21.4 Transport properties of a perfect gas

1. Step 1: Derive an expression that shows how the pressure of a gas inside an effusion oven varies with time if the oven is not replenished as the gas escapes. • step 2: calculate the derivative of P: step 3: the rate of change of the number of molecules is equal to the collision frequency with the hole multiplied by the area of the hole: so step 4: integrate the above equation P = P0e-t/τ (Remark: how does the temperature (or the size of the hole) affect the decrease of pressure?)

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

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

5. Table 21.3

6. Diffusion

7. 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)

8. A factor of 2/3 needs to be introduced. So we get D =

9. 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.

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

11. The viscosity is independent of the pressure. Proportional to T1/2 Viscosity

12. Measuring the viscosity • Poiseuille’s formula:

13. 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 wa s measured at the latter pressure. In 100s, 90.2cm3 of air passed through the tube. • Solution: Reorganize Poseuille’s equation: