Question

a) What is transport phenomenon? Obtain an expression for:
i) the average number of molecules crossing an arbitrary plane from either side per
unit area per second is n ,v
ii) the average height at which a molecule makes its last collision before crossing
the plane is .
3
2
λ
iii) Using the results in (i) and (ii) obtain an expression for viscosity of the gas.
iv) Discuss its pressure and temperature dependence.

Answer 1

Question #75494

Description:

a) What is transport phenomenon? Obtain an expression for:

i) the average number of molecules crossing an arbitrary plane from either side per unit area per second is N, v

ii) the average height at which a molecule makes its last collision before crossing the plane is.

3

2

λ

iii) Using the results in (i) and (ii) obtain an expression for viscosity of the gas.

iv) Discuss its pressure and temperature dependence.

Solution.

1. basic equation of molecular kinetic theory

$$p = n k T$$

where n concentration, p pressure, T temperature, the required number of molecules is (taking into account the probable movement of 3 axes)

$$N = \frac{n(\mathrm{v}^2) S}{6}$$

Where $<\mathrm{v}^2>$ the mean quadratic velocity, S the area of the plane and equal condition $1\ \mathrm{m}^2$, taking into account the motion in both directions through the plane

$$N = \frac{n(\mathrm{v}^2) S}{6} \cdot 2 = \frac{n(\mathrm{v}^2) \cdot 1}{3} = \frac{n}{3} \sqrt{\frac{3 R T}{M}}$$

Where T temperature, R universal gas constant, M molar mass.

2. According to Maxwell's distribution, the free run will be

$$\lambda = \frac{1}{\sqrt{2} \pi d^2 n}$$

d the effective radius of the molecule if $\lambda = 2$ or $3\ \mathrm{m}$

$$d = \sqrt{\frac{1}{\sqrt{2} \pi 2 n}}, \sqrt{\frac{1}{\sqrt{2} \pi 2 n}}$$

3 Gas viscosity

$$\eta = \frac{1}{3} \rho(\mathrm{v}^2) \lambda$$

taking into account the formula of the Mendeleev - Clapeyron, the length of the free path of molecules and required number N, and basic equation of molecular kinetic theory

$$\eta = \frac{1}{3} \frac{p M}{R T} \sqrt{\frac{3 R T}{M}} \cdot \frac{1}{\sqrt{2} \pi d^2 n} = \frac{1}{3} \frac{p M}{R T} \sqrt{\frac{3 R T}{M}} \cdot \frac{k T}{\sqrt{2} \pi d^2 p} = \sqrt{\frac{T M}{3 R}} \cdot \frac{k}{\sqrt{2} \pi d^2}$$

4 Gas viscosity viscosity is virtually independent of pressure for gases subject to Maxwell statistics that is, it means

$$\frac{\partial \eta}{\partial p} = 0$$

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