Transport Phenomena in Gases

Derivation of Coefficient of Viscosity of a Gas

Conditions: Steady state, steady velocity gradient along the z-axis, low density gas where mean free path λ is much smaller than macroscopic dimensions.

• Consider a gas with density ρ, mean molecular velocity v̄, and mean free path λ. Let the flow velocity u_x vary along the z-axis such that the velocity gradient is du_x/dz.
• Molecules crossing a horizontal plane at z = 0 from a distance equal to the mean free path above ($\bar{z} = +\lambda)$ and below ($\bar{z} = -\lambda) transport momentum parallel$ to the x-axis.
• The net viscous drag (force per unit area) F/A across the plane is equated to η (du_x/dz), yielding $\eta = (1/3) \rho \lambda v̄$.

Result: \eta = \frac{1}{3} \rho \lambda \bar{v}

Derivation of Coefficient of Thermal Conductivity

Conditions: Stationary gas with a temperature gradient dT/dz along the z-axis.

• Consider layers at distance λ above and below the reference plane $z = 0 with temperatures T + \lambda(dT/dz)$ and T - λ(dT/dz) respectively.
• The net rate of heat energy transport per unit area across the plane is given by Fourier's law of heat conduction: Q/$A = -K (dT/dz)$.
• Using the molecular flux and average energy per molecule (Cv T), the thermal conductivity K is derived as $K = (1/3) \rho \lambda v̄ C_v$, or in terms of viscosity, $K = \eta C_v (neglecting internal degrees of freedom correction factors)$.

Result: $K = \frac{1}{3} \rho \lambda \bar{v} C_v$

Viscosity vs Pressure of a Gas

Draw: A plot of viscosity (η) on the y-axis versus pressure (P) on the x-axis, showing a constant horizontal line over a wide range of moderate pressures, dropping off only at very low vacuums.

Interpret: Demonstrates the surprising kinetic theory result that gas viscosity is independent of pressure because increased molecular density is exactly compensated by a decreased mean free path.

Exam use: Frequently tested in conceptual MCQs asking about the effect of compressing a gas on its viscosity.

Viscosity vs Temperature

Draw: A parabolic-like curve showing viscosity increasing with absolute temperature, typically plotted as η versus T^(1/2).

Interpret: Reflects that higher temperatures increase molecular thermal speeds v̄, thereby increasing the rate of momentum transfer across layers.

Exam use: Distinguishing liquid viscosity (which decreases with temperature) from gas viscosity (which increases with temperature).

Maxwell's Determination of Viscosity (Experimental concept)

Draw: Schematic of oscillating disks or concentric cylinders where an outer rotating cylinder drags an inner suspended cylinder.

Interpret: Used to measure gas viscosity by observing the damping of rotational oscillations due to momentum transfer to the surrounding gas.

Exam use: Understanding experimental verification of kinetic theory transport coefficients.

If the temperature of an ideal gas is increased from 27 °C to 327 °C, by what factor does its coefficient of viscosity change?

• Recall the temperature dependence of viscosity: η ∝ \sqrt{T}.
• Convert temperatures to Kelvin: $T_1 = 27 + 273 = 300 K$, $T_2 = 327 + 273 = 600 K$.
• Calculate the ratio: \frac{\eta_2}{\et$a_1 = \sqrt{\frac{T_2}{T_1}} = \sqrt{\frac{600}{300}} = \sqrt{2}$.

Answer: \sqrt{2} (or approximately 1.414)

Calculate the coefficient of thermal conductivity of helium gas at NTP, given that its coefficient of viscosity is 2.0 × 10^-5 kg m^-1 s^-1 and its specific heat at constant volume C_v is 3120 J kg^-1 K^-1.

• Use the kinetic theory relation between thermal conductivity and viscosity: $K = \eta C_v$.
• Substitute the given values: $K = (2.0 \times 10^-5 kg m^-1 s^-1) \times (3120 J kg^-1 K^-1)$.
• Perform the multiplication: $K = 6.24 \times 10^-2 W m^-1 K^-1$.

Answer: 6.24 \times 10^{-2} \text{ W m}^{-1} \text{ K}^{-1}

Explain how the coefficient of self-diffusion D in a gas depends on pressure P and absolute temperature T. Find the combined proportionality.

• The diffusion coefficient is given by $D = (1/3) \lambda v̄$.
• Mean free path λ is inversely proportional to pressure P (or number density n), and average speed v̄ is proportional to \sqrt{T}.
• Density ρ is proportional to P/T. Since λ ∝ T/P and v̄ ∝ T^(1/2), combining gives D ∝ \frac{T^{3/2}}{P}.

Answer: D \propto \frac{T^{3/2}}{P}