21

TOPIC 6 – TRANSPORT PROPERTIES

Examples of Solved Problems

1. What is the diffusion coefficient, D of CO2 at 400 bar and 300 oC? dCO2= 0.40 nm.

Answer:

Use simple formula for D = ⅓ λ<v> = cλ3

1

111122226666 ssssmmmm101010104.874.874.874.87 −−−−

−

−−

−

−−−

−

−

−

×=×=

×=××

=

=><

=

×=××

××===

=

××

××=

>=<

)525)(1078.2(3

1

1078.2)1006.5()104.0(2

1

2

1

1006.510

1

5733145.8

10022.6400

5251044

573314.888

18

8

32529

211

325

3311

123

1

2/1

3

2/1

smmD

mmm

dz

v

mm

L

KmolKbarL

molbar

RT

PN

V

N

msM

RTv

A

π

ρπλ

ρ

ππ

2. Calculate the viscosity of methane (16.04 g mol–1

) vapor at 273 K. Take πd2

= 0.46 nm2

.

Formula given: 2

2/1

16

5

d

m

m

kT

ππ

πη

= and k =1.38×10

-23 J K

-1.

Answer:

Use the formula given for the calculation:

....ssssmmmmkgkgkgkg101010103.613.613.613.61 111111115555 −−−

−

−

−

−−

−

−

−−

×=×

×

××

×=

=

×=×

×==

218

262/1

26

123

2

2/1

26

123

13

1046.0

1066.2

1066.2

)273)(10381.1(

16

5

16

5

1066.21002.6

1004.16

m

kg

kg

KKJ

d

m

m

kT

kgmol

molkg

N

Mm

A

π

π

ππ

πη

22

3. Calculate the thermal conductivity of neon gas at 300 K and 15 mbar. The gas in

confined a cubic vessel of side 15 cm. The temperature of one side of the wall is 305 K

and the opposite is 335 K. What is the rate of flow of energy (as heat) from one wall to

the other ? Given for Ne: Mr=20.18, Cv,m=1.5 R , d= 0.439 nm.

Answer:

Use the approximation formula for K = ⅓ λ<v> ][, AC mv = ][3

1, ACc mv λ

)314.85.1)(561)(1094.1(3

1

1094.1)1002.6()10439.0(2

1

1

2

1

2

1

2

1][][

5611018.20

300314.888

11126

26

12329

22211

1

2/1

3

2/1

−−−−−

−−

−−

−

−

××=

×=××

=

×=××=×=×><

=×

==

=

××

××=

>==<

molJKsmmolmK

molmmolm

NdV

n

nN

V

dV

n

dA

z

vA

V

nN

V

N

smM

RTvc

AA

A

π

ππρπλ

ρ

ππ

= 0.0453 J m-1

s-1

K-1

.

The rate of flow of energy (as heat) = qz× A= - KdZ

dT× A

= -(0.0453 J m-1

s-1

K-1

) m

K21015

)305335(−×

−×(15×10

-2 m)

2= - 0.204 J s

-1

Exercise 6a

1. The viscosity, η of H2 at STP is 5.40 ×10-5

poies. Calculate the mean free path of

H2. (1.07×10-7

m )

2. Calculate the viscosity, η of molecular oxygen at 373 K and 1 atm, dO2= 0.361

nm. (1. 98×10-4

poies).

3. The viscosity, η of helium is 1.88×10-4

P at 0 oC. Calculate the collision diameter.

(0.179 nm)

4. For Ne (20.18 g mol-1

) at 1 atm and 0 oC , η =2.97×10

-4 poies. Predict the

diffusion coefficient, D of Ne ? ( 0.33 cm2s

-1).

5. Calculate the flux of energy arising from a temperature gradient of –30 K/m

in a

sample of krypton at a mean temperature for which κ = 0.0087 J K–1

m–1

s–1

.

(0.261 J m-2

s-1

)

6. Calculate the thermal conductivity, κ of argon. Given Cv,m = 12.5 J K

−1

mol−1

, πd2

= 0.36 nm2

, M = 39.95 g/mol at room temperature (25°C). (5.4×10-3

J K-1

m-1

s-1

)

23

7. Calculate the rate of diffusion, Jz of a gas at 300 K with λ=1.00×10-5

cm, d =

3.16×10-8

cm, and M = 30.0 g/mol, if a concentration gradient is 1.00×10-7

mol

cm-4

? (9.2×1019

m-2

s-1

)

8. The sheets of copper of area 1.50 m2 are separated in air by 10.0 cm. What is the

rate of transfer of heat by conduction from the warm sheet, 50 oC to the cold

sheet, -10 oC? Given : κ (air) = 0.1241 J K

-1 m

-1 s

-1. (21.69 J s

-1)

Exercise 6b (Objective questions)

1. The statement below are true concerning the mean free path EXCEPT..

A λ is the everage distance travelled by a molecule between collisions.

B The average mass between collisions is c / λ

C At the equal molecule density ρ, the λ value of each gas is different.

D Mean free path , λ is proportonal to temperature.

2. Diffusion coefficient, D depends on

I Size of molecules

II Mass of molecule

III Number of molecules

IV Molar heat capacity

A. I and II B. I, III , and IV C. I, II , and III D. All

3. Which of the following statements are true about the properties of gases?

I Diffusion occurs due to the difference in velocities

II The viscosity, is independent of pressure

III The thermal conductivity, K, is direcly propotional to T1/2

IV The rate of thermal transport along xy axis is directly proportinal to the

temperature.gradient,dT/dz, along z axis.

A. I and II B. III , and IV C. I, II , and III D. IV only

4. Choose the correct statement(s) about the kinetics of gases and the trasport

properties of gases?

I In a diffusion process, the transported quantity is matter

II The rate of effusion is inverly proportional to the molar mass.

III If the pressure of a gas in a closed container is doubled, the mean free

path will be halved.

A. I only B. I and II C. II , and III D. I, II , and III

24

Test questions

Feb., 2010

1. Assume that an imaginary gas exhibits the following behavior;

o At constant T, the P is inversely proportional to the square of the V.

o At constant P, the V varies directly with the 2/3 power of the T.

Show that, under these conditions, 4

63

T

VP= a constant.

2. (Add in the seperated copy).

3. A 100.0-mL flask contains 1.5 mol of pure O2 gas. If the mean free path

O2 is 7.1057×10-8

m and the collision frequency for one particular O2

molecule is 6.25×109 s

-1, calculate (a) Mean speed,(b) Molecular density,

(c) Average kinetic energy,and (d) Collision diameter.

4. A container of fixed volume contains 2 ideal gases, A and B, of unknown

quantity.The mole fraction of A in the mixture is 1/3 and the total pressure

in the container at a given temperature is P1. Two additional mole of one

of the gases are then injected into the container at the same

temperature.The new total pressure, P2 is such that the ratio P2/P1 =11/9.

Determine the number of moles of A and B originally present in the

container.

Sept., 2009

1. For molecules colliding with a wall an area A, show that the flux,

4

><=

vJ N

ρ. Given that

adxex

axn

2

1)(

0

2

=−∞

∫ , where n=1 and

f(vx) =

−

kT

mvx

ekT

m 22/1

2

2π

2. At 25 oC, the mean free path for nitrogen molecules is 65.9 nm. Find (i) the

pressure in atm, (ii) Time between collisions for nitrogen molecules.

Given dN2= 0.375 nm; 1 atm =1.01325×105 Nm

-2).

3. (a) The density of dry air at 0.986 bar and 27 oC is 1.145 g dm

-3. Calculate

the composition of air assuming only N2 and O2 to be present.

(b) A 20.5 L contains 2 mol of H2 and 1.5 mol of N2 at 25 oC. All of the H2

reacted with N2 to form NH3. Calculate the partial pressure (in atm) of

H2, N2 and NH3 in the container.

25

Feb, 2009

1. (a) Derive an expression for the compression factor of a van der Waals gas.

(b) One mole of CO(g) at 300.5 K occupies a volume of 137.69 cm3. If it

obeys the van der Waals equation, calculate the compression factor of the

gas. Which intermolecular forces are dominating in the sample?(Given:

Tc=132 K; Pc=35.9 atm, a =1.485 L2atm mol

-2).

2. The mole percentage composition of oxygen in dry air is 20.97%.Calculate

the number of collisions made by oxygen against a wall with an area of

2.0 cm2 in 5 seconds, at 1 bar(air pressure) and 25

oC.

3. Two sheets of copper of area 1.50 m2 are separated in air by 10.0 cm. What is

the rate of transfer heat by conduction from the warm sheet (50 oC) to the

cold sheet(-10 oC)?[Given κ (air) = 0.024 1 J K

-1s

-1]

4. Consider samples of pure He(g) and pure Ar(g), each at 100 K and 1 atm. For

each of the following properties, state which gas (if any) has the greater

value. Give a short reason.(a) vrms (b) ε k (c) λ (d) JN.

August, 2008

1. The foiling gases are produced from exploding 5.0 mL of nitroglycerine

C3H5(NO3)3 at 25 oC and a final pressure of 1 atm, 4C3H5(NO3)3 (l)→

6N2(g) + O2(g) + 12CO2(g) + 10H2O(l). Calculate the volume occupied

by the gases and the partial pressure (in torr) of CO2(g) if the density of

nitroglycerine is 1.59 g mL-1

. Assume all gases behave ideally.

2. What is the critical temperature for a van der Waals gas that has a critical

pressure, Pc=100 atm and b= 50 cm3 mol

-1?

3. The diffusion coefficient, D, for nitrogen gas at 300 K and 1.2 atm is

8.89×10-6

m2s

-1. Calculate the mean free path, λ , and the collision

diameter, d. [Used the simplified form for D]. Indicate how D varies with:

(i) Pressure, (ii) Temperature at constant P.

March, 2007

1. A gas is found to obey the following equation of state: V

a

bV

RTP −

−= ,

where a and b are constants not equal to zero. Determine whether this gas

has a critical point, if it does, express the critical constants in term of a and

b. If it does not, explain how do you determined this and the implication

for the statement of the problem.

2. (a) A solid surface with a 1.5 mm × 3.2 mm dimension is exposed to neon gas

at 111 Pa and 1500 K. How many collisions do the Ne atoms make with

this surface in 10 s?

26

(b) Calculate the thermal conductivity of neon gas ata300 K and 15 mbar. The

gas in confined a cubic vessel of side 15 cm. The temperature of one side

of the wall is 305 K and the one at the opposite is 335 K. What is the rate

of flow of energy (as heat) from one wall to the other?

[Given.RMM Ne=20.18 g mol-1

, Cv,m=3R/2, dNe=0.439 nm].

3. A gas mixture is made of H2 and NO2 at the pressure of 1 bar and

temperature of 25 oC. If the mole fraction of NO2 is 0.32, calculate (a)

molecular density of H2 (b) reduced mass of the mixture,(c) relative speed

H2 respect to NO2 (d) collision frequency, z(H2-H2) (e) collision density,

ZH2-NO2.[Given dH2= 0.361 nm and dNO2=0.562 nm].