collision.

For the case of two gases, a and b, the mean free path of a in b is

2.1.4 Particle Flux

The concept of particle flux is helpful in understanding gas flow, pumping, and evaporation. According to kinetic theory the flux r of an ideal gas striking a unit surface or crossing an imaginary plane of unit area from one side is

r(partic1es - m2 - s" ) = nv / 4 b (2.8)

where n is the particle density and v, the average velocity. On substituting (2.2) we see that

14

112

r=n(')

GAS PROPERTIES

The particle flux is directly proportional to the particle density and the square root of T/m.

2.1.5 Monolayer Formation Time

The time to saturate a surface with one layer of molecules is a function of the molecular arrival rate r, an molecular size. Assuming each molecule sticks and occupies surface area d:, the time to form a monolayer is

4 --

1 t =-- I'd: nvd: mi (2.10)

At ambient temperature, a monolayer of air (do = 0.372 nm, v = 467 d s ) will form in about 2.5 s at a pressure of lo4 Pa. The formation time will be longer, if the sticking coefficient is less than unity.

2.1.6 Pressure

The absolute pressure on a surface is defined as the rate at which momentum mv, is imparted to a unit surface. A molecule incident on a surface at an angle 'p from the normal will impart a total impulse or pressure of 2mv cos 9. By integrating over all possible angles in the half- plane we find that the pressure is

1 2 P = -nmv,, 3

(2.1 1)

The total energy of a molecule, however, is proportional to its temperature

(2.12)

Equations (2.1 1) and (2.12) may be combined to form the ideal gas law.

P = nkT b (2.13)

If n is expressed in units of m-3, k in joules per kelvin, and Tin kelvin, then P will have units of pascal (Pa). A pascal is a newton per square meter and the fundamental unit of pressure in System International (SI). Simply divide the number of pascals by 133.32 to convert to units of Torr, or divide by 100 to convert to units of millibars. A conversion table is included in Appendix A.3. Values of n, d', A, and r for air at 22°C are

2.1 THE KINETIC PICTURE OF A GAS 15

Table 2.1 Low Pressure Properties of Air"

Pressure (Pa)

l.01x105(760Torr) 2.48~10" 3.43~ 6.5~10-~ 2 . 8 6 ~ loz7 100 (.75 Torr) 2 . 4 5 ~ 1 0 ~ ~ 3.44~10' 6 .6~10-~ 2.83~10'~ 1 (7.5 mTorr) 2 . 4 5 ~ 1 0 ~ ~ 1 . 6 ~ 1 0 ~ 6 . 6 ~ 1 O5 2 . 8 3 ~ 10''

(7.5~10" TOK) 2 .45~10 '~ 1 . 6 ~ 10" 6.64 2.83~10'~ lo5 (7.5x10-* Torr) 2.45~10'~ 7.41 xlOd 664 2 .83~10 '~

( 7 . 5 ~ 1 0 ' ~ Torr) 2 .45~10'~ 3 . 4 4 ~ 1 0 ~ 6 . 6 ~ 1 0 ~ 2.83~10'~

Particle density n; average molecular spacing d; mean free path I; and particle flux on a surface r, for T = 22°C.

tabulated in Table 2.1 for pressures ranging from atmospheric to ultrahigh vacuum. The pressure dependence of the mean free path is given for several gases in Appendix B. 1.

2.2 GASLAWS

Kinetic theory, as expressed in (2.13), summarizes all the earlier experimentally determined gas laws. However, we review several of the experimentally verified laws here, because they are especially helpful to those with no experience in gas kinetics. When using kinetic theory, we need to remember that the primary assumption of a gas at rest in thermal equilibrium with its container is not always valid in practical situations. For example, a pressure gauge close to and facing a high vacuum cryogenic pumping surface will register a lower pressure than when it is close to and facing a warm surface in the same vessel [l]. This and other non-equilibrium situations will be discussed as required.

2.2.1 Boyle's Law

In 1662 Robert Boyle demonstrated that the volume occupied by a given quantity of gas varied inversely as its pressure, when the gas temperature remained the same.

&& = p2V2 (N,T constant) (2.14)

This is easily derived from the general law by multiplying both sides by the volume V and noting that N = n V.

16 GAS PROPERTIES

2.2.2 Amontons’ Law

Amontons discovered the pressure in a confined chamber increased as the temperature increased. Amontons’ law can be expressed as

(N, V constant) P P 1.=1 T T2

(2.15)

In 1703 he constructed an air thermometer based on this relationship. This later came to be known as the law of Gay-Lussac.

2.2.3 Charles’ Law

The French chemist Charles found in 1787 that gases expanded and contracted to the same extent under the same changes of temperature provided that no change in pressure occurred. Again by the same substitution in (2.13) we obtain

v, - v2 (N, P constant) T T2

(2.16)

2.2.4 Dalton’s Law

Dalton discovered in 1801 that the total pressure of a mixture of gases was equal to the sum of the forces per unit area of each gas taken individually. By the same methods for a mixture of gases, we can develop the relation

P, =nkT=n,kT+n,kT+n,kT+- . - (2.17)

which reduces to

P, = 4 + P 2 + p 3 + - . (2.18)

pressures and densities, respectively. Equation (2.1 8) is called Dalton’s law of partial pressures and is valid for pressures below atmospheric [2].

2.2.5 Avogadro’s Law

In 18 1 1 Avogadro observed that pressure and number of molecules were proportional for a given temperature and volume:

(2.19)

Two terms, “standard temperature and pressure” and “mole” often cause confusion. Standard temperature and pressure (STP) conditions refer to a

2.2 GASLAWS 17

gas with a temperature of OOC, at pressure of 1 atmosphere. However, in Chapter 6, we note that some manufacturers of thermal mass flow controllers assume a temperature of 20°C when calibrating flow meters; this is called "IT, or normal temperature and pressure. In SI, Avogadro's number of any gas species is No = 6 . 0 2 2 5 2 ~ 1 0 ~ ~ and occupies a molar volume of V, = 22.4136 m3. Avogadro's number of molecules is known as a mole. In SI, the unit of volume is the m3 and the unit of mass is the kg. To avoid confusion with cgs units, we will use "kg-mole." For example one kg-mole of oxygen contains 6 . 0 2 2 5 2 ~ 1 0 ~ ~ molecules and weighs 32 kg. Its density at STP is therefore 32 kd22.4136 m3 or 1.45 kg/m3.

2.2.6 Graham's Law

In the nineteenth century Graham studied the rate of efksion of gases through very small holes in porous membranes. He observed the rate of effusion to be inversely proportional to the square root of the density of the gas provided that the pressure and temperature were held constant. Since the density of a gas is proportional to its molecular weight, Grahams' law can be stated as

112

(2.20) effusion rate, - - - [ ::) (P,T constant) effusion rate,

Grahams' law describes how a helium-filled balloon loses its gas more quickly than an air-filled balloon.

Fig. 2.5 Dalton's law states that the total pressure is the sum of the partial pressures.

18 GAS PROPERTIES

2.3 ELEMENTARY GAS TRANSPORT PHENOMENA

In this section approximate views of viscosity, thermal conductivity, diffusion, and thermal transpiration are discussed. We state results from kinetic theory without derivation.

2.3.1 Viscosity

A viscous force is present in a gas when it is undergoing shear. Figure 2.5 illustrates two plane surfaces, one fixed and the other traveling in the x direction with a uniform velocity. The coefficient of absolute viscosity q is defined by the equation

du -- Fx -q- A, 4

(2.21)

where F, is the force in the x direction, A, is the surface area in the x-z plane, and d d 4 is the rate of change of the gas velocity at this position between the two surfaces. Because the gas stream velocity increases as the moving plate is approached, those molecules crossing the plane Ax, from below (1 in Fig. 2.5) will transport less momentum across the plane than will those crossing the same plane from above (2 in Fig. 2.5). The result is that molecules crossing fiom below the plane will, on the average, reduce the momentum of the molecules from above the plane, in the same manner molecules crossing from above the plane will increase the momentum of those molecules below the plane. To an observer this viscous force appears to be frictional; actually it is not. It is merely the result of momentum transfer between the plates by successive molecular collisions. Again, from kinetic theory the coefficient of viscosity is

1 3

q = -nmvh

Y

- 1 I Moving Surface

"1 2

/ -X

2 Fixed Surface

(2.22)

Fig. 2.6 Origin of the viscous force in a gas.

2.3 ELEMENTARY GAS TRANSPORT PHENOMENA 19

When the gas density is measured in units of m'3, the molecular mass in kg, the velocity in d s and the mean free path in m, q will have units of (N- s)/m2, or Pa-s. One Pa-s is equal to 10 poise. A more rigorous treatment of viscosity 13 J yields a result with a slightly different numerical coefficient:

q = 0.499nmvh b (2.23)

Substituting (2.2) and (2.4) into this result yields

(2.24)

From (2.24) we see that kinetic theory predicts that viscosity should increase as (rnT)'" and decrease as the square of the molecular diameter. An interesting result of this simple theory is that viscosity is independent of gas density or pressure. This theory, however, is valid only in a limited pressure range. If there were a perfect vacuum between the two plates, there would be no viscous force because there would be no mechanism for transferring momentum from one plate to another. This understanding leads to the conclusion that (2.24) is valid as long as the distance between the plates is greater than the mean free path-for example, the gas is viscous.

For a rarefied gas in which the ratio of the mean free path to plate separation Wy >> 1 the viscous force can be expressed as

1 O2 I I I I I I

-

Pressure Difference Between Plates (Pa)

(2.25)

Fig. 2.7 Viscous shear force between two plates at 22°C.

20 GAS PROPERTIES

where the term in parentheses is referred to as the free-molecular viscosity. The free-molecular viscosity is directly proportional to the molecular density (n = P/kT) available to transfer momentum between the plates. It is valid in the region h >> y . The constant j3 in (2.25) is related to the slip of molecules on the plate surfaces. For most vacuum conditions j3 - 1.

Figure 2.6 illustrates the magnitude of the viscous force caused by air at 22°C between two plates moving with a relative velocity of 100 m/s for three plate separations. Equation (2.24) was used to calculate the asymptotic value of the viscous drag at high pressures and (2.25) was used to calculate the free molecular limit. A more complete treatment of the intermediate or viscous slip region is given elsewhere [4]. The viscous shear force is independent of the plate spacing as long as the mean free path is larger than the spacing. This idea was used by Langmuir [5] to construct a viscosity gauge in which damping was proportional to pressure.

2.3.2 Thermal Conductivity

Heat conductivity between two infinite parallel plates is explained by kinetic theory in a manner analogous to that used to explain viscosity. The diagram in Fig. 2.6 could be relabeled to make the top plate stationary at temperature T2, and the lower plate a stationary plate at a temperature TI , where TI < T2. Heat conduction can be modeled by noting that the molecules moving across the plane toward the hotter plate carry less energy than those moving across the plane toward the cooler surface. The heat flow can be expressed as

dT H = A K -

4Y (2.26)

where H is the heat flow and K is the heat conductivity. The simple theory predicts that the heat conductivity K is expressed by K = qc,, where q is the viscosity and c, is the specific heat at constant volume. This simple theory is correct only to an order of magnitude. A more detailed analysis, which accounts for molecular rotational and vibrational energy, yields

1 4

K = -(9y - 5 ) q q b (2.27)

where y is the ratio of specific cdcy. When q has the units of Pa-s and c, has units (J/kg)/K, then K will have units of (W/m)/K. At room temperature, the heat conductivity and viscosity both increase as (rnT)''/d. For infinite parallel plates, K does not depend on pressure, as long as the mean free path is smaller than the plate spacing. In the low-pressure region, the heat transfer between parallel plates 161 has been calculated as

2.3 ELEMENTARY GAS TRANSPORT PHENOMENA 21

E, = aAP(T2 - q ) where

(2.28)

This equation has the same general form as (2.25) for fi-ee-molecular viscosity. A is the free-molecular heat conductivity and al, a 2 , a are the accommodation coefficients of the cold surface, hot surface, and system, respectively. If molecules can thermally equilibrate with the surface, say by making many small collisions on a rough surface, a will have a value that approaches unity. If however, the same surface is smooth and the molecule recoils without gaining or losing energy, a will approach zero. The kinetic picture of heat conductivity is rather like viscosity except that energy transfer determines the thermal conductivity, and momentum transfer determines the viscous drag. Even so, Fig. 2.5 can be sketched for the thermal conductivity of a gas between two parallel plates, where the vertical axis has dimensions of heat flow. In SI A has units of W-rnS-K1- Pa-’, whereas E, has units of W/m2. Tables of the accommodation coefficient are given elsewhere [3,7]. The accommodation coefficient of a gas is not only dependent on the material but on its cleanliness, surface roughness, and gas adsorption as well.

When the heated parallel plate is replaced by a heated fine wire, the situation changes. In the case of a heated fine wire, the upper “knee” of the curve is not dependent on the ratio of h to plate separation, but rather on the ratio of h to hot wire diameter [8,9]. Both thermocouple and Pirani gauges operate in a region in which the heat conduction from the heated wire is linearly dependent on pressure. However, the “knee” in their linear range begins at a mean free path equal to a few multiples of the wire diameter and not at a mean free path related to the wire-envelope distance.

2.3.3 Diffusion

Diffusion is a complex phenomenon. This discussion has been simplified by restricting it to the situation in a vessel that contains two gases whose compositions vary slowly throughout the vessel but density is everywhere the same. The coefficient of gases is defined in terms of their particle fluxes r1,2

dn r --~2 ak

dn

ak r, = -D- 2 -

whose total number diffusion D, of two

(2.29)

22 GAS PROPERTIES

These fluxes result fiom the partial pressure gradient of the two gases. The result from kinetic theory, when corrected for the Maxwellian distribution of velocities and for velocity persistence, is [9]

where D12 is the constant of inter-diffusion of the two gases. In SI it has units of m2/s. For the case of self diffusion, the coefficient is

1 /2 Dll = T( 4 E)

3nnd0 nm b ( 2 . 3 1 )

If the density n is replaced by P/kT, it becomes apparent that the diffusion constant is approximately proportional to T3” and P-’.

The diffusion equation, dC/dt = -Dd2C/dt2, whose solutions we do not describe here, contains the term (Dt)’”. This term has the dimensions of length and is called “diffision length.” For lon times the diffusion “front” moves through the gas in proportion to (Dtf2 . Values of the diffusion constant for several gases in air are given in Appendix B.2.

Examination of (2.3 1 ) shows the diffusion coefficient will become infinitely large as the density of molecules goes to zero. This does not happen. When the pressure becomes low enough so that the mean free path is much larger than the dimensions of the container, say the diameter of a pipe, gas diffusion is limited by molecules recoiling from walls, rather than from collisions with each other. At low pressures, where h >> d, the diffusion coefficient is given by

b(2.35) D=--w

where r is the radius of the pipe and v is the thermal velocity. This is called the Knudsen diffusion coefficient for a long capillary [ l o ] .

2 3

2.3.4 Thermal Transpiration

When a tube or orifice connects two chambers of different temperatures, their relative pressures depend on the nature of the gas flow in the connecting tubing. The nature of the gas flow in the tubing or orifice is characterized by (Wd), where d is the diameter of the connecting tube or orifice. For h << d the pressure is everywhere the same in both chambers, PI = P2. The densities in the two chambers are related by

2.3 ELEMENTARY GAS TRANSPORT PHENOMENA 23

b (2.36)

When the orifice diameter is such that h >> d, the flux of gas through the orifice is given by (2.8).

b(2.37)

In steady state the net flux between the two chambers must be zero-for example, (r1+2 = )-with the result

b(2.38)

Equations (2.36) and (2.38) can be used to calculate the pressures within furnaces or cryogenic enclosures when the pressure gauge is located outside the enclosure at a different temperature. Equation (2.36) is used at high pressure (A < d/lO), and (2.38) is used at low pressure (h > 106).

Thermal transpiration was discovered by Neumann [ 1 13 and studied by Maxwell [12], who predicted the square-root dependence given in (2.38). The geometry and reflectivity from walls of the connecting tubing introduce deviations from the theory. Siu [ 131 has studied these effects and has predicted that (2.38) is obtained in short tubes only for specular reflection and in long tubes only for diffuse reflection.

REFERENCES

1. R. W. Moore, Jr., Proc. 8th Nat’l. Vac. S’p. 1961, 1, Pergamon, New York, 1962, p. 426.

2. E. H. Kennard, Kinetic Theory of Gases, McGraw-Hill, New Yo& 1938, p. 9. 3. Ref. 2, pp. 135-205 and 291-337. 4. S. Dushman, Scientific Foundations of Vacuum Technique, 2nd ed., J. M. Lafferty, Ed.,

Wiley, New York, 1962, p. 35. 5. Langmuir, Phys. Rev., 1, 337 (1913). 6. Ref. 4, p.6. 7. Ref. 4, p. 68. 8. H. von Ubisch, Vak. Tech., 6, 175 (1 957). 9. M. Pirani and J. Yarwood, Principles of Vacuum Engineering, Reinhold, New York,

1961. p. 100. 10. For example, see L. M. Lund and A. S. Berman, J. Appl. Phys., 37,2489 (1966). 11. C. Neumann, Math Phys. K., 24,49 (1 872). 12. J. C. Maxwell, Philos. Trans. R. SOC. London, 170,231 (1879). 13. M. C. I. Siu, J. Vac. Sci. Technol., 10,368 (1973).

24

2. I 2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

GAS PROPERTIES

PROBLEMS

t State the assumptions that form the basis of kinetic theory.

Consider a 1-cm-diameter pipe 10 meters long. (a) Sketch the self- diffusion constant of air in this pipe over the pressure range 0.01 Pa to lo5 Pa. (b) Knowing that the diffusion fiont moves as d2 oc Dt, estimate the time required for air to diffuse to the end of the 1 O-meter- long pipe over the pressure range given in (a).

The diffbsion constant for gas (1) in gas (2), 0 1 2 describes how gas a gas of one molecular weight diffuses in the background of a second gas. For the case where the d i fh ing gas (1) is light (ml << MZ), and is present in a small concentration (nl << n2), derive a simplified relationship for (2.30). This happens during helium leak checking.

t Room temperature N2 molecules are directed toward a surface 100- cm distant. To what pressure must a chamber be evacuated in order that the molecules reach the surface, on average, without first colliding with another N2 molecule?

j - What is the advantage of low-pressure chemical vapor deposition over atmospheric pressure chemical vapor deposition?

Explain why the viscosity of a gas should increase with increasing particle mass and temperature. (The viscosity of a liquid decreases with increasing temperature.)

j - The air surrounding it cools an object that is heated by a constant- power source. What would happen to its temperature, if the surrounding air were replaced (a) by helium and (b) by argon?

What is the heat flow by thermal conduction between two 0.1 m2 sheets of copper with a temperature difference of 100°C and which are separated by 0.1 cm of C02 at pressures of (a) lo5 Pa, (b) lo3 Pa, and (c) 100 Pa.

A 0.5-cm-diameter tube interconnects two chambers. The left-hand chamber is heated to a temperature of 250°C, while the right-hand chamber remains at room temperature. Over what pressure range is the pressure in the two chambers the same? Over what pressure range does the transpiration equation apply? Sketch a plot of Pha/Pmld versus the pressure of the cold chamber. Assume diffuse scattering.

2.10111 Problem 2.9, nitrogen is replaced by helium. Sketch a plot of Phot/Pcold versus the pressure of the cold chamber in the transition region. How does this compare to the curve in Problem 2.9 for helium?