Transport Phenomena in Gases & Real Equations of State

1. The Concept of Mean Free Path

In the kinetic theory of gases, molecules are in constant, randomized rectilinear motion interrupted by elastic collisions. The Mean Free Path $\lambda$ is the average distance traversed by a molecule between two successive collisions.

Consider a gas of spherical molecules with collision diameter $d$ and number density $n = N/V$. If one test molecule moves with speed $\bar{v}$ while all others are stationary, its effective collision cross-section is $\sigma = \pi d^2$. In time $\Delta t$, it sweeps out a collision cylinder of volume $\sigma \bar{v} \Delta t = \pi d^2 \bar{v} \Delta t$, encountering $n \pi d^2 \bar{v} \Delta t$ target molecules. The collision frequency is $Z_1 = n \pi d^2 \bar{v}$, yielding Clausius's initial estimate:

2. Maxwell's Relative Velocity Correction

James Clerk Maxwell (1860) showed that other molecules are also moving with a Maxwell-Boltzmann velocity distribution. The average relative velocity between two colliding molecules is $\bar{v}_{\text{rel}} = \sqrt{2} \bar{v}$. Hence the true collision frequency is $Z = \sqrt{2} n \pi d^2 \bar{v}$, yielding:

Using the ideal gas law $P = n k_B T \implies n = \frac{P}{k_B T}$:

Key Dependences:

• At constant temperature, mean free path is inversely proportional to pressure: $\lambda \propto 1/P$.
• At constant volume ($n = \text{const}$), mean free path is independent of temperature.
• In atmospheric air ($P = 1\text{ atm}$, $d \approx 0.37\text{ nm}$), $\lambda \approx 68\text{ nm}$. In ultra-high vacuum ($P \sim 10^{-10}\text{ torr}$), $\lambda$ exceeds several kilometers.

3. Collision Probability

The probability that a molecule travels a distance $x$ without undergoing any collisions is given by the Poisson survival distribution:

1. Viscosity (Transport of Momentum)

Consider a gas with a macroscopic velocity gradient $du/dz$ along the $z$-axis. Molecules crossing a plane $z = z_0$ from layer $z_0 + \lambda$ carry momentum $m [u(z_0) + \lambda du/dz]$, while molecules from $z_0 - \lambda$ carry $m [u(z_0) - \lambda du/dz]$. By kinetic theory, the molecular flux crossing in each direction is $\frac{1}{6} n \bar{v}$. The net rate of momentum transfer per unit area (shear stress $\tau$) is:

Comparing with Newton's law of viscosity $\tau = \eta \frac{du}{dz}$:

Substituting $\rho = n m$ and $\lambda = \frac{1}{\sqrt{2} n \pi d^2}$:

Remarkable Conclusions:

1. Independence of Pressure: Over a wide pressure range ($10^{-3}\text{ atm} \le P \le 10\text{ atm}$), the viscosity of an ideal gas is completely independent of pressure/density (experimentally verified by Maxwell in 1866).

2. Temperature Dependence: Unlike liquids (whose viscosity decreases with $T$), the viscosity of gases increases with temperature: $\eta \propto \sqrt{T}$.

2. Thermal Conductivity (Transport of Energy)

When a temperature gradient $dT/dz$ exists, molecules carry internal kinetic energy across planes. The conductive heat flux is:

Like viscosity, gaseous thermal conductivity is independent of pressure over moderate ranges.

3. Diffusion (Transport of Mass)

When a concentration gradient $dn/dz$ exists, net particle flux is $J = -D \frac{dn}{dz}$. The coefficient of self-diffusion is:

Because $\rho \propto P$ and $\lambda \propto 1/P$, diffusion is inversely proportional to pressure: $D \propto 1/P$.

1. Robert Brown's Discovery & Kinetic Interpretation

In 1827, Scottish botanist Robert Brown observed under a microscope that microscopic pollen grains suspended in water perform perpetual, erratic, jittery zigzag motions. In 1905, Albert Einstein and Marian Smoluchowski proved that Brownian motion is the direct macroscopic consequence of relentless, unbalanced thermal molecular collisions.

2. Langevin Equation & Mean Squared Displacement

Paul Langevin (1908) formulated the equation of motion for a Brownian particle of mass $m$ and radius $r$ suspended in a fluid of viscosity $\eta$:

where $-6\pi \eta r v$ is Stokes drag and $F_{\text{random}}(t)$ is a zero-mean Gaussian white-noise thermal bombardment force. Multiplying by $x$ and taking the statistical ensemble average $\langle \dots \rangle$:

Using $x \frac{d^2 x}{dt^2} = \frac{1}{2} \frac{d^2 (x^2)}{dt^2} - \left(\frac{dx}{dt}\right)^2$ and the equipartition theorem $m \langle v_x^2 \rangle = k_B T$, with $\langle x F_{\text{random}} \rangle = 0$:

For observational timescales ($t \gg m / 6\pi\eta r \sim 10^{-7}\text{ s}$), inertial term vanishes:

For 3D motion: $\langle r^2(t) \rangle = 6 D t = \frac{k_B T}{\pi \eta r} t$.

3. Perrin's Determination of Avogadro's Number

In 1908, Jean Perrin tracked individual mastic and gamboge colloidal particles under a microscope, plotting $\langle r^2 \rangle$ vs $t$ and measuring sedimentation equilibrium. Using $k_B = R / N_A$, Perrin measured $N_A \approx 6.5 \times 10^{23}\text{ mol}^{-1}$, providing undeniable empirical proof for the reality of atoms and winning the 1926 Nobel Prize in Physics.

1. The Van der Waals Equation of State

Johannes Diderik van der Waals (1873) modified the ideal gas law to account for two microscopic realities of real gas molecules:

1. Finite Molecular Volume (Co-volume $b$): Molecules are hard spheres; the free volume available for motion is $(V_m - b)$, where $b = 4 N_A \left(\frac{4}{3}\pi r^3\right)$ is four times the true molecular volume.

2. Intermolecular Attractive Forces ($a/V_m^2$): Molecules near the container wall experience an inward net pull from interior molecules, reducing impact pressure by an internal cohesion pressure proportional to density squared: $P_{\text{int}} = a / V_m^2$.

Combining these corrections yields:

2. Critical Point Constants

On a $P$-$V$ diagram, isotherms exhibit cubic behavior. At the Critical Temperature $T_c$, the liquid-gas coexistence curve terminates at an inflection point with horizontal tangent:

Differentiating $P = \frac{R T}{V_m - b} - \frac{a}{V_m^2}$:

Equating both expressions gives:

Substituting $V_c = 3b$ back:

3. Law of Corresponding States

The critical compressibility factor is a universal dimensionless number:

Defining reduced coordinates $P_r = P/P_c$, $V_r = V/V_c$, and $T_r = T/T_c$:

In reduced coordinates, all gases obey the identical universal equation of state, independent of parameters $a$ and $b$.

1. Gibbs' Phase Rule

Josiah Willard Gibbs (1876) derived the thermodynamic degree of freedom $F$ for a multi-phase, multi-component heterogeneous system in equilibrium:

where $C$ is the number of independent chemical components and $P$ is the number of coexisting phases. For a single-component system ($C = 1$, e.g., pure water):

• Single phase ($P = 1$, liquid or vapor): $F = 1 - 1 + 2 = 2$ (can independently vary both $T$ and $P$).
• Two coexisting phases ($P = 2$, liquid-vapor boiling curve): $F = 1 - 2 + 2 = 1$ (univariant; fixing $T$ automatically fixes equilibrium vapor pressure $P$).
• Three coexisting phases ($P = 3$, Triple Point): $F = 1 - 3 + 2 = 0$ (invariant; triple point occurs at a unique, immutable temperature and pressure: $T_t = 0.01^\circ\text{C}$, $P_t = 611.65\text{ Pa}$ for water).

2. Regenerative Gas Liquefaction (Linde and Claude Cycles)

Because the Joule-Thomson temperature drop $\Delta T = \mu_{\text{JT}} \Delta P$ across a single throttle valve is typically $20\text{ K}$ to $40\text{ K}$, liquefaction from room temperature requires regenerative heat exchange:

• Linde Cycle: High-pressure gas ($200\text{ atm}$) flows through the inner tube of a counter-current heat exchanger, throttles through a porous nozzle, cools, and the unliquefied cold gas returns via the outer annular tube to pre-cool incoming compressed gas. As the cycle repeats, the pre-throttling temperature drops progressively until it enters the liquid-vapor dome, yielding continuous liquid air, nitrogen, or oxygen.
• Claude Cycle: Replaces a portion of the throttling valve with an adiabatic expansion engine doing external mechanical work against a piston, achieving greater cooling ($\Delta T_{\text{rev, ad}} \gg \Delta T_{\text{JT}}$) and higher liquefaction efficiency.