CD

Kinetic Theory and Real Gases — Vocabulary Flashcards

Administrative announcements

  • Homework one: likely assigned Monday, due Wednesday, September 10.
    • Official deadline communicated via Canvas or email announcements.
  • Next week: no in-person classes due to Labor Day and university travel.
    • Lectures will be recorded and posted on Canvas for Wednesday and Friday.
    • Students should watch recordings before returning to class.
    • No in-person office hours next week; Zoom may be arranged if needed.
  • Plan: continue with material via recorded lectures; attendance updates will be posted.

Recap: Maxwell distribution, temperature, and molecular speeds

  • Maxwell distribution describes speeds in a gas; speeds depend on temperature and molecular weight.
  • Relative speed (v_rel) is the speed of a given particle relative to others in the gas.
  • Consider a single moving particle in a sea of static particles; collisions occur if other particles enter a certain region around the moving particle.
  • Cross-sectional/collision area concept: define a cylindrical (or column-like) collision region as the particle moves.
  • Distance traveled during a time interval Δt is the speed times time: L = v_{\text{rel}} \; \Delta t
  • The collision region expands along the path of the particle; any particle within this region may collide.

Collision geometry and key quantities

  • Cross-sectional area: denote as σ (collision cross-section).
  • Volume of the collision cylinder: V{\text{cyl}} = σ \; v{\text{rel}} \; Δt
  • Number of particles in the collision cylinder: N{\text{cyl}} = n \; V{\text{cyl}} = n \; σ \; v_{\text{rel}} \; Δt
  • Collisions during time Δt: approximately equal to the number of particles in the cylinder, so
    N{\text{coll}} \approx n \; σ \, v{\text{rel}} \; Δt

Collision frequency and mean free path

  • Collision frequency (collisions per unit time):
    z = \frac{dN{\text{coll}}}{dt} = n \; σ \; v{\text{rel}}
  • Mean free path (average distance traveled before a collision): \lambda = \frac{v{\text{rel}}}{z} = \frac{v{\text{rel}}}{n \; σ \; v_{\text{rel}}} = \frac{1}{n \; σ}
    • Note: in this derivation speed cancels out, so the mean free path does not depend on speed in this simplified picture.
  • Additional context from the lecture:
    • Typical collision frequency: about once every nanosecond in typical gases.
    • Mean free path is roughly the average distance traveled between collisions; typical value is on the order of ~1000 particle sizes (a rough qualitative rule mentioned in class).
  • Relationship to density and pressure:
    • z increases with pressure (higher density means more collisions).
    • Higher density or higher pressure ⇒ higher collision frequency z.

From density to pressure: connecting to experimental parameters

  • Collision frequency can be expressed in terms of density (or number density) and temperature through kinetic theory relations.
  • Ideal-gas form to connect to experiments:
    • Ideal gas law (depending on formulation):
      P V = n R T
      or equivalently with number of molecules N:
      P V = N kB T where R = NA kB and n = N / NA.
  • The lecture notes mention converting between density-based expressions and pressures/temperatures to connect theory to experimental observables.
  • Units note: in the collision-frequency expression, the unit of z is s^{-1} (per second).

Real gas behavior vs. ideal gas behavior

  • When gases are compressed (higher density, smaller volume), molecules get closer and deviate from ideal (perfect) gas behavior.
  • At lower temperatures, kinetic energy decreases; deviations from ideal gas behavior become more pronounced.
  • Potential energy landscape between molecules:
    • Intermolecular potential energy vs. interparticle distance: as particles get closer, interactions become significant.
    • Far apart: negligible interactions (ideal-gas-like).
    • As distance decreases: attractive forces dominate first (negative potential energy), lowering potential energy.
    • At very short distances: repulsive forces dominate (positive potential energy) due to strong repulsion between closely packed particles.
  • Lennard-Jones picture (as discussed): captures both attractive and repulsive regions of interaction; the balance of these forces governs deviations from ideal gas behavior.
  • Important qualitative points shared:
    • The zero-crossing point of the potential is associated with the Van der Waals radius: a distance where attractive and repulsive contributions cancel.
    • The equilibrium distance is the distance at which the potential energy is minimized (net force is zero at that distance).
  • Isotherms (P vs V at constant T):
    • Under high temperature, the P–V curve resembles the ideal-gas behavior; pressure rises with compression roughly as Boyle’s law.
    • At lower temperatures, deviations become visible; the curve bends, showing non-ideal behavior.
    • As temperature decreases further and compression increases, the gas can condense into a liquid; this is the gas–liquid equilibrium region.
    • Liquids are much less compressible than gases.
  • Experimental isotherms for CO₂ (as discussed):
    • Lower temperatures widen the liquid–gas coexistence region on the P–V diagram.
    • The point where the liquid–gas boundary ends is the critical point characterized by critical temperature Tc, critical pressure Pc, and critical volume V_c.
  • Practical takeaway: real-gas corrections become necessary when compressing gases or lowering temperatures; we move beyond the ideal gas law to describe pressure for a real gas.
  • The lecturer noted that these real-gas corrections are often introduced via “fudge factors” to go from the ideal gas value to a real-gas pressure expression; more details would be discussed in a future session (Equation of state for real gases).

Key concepts and implications to remember

  • Maxwell distribution governs speeds, with temperature and molecular weight determining the typical speeds (
    speed increases with temperature and decreases with mass).
  • Relative speed v_{rel} is central to collision geometry and collision frequency.
  • Collision cross-section σ and number density n determine how often molecules collide; their product n σ v_{rel} sets the collision frequency z.
  • Mean free path λ = 1/(n σ) in the simplified treatment; speed cancels out, so λ is independent of v_{rel} in this derivation.
  • Higher pressure/density → higher collision frequency; gas behaves more non-ideally as density increases.
  • Real gases exhibit attractions at longer ranges and repulsions at short ranges; these interactions modify the P–V behavior, especially at lower temperatures, leading to condensation and a well-defined critical point.
  • The practical modeling approach for real gases involves adjusting the ideal-gas equation of state with corrections (the lecture hints at Van der Waals-like corrections) to account for finite molecular size and intermolecular forces.

Next steps mentioned in the session

  • More detailed discussion on the equation of state for real gases (how to modify pressure calculations beyond PV = nRT) in upcoming lectures.
  • Next class planned topics likely include critical constants (Tc, Pc, V_c) and further analysis of CO₂ isotherms and their implications.