Kinetic Theory Of Gases

Critical temperature ($T_c$) is the temperature above which a gas cannot be liquefied regardless of applied pressure.

Relationship to molecular interactions:

• At low temperatures: attractive forces between molecules can overcome kinetic energy
• At high temperatures: kinetic energy dominates over attractive forces
• Critical temperature represents the balance point where:
  • Average kinetic energy = Critical attraction energy
  • $\frac{3}{2}kT_c \approx$ potential energy of molecular attraction

Liquefaction conditions:

• Below $T_c$: gas can be liquefied by pressure alone (called a vapor)
• At $T_c$: gas and liquid properties become indistinguishable
• Above $T_c$: no amount of pressure can cause liquefaction

Critical point characteristics:

• Defined by critical temperature ($T_c$), critical pressure ($P_c$), and critical volume ($V_c$)
• At critical point, the distinction between liquid and gas phases disappears
• Example: Water's critical temperature is 374.1°C

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 critical temperature of water is 374.1°C.

Physical significance:

• Below 374.1°C: Water vapor can be liquefied by increasing pressure without changing temperature
• At 374.1°C: The distinction between liquid and gas phases vanishes
• Above 374.1°C: No amount of pressure can cause water to separate into distinct gas and liquid phases
• At critical temperature, the density of saturated vapor equals the density of liquid
• The meniscus between phases disappears at this temperature
• Water above critical temperature but below critical pressure is properly called "water gas" rather than "water vapor"

This represents the temperature above which the kinetic energy of molecules always exceeds the intermolecular attractive forces that would allow liquid formation.

Saturated vapor represents a dynamic equilibrium where:

• The space above a liquid contains the maximum possible amount of vapor molecules at a given temperature

• The rate of molecules escaping the liquid (evaporation) exactly equals the rate of molecules returning to the liquid (condensation)

• Adding more vapor molecules forces excess vapor to condense into liquid

• The maximum vapor content increases with temperature because higher molecular kinetic energy increases evaporation rate, requiring more vapor molecules for equilibrium

• This equilibrium pressure exerted by the vapor is called the "vapor pressure" of the liquid at that temperature

This process underlies phenomena such as humidity limits and pressure cooker operations.

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

Boiling vs. Evaporation:

• Evaporation: Only molecules near the surface with kinetic energy greater than average escape the liquid
• Boiling: Molecules throughout the entire liquid gain enough energy to form vapor bubbles that rise and escape

Relationship with pressure:

• Boiling occurs when SVP equals external pressure
• Higher external pressure requires higher temperature to achieve sufficient SVP
• Lower external pressure allows boiling at lower temperatures

Example: Water boils at 100°C at 1 atm, but at only 82°C at 0.5 atm pressure because less thermal energy is needed for molecules to overcome the lower external pressure

Saturation vapor pressure (SVP):

• Maximum pressure exerted by vapor in equilibrium with its liquid phase at a given temperature
• Represents dynamic equilibrium where rate of evaporation equals rate of condensation

Why SVP increases with temperature:

• Higher temperature means higher average molecular kinetic energy
• More molecules have sufficient energy to escape liquid phase
• Greater number of vapor molecules creates higher pressure

Why SVP is independent of liquid amount:

• Depends only on rate processes at the liquid-vapor interface
• Adding more liquid increases surface area slightly but doesn't change energy distribution of molecules
• Dynamic equilibrium is reached at the same pressure regardless of liquid volume

Example: Water at 20°C has SVP of 17.5 mmHg whether there's a drop or a liter present

• Methyl alcohol reaches a saturation vapor pressure of 760 mm Hg at a lower temperature than water • This occurs because methyl alcohol has weaker intermolecular forces (primarily hydrogen bonding) than water • Water molecules form stronger and more extensive hydrogen bonding networks • Practical implications:

• Methyl alcohol evaporates more quickly than water at room temperature
• Methyl alcohol is more volatile and has higher vapor concentrations in air
• In industrial distillation, methyl alcohol can be separated from water by heating to temperatures between their boiling points
• Methyl alcohol works more effectively as a quick-drying solvent
• When designing cooling systems, methyl alcohol reaches vapor state more easily than water

The relationship between dew point, saturation vapor pressure, and relative humidity:

• Relative Humidity (RH) = (SVP at dew point / SVP at air temperature) × 100%

Where:

• SVP = Saturation Vapor Pressure
• The SVP at dew point equals the actual vapor pressure in the air
• At the dew point, air becomes saturated with water vapor (100% relative humidity)

Example: If dew point is 10°C (SVP = 8.94 mmHg) and air temperature is 20°C (SVP = 17.5 mmHg), then: RH = (8.94/17.5) × 100% = 51.1%

This works because vapor pressure at dew point represents the actual amount of water vapor present, while SVP at air temperature represents the maximum possible amount at current temperature.