21.4 Transport properties of a perfect gas

• Experimental observations on transport properties shows that the flux of a property is proportional to the first derivative of other related properties.

• The flux of matter is proportional to the first derivative of the concentration (Fick’s first law of diffusion): J(matter)

• The rate of thermal conduction is proportional to the temperature gradient: J(energy)

• J(matter) = D is called the diffusion coefficient (m2s-1);

dz

dN

dz

dT

dz

dND

• J(energy) = - k dT/dz k is called the coefficient of thermal conductivity (J K-1 m-1 s-1)

• J(x-component of momentum) = , η is the coefficient of viscosity.

•

dz

dvx

Table 21.3

Diffusion

_cD

3

1

As represented by the above Figure, on average the molecules passing through the area A at z = 0 have traveled about one free path.

The average number of molecules travels through the imaginary window A from Left to Right during an interval Δt is

ZwA Δt (L→R) Because Zw = So A Δt (L→R) The average number of molecules travels through the imaginary window A from Right to

Left during an interval Δt is A Δt (R →L)

The net number of molecules passing through the window A along the z direction is:

A Δt - A Δt

By definition the flux of molecules along z direction can be calculate as J(z) = ( A Δt - A Δt )/(A Δt )

J(z) = The number density N(-λ) and N(λ) can be represented by number density N(0) at z =0

N(-λ) = N(0) - λ N(λ) = N(0) + λ

Therefore: J(z) =

then we get D = (different from what we expected)

cN )( 4

1

cN )( 4

1

cN )(4

1

cN )( 4

1

cN )(4

1

cN )( 4

1 cN )(4

1

cNcN )()( 4

1

4

1

0)(dz

dN0)(

dz

dN

02

1)(

dz

dNc

c2

1

A factor of 2/3 needs to be introduced.

So we get

D =

c3

1

Thermal conduction

k =

where CV,m is the molar heat capacity at constant volume.

Because λ is inversely proportional to the molar concentration of the gas, the thermal conductivity is independent of the concentration of gas, and hence independent of the gas pressure.

One exception: at very low pressure, where the mean free path is larger than the size of the container.

][,_

ACc mV3

1

dz

dTkenergyJ )(

• J(x-component of momentum) = , η is the coefficient of viscosity.

dz

dvx

Viscosity

• The viscosity is independent of

the pressure.• Proportional to T1/2

][_AcM

3

1

Measuring the viscosity

• Poiseuille’s formula: 0

422

21

16 pl

rpp

dt

dV

)(

Calculations with Poiseuille’s formula

• Example: In a poiseuille flow experiment to measure the viscosity of air at 298K, the sample was allowed to flow through a tube of length 100cm and internal diameter 1.00mm. The high-pressure end was at 765 Torr and the low-pressure end was at 760Torr. The volume was measured at the latter pressure. In 100s, 90.2cm3 of air passed through the tube.

• Solution: Reorganize Poseuille’s equation:

)/(

)(

dtdVlp

rpp

0

422

21

16

)(.

).

().().(

).(}).().{(

21114

351

4422

1110821

10010029

31337601000116

1000531337603133765

skgmPaskgm

sm

Pam

mPaPa