We elaborate here on the material of Section 2-7, as applied to a mixture of gases 1 and 2. Let Ζ / = 4σί 2(π*Γ//ιι,)1 / 2, i = 1 or 2, and Z'12 = 2ν^σ?2(πΑ;Γ/μ1 2)1 / 2. There are four kinds of molecular collision frequencies:
^i(i) = Ζιηι, Z2 ( 2) = Z2' n2, ^
^ 1 ( 2 ) = ^ 1 2N2 , Z2 ( 1 ) = Z[2^i ·
There are then three kinds of bimolecular collision frequencies (collisions per unit volume per second):
Zn = ^Ζχ'πχ2, Z2 2 = £ Z2' n2 2, and Z1 2 (or Z2 1) = Z'12\k^i2 . The total collision frequency is the sum of these three.
Mean free paths in a mixture may also be formulated. Thus λχ = C I/ [ Z1 ( 1 ) + Z1 ( 2J and λ2 = c2/ [ Z2 ( 2 ) + Z2 ( 1 )] . The extension to mixtures of more than two components is straightforward.
G E N E R A L R E F E R E N C E S
HIRSCHFELDER, J . O . , C U R T I S S , C. F . , A N D B I R D , R. B . ( 1 9 6 4 ) . "Molecular T h e o r y o f G a s e s a n d Liquids," corrected ed. Wiley, N e w York.
K A U Z M A N N , W . ( 1 9 6 6 ) . "Kinetic Theory o f Gases." Benjamin, N e w York.
C I T E D R E F E R E N C E S
BLINDER, S. M . (1969). "Advanced Physical Chemistry." Macmillan, N e w York.
BRIDGMAN, P. (1946). "The Nature o f Modern Physics." Macmillan, N e w York.
HERZFELD, K. F., A N D SMALLWOOD, H . (1951). " A Treatise o n Physical Chemistry" ( H . S. Taylor and S. Glasstone, eds.), 3rd ed., Vol. 2. V a n N o s t r a n d - R e i n h o l d , Princeton, N e w Jersey.
M O E L W Y N - H U G H E S , E . A . (1961). "Physical Chemistry." Pergamon, Oxford.
EXERCISES
The student is reminded t o m a k e a consistent c h o i c e o f units in carrying o u t t h e following calculations. T a k e a s exact numbers given t o o n e significant figure.
2-1 Calculate the fraction o f OA molecules per unit velocity interval, (\(N0) dN(u)ldu9 having velocity u such that their kinetic energy in o n e dimension is equal t o £ Γ at 25°C. Repeat the calculation in t w o and in three dimensions where, in each case, the velocity c is such that the corresponding kinetic energy is Α:Γ at 25°C.
Ans. 0 . 5 2 7 X 1 0 ~5, 1 . 8 7 X 1 0 ~5, 2 . 1 1 X 1 0 ~5.
2-2 Repeat the calculations in Exercise 2-1, but taking the one-, two-, and three-dimensional velocities to be 2 χ 10* c m s e c- 1.
Ans. 8.792 x 1 0 "1 7, 1 . 5 8 3 9 x 1 0 "1 6, 9 . 0 8 1 χ 1 0 "1 6. 2-3 Calculate the values of cp, c, and the root mean square velocity for argon gas at (a) 25°C
and (b) 2 0 0 ° C
Ans. (a) 3.523 χ 1 04, 3.975 χ 1 04, 4.315 χ 1 04 and (b) 4.438 χ 1 04, 5.008 χ 1 04, and 5.435 χ 1 04 (all in c m s e c "1) . 2-4 Consider N2 gas at 77 Κ and 0.1 atm pressure. Calculate (a) c, (b) n, (c) Ζ in moles per square centimeter per second hitting a surface, and (d) Zm in grams per square centimeter per second hitting a surface. 2-7 A n astronaut's spacesuit has a pinhole leak. If the suit contains air of normal composition,
what should be the composition of the gas escaping into space from the pinhole, according t o Graham's law? 2-10 For g o o d insulation the pressure in the evacuated space of a Dewar or thermos flask
should be reduced to the point that the mean free path is greater than the distance between the walls. What should the pressure of air be at 25°C if the mean free path is to equal 0.5cm?
Ans. 1.336 χ 1 0 "5 atm. ( W e estimate from Exercise 2-9 that aa v is 3.70 A.) 2-11 Calculate the mean free path of argon gas at 1 0 "1 1 atm (an obtainable pressure under modern
ultrahigh-vacuum conditions) and 0°C, taking a to be 3.64 A.
Ans. 6.32 χ 1 05c m . 2-12 The viscosity o f air at 0 ° C is 1.71 χ 1 0- 4 P. What should the volume flow rate o f air be,
in liter s e c- 1 through a capillary tube 10 m long and 0.1 m m in radius if the pressure drop is 15 a t m ?
Ans. 3.490 χ 1 0 "4 liter s e c "1. 2-13 Calculate σΛ ν for air using information supplied in Exercise 2-12.
Ans. 1J62 A. ( W e use 1.71 χ 1 0 " 4 Ρ as the viscosity o f air at 0°C.) 2-14 Calculate the self-diffusion coefficient for helium at STP taking a to be 2.18 A. Also cal
culate the viscosity and the thermal conductivity coefficient taking Cv t o be 3.00 cal K "1 m o l e "1.
Ans. 0 = 1.059 c m2 s e c "1, ^ = 1.892 χ Ι Ο "4 P, * = 1 . 4 1 8 x 1 0 "4 cal K "1 c m "1 s e c "1. 2-15 Referring to Exercise 2-14 for data, h o w far should a helium atom at STP diffuse in 1 sec?
H o w long should it take to diffuse 1 c m ?
Ans. 1.46 c m , 0.472 sec.
PROBLEMS 73 P R O B L E M S
Problems marked with a n asterisk require fairly lengthy c o m p u t a t i o n s .
2-1 Extend the example o f Table 2-1 to find the possible distributions a n d their probability weights if w e h a v e 3 0 m o l e c u l e s with a n average energy o f 2 units per molecule. accurate plot o f probability versus velocity. Verify by a graphical or an analytical m e t h o d that the plot is normalized. Locate cp , c, and ( c2)1 / 2 o n the plot.
2-5 Make the change of variable Ε = \mci to derive from Eq. (2-29) the expression for the fraction dN/N0 of molecules having energy in two dimensions between Ε and Ε + dE, as a function of E. Calculate the fraction of molecules of a gas which have an energy equal to or greater than ten times their average kinetic energy (which is 3kT/2).
2-6* Calculate a n d plot the o n e - , t w o - , a n d three-dimensional velocity distributions for H e gas at 100°C.
2-7 Calculate cp , c, and ( c2)1'2 for A r gas at 100°C a n d at 1000°C.
2-8 Calculate p(c) for c = c in the c a s e o f Na g a s at 25°C.
2-9 Derive the expressions for c and c2 for a two-dimensional gas.
2-10 W o r k Exercise 2-3 d o i n g the entire calculation in SI units.
2-11 Argon gas is present in a flask at 25°C and pressure P. T h e flask has a pinhole which may be regarded as a cylindrical hole of 2 χ 1 0- 4 c m2 area and 0.20 c m length. Calculate the rate of escape of the argon assuming that the process is o n e of (a) effusion and (b) diffusion, and for Ρ = 1 atm and Ρ = 1 0- 7 atm. Express your answer in moles of gas per second.
2-12 A flask contains a 3:1 m o l e ratio o f H2 t o H e at 100°C and 2 a t m total pressure, a n d has a pinhole of 1 x 1 0 "4 c m2 area. Calculate (a) the total m o l e s o f gas escaping per s e c o n d a n d (b) the total m a s s o f gas escaping per s e c o n d .
2-13 Calculate the mass per second of C 02 striking each square centimeter of a leaf in air containing C 02 at a partial pressure of 0.0010 atm at 2 5 ° C .
2-14 What quantity of heat in calories per centimeter length per second will be lost through conduction by gas molecules by a filament 0.2 m m in diameter heated electrically to 200°C in a light bulb containing argon at 0.01 m m pressure, the wall temperature of the bulb being 2 5 ° C ? [Note: This pressure m a y be assumed to be l o w enough s o that gas molecules travel from filament to wall and back again without hitting each other o n the way. There
fore argon atoms hit the filament, then leave it with a kinetic energy corresponding to 200°C, and fly to the wall to be "cooled d o w n " t o 25°C. F o r steady-state conditions, the rate of leaving the filament should be the same as the rate of hitting it.]
2-15 Calculate the collision frequency for H2 at S T P . T a k e the collision diameter to be 2.5 A.
2-16 Calculate the collision frequency for C l2 at S T P . T a k e the collision diameter t o be 3.5 A.
2-17 Calculate the initial rate for the reaction H2 + C l2 = 2 HC1 in a n equimolar mixture at S T P assuming that reaction occurs with each collision (note Problems 2 - 1 5 a n d 2 - 1 6 ) . T h e reaction is actually quite s l o w ; w h y might this b e ?
2-18 The collision diameter for argon is 3 . 6 4 A . Calculate ΖΑΓ_ Α Γ at STP. The average concentra
tion of molecules which are in the act o f colliding m a y be estimated as given by their collision frequency times the collision lifetime o f about 1 0 ~1 8 sec. Calculate the frequency, in moles of events per liter per second, with which such a collision pair will undergo colli
sion with an argon atom—that is, the frequency of triple collisions.
2-19 The coefficient for the interdiffusion o f t w o chemical species is given by
* „ - 2 . 6 2 8 0 χ . 0 - + (CM, S E C-1 } >
Pa2
12
where Μ denotes molecular weight. D e d u c e what the units of Ρ and Σ1 2 must be and calculate for the interdiffusion of oxygen and nitrogen in air at STP.
2-20 Three of the experimentally measurable quantities listed in Table 2 - 2 m a y be combined as a dimensionless product (or quotient). Find o n e such combination, calculate its theoretical value, and compare with experiment.
2-21 Show that the average volume occupied by a molecule can be regarded as a cylinder o f area nr2 (r being the actual molecular radius) and length 4 \ / 2 λ .
2-22 Calculate Cv for methane at S T P from the data o f Table 2 - 3 .
2-23 T h e viscosity o f H2 at S T P is 8 . 4 0 χ 1 0 "6 P. Calculate the m e a n free path.
SPECIAL T O P I C S P R O B L E M S
2-1 Calculate the viscosity of C H4 at 280 Κ using Eq. (2-82) and compare your result with the experimental value.
2-2 The viscosity of X e is 2.235 χ 1 0 "4P at 290 Κ and 3.954 χ 1 0 "4P at 550 K. Devise a trial and error method s o as to calculate reasonably matching values o f φ* Ik and Σ.
2-3 Suppose w e have an equimolar mixture of 1 3C H4 and 1 2C H3D (species A and B, respectively) at S T P . Calculate ZA a n d ZAiB}. Explain if the answers are different. ( T h e collision diameter which is needed is implicit in the data of Table 2-3.)
2-4 W e have a mixture of H2 a n d 02 at 25°C, t h e respective partial pressures being f a t m and i a t m . Calculate the moles per liter per second o f (a) H2- H2, (b) H2- Oa, a n d (c) 02- 02 collisions. Calculate also the average distance traveled by a hydrogen molecule before making a collision with (d) another hydrogen molecule, (e) a n oxygen molecule, and ( J ) any molecule. T h e cross sections for H2 a n d Oa are 2.74 A a n d 3.61 A, respec
tively.