Kinetic Theory Of Gases

Starting with a molecule hitting a wall:

1. Change in momentum of one molecule: $\Delta p = 2mv_x$
2. Time between collisions with same wall: $\Delta t = 2L/v_x$
3. Force from one molecule: $F_1 = \frac{mv_x^2}{L}$
4. Total force on wall: $F = \frac{m}{L}\sum v_x^2$
5. Since all directions are equivalent: $\sum v_x^2 = \frac{1}{3}\sum v^2$
6. Force becomes: $F = \frac{1}{3}\frac{mN}{L}\frac{\sum v^2}{N}$
7. Pressure: $p = \frac{F}{L^2} = \frac{1}{3}\rho\overline{v^2}$

Final expression: $p = \frac{1}{3}\rho\overline{v^2}$ or $pV = \frac{1}{3}Nmv_{rms}^2$

Upon elastic collision with a wall perpendicular to the x-axis:

• vₓ reverses direction: vₓ → -vₓ
• vᵧ remains unchanged
• vᵣ remains unchanged

This selective reversal occurs because:

• Only the momentum component perpendicular to the surface changes
• The collision force acts normal to the surface
• Tangential components experience no force in an ideal, frictionless collision

The momentum change equals Δp = 2mvₓ, which contributes to pressure when multiplied by collision frequency.

Temperature effects on Maxwell speed distribution:

• As temperature increases:

  • The distribution curve flattens and widens
  • The peak shifts to higher speeds
  • The "tail" extends further to higher speeds

• Quantitative relationships:

  • Most probable speed: $v_p = \sqrt{\frac{2kT}{m}} = \sqrt{\frac{2RT}{M}}$
  • Average speed: $\bar{v} = \sqrt{\frac{8kT}{\pi m}} = \sqrt{\frac{8RT}{\pi M}}$
  • RMS speed: $v_{rms} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}}$

• All three characteristic speeds are proportional to $\sqrt{T}$

• Doubling the absolute temperature increases all speeds by a factor of $\sqrt{2}$ ≈ 1.41

• Relationship between speeds: $v_p < \bar{v} < v_{rms}$

Example: Nitrogen molecules at 20°C have an rms speed of about 500 m/s, but heating the gas to 500°C would increase this to approximately 810 m/s.

The shape variation of Maxwell speed distribution with temperature reflects fundamental thermodynamic principles:

• Physical explanation for shape changes:

  • Higher temperature means greater average kinetic energy (KE = 3kT/2)
  • Energy is randomly distributed among molecules through collisions
  • More thermal energy creates wider spread of possible molecular speeds

• Fast-moving molecules in the distribution "tail":

  • Result of random energy exchange during molecular collisions
  • Some molecules gain energy from multiple consecutive collisions
  • Statistical probability predicts some molecules will have energies many times the average
  • These high-speed outliers are crucial for chemical reactions with activation energies

• Boltzmann factor (e^(-E/kT)) governs the probability of finding a molecule with energy E

  • Explains the exponential decay in the high-speed tail
  • Higher temperatures make high-energy states more accessible

Example: At room temperature, while the most probable speed of oxygen molecules is around 400 m/s, a small but significant fraction moves faster than 1000 m/s, enabling reactions that wouldn't occur if all molecules moved at the average speed.

At 350°C (below critical temperature):

• Initially exists as gas at low pressure
• At 163 atm, liquefaction begins
• Volume rapidly decreases during phase transition
• Eventually becomes completely liquid with high resistance to further compression

At 380°C (above critical temperature):

• No distinct phase transition occurs regardless of pressure
• The substance continuously compresses without suddenly changing volume
• Cannot be liquefied by pressure alone
• Exhibits supercritical fluid behavior

This difference demonstrates the concept of critical temperature (374.1°C for water), above which the distinction between liquid and gas phases disappears.

When temperature falls below the dew point:

• Vapor begins to condense
• Water droplets form on surfaces or dust particles

The mathematical relationship to relative humidity:

• RH = (SVP at dew point / SVP at air temperature) × 100%
• Where SVP = saturation vapor pressure

Example: If dew forms at 10°C when air temperature is 40°C, and SVP at 10°C is 8.94 mmHg while SVP at 40°C is 55.1 mmHg, then: RH = (8.94/55.1) × 100% = 16.2%

Triple point: The unique pressure and temperature where solid, liquid, and vapor phases of a substance coexist in equilibrium.

Water triple point:

• Temperature: 273.16 K (0.01°C)
• Pressure: 4.58 mmHg
• All three phases can exist at this specific combination

Carbon dioxide triple point:

• Temperature: 216.55 K (-56.6°C)
• Pressure: 5.11 atm (much higher than atmospheric pressure)
• At atmospheric pressure, CO₂ can only exist as solid or gas
• This explains why solid CO₂ (dry ice) sublimates directly to vapor without passing through liquid phase

Molecular mechanism leading to saturation:

1. Initially: Molecules are far apart; attraction energy << kinetic energy
2. As more vapor is added: Average separation decreases
3. Critical point: Attraction energy becomes sufficient to draw molecules close enough to form liquid
4. Result: Some vapor condenses to liquid phase (saturation)

Pressure stabilization explanation:

• Once saturation occurs, any additional vapor injected will condense to liquid
• The number of molecules in vapor phase remains constant
• Pressure stabilizes at the saturation vapor pressure (SVP)
• Dynamic equilibrium: Rate of evaporation equals rate of condensation

This explains why vapor pressure cannot exceed SVP at a given temperature, regardless of how much additional vapor is injected

Dew formation:

• Occurs when surface objects (grass, windows, flowers) cool below air's dew point
• Cooling typically happens through radiation heat loss at night
• Condensation forms directly on the cooled surfaces
• Requires temperature gradient where surfaces are colder than surrounding air

Fog formation:

• Occurs when the entire air mass cools to below its dew point
• Water vapor condenses around suspended dust/aerosol particles
• Droplets remain suspended in air rather than depositing on surfaces
• Creates visible mist that reduces visibility

Common thermodynamic principle: Both phenomena involve cooling air to saturation (100% relative humidity) followed by condensation, but differ in the spatial distribution of cooling

The triple point is the unique combination of temperature and pressure at which all three phases of matter (solid, liquid, and vapor) can coexist in thermodynamic equilibrium.

Key characteristics:

• Represents a single point on a pressure-temperature phase diagram
• Each substance has its own unique triple point values
• For water: 273.16 K (0.01°C) and 4.58 mm of mercury
• For carbon dioxide: 216.55 K and 5.11 atm

At the triple point, any slight change in temperature or pressure will cause some portion of the substance to transform into another phase while maintaining equilibrium.