Kinetic Theory Of Gases

Relationship: $T \propto \overline{v^2}$ for a given gas

Key equations:

• Average kinetic energy per molecule: $\frac{1}{2}mv_{rms}^2 = \frac{3}{2}kT$
• Total kinetic energy: $K = \frac{3}{2}NkT = \frac{3}{2}nRT$

Physical significance:

• Absolute temperature is directly proportional to average translational kinetic energy
• Different gases at same temperature have same average molecular kinetic energy
• Heavier molecules move with smaller rms speeds to maintain the same kinetic energy
• Temperature is a measure of the intensity of random molecular motion
• At thermal equilibrium, energy is shared evenly among all molecules regardless of mass

Derivation:

1. From kinetic theory: $pV = \frac{1}{3}Nmv_{rms}^2$
2. From ideal gas law: $pV = NkT$
3. Equating these: $\frac{1}{3}Nmv_{rms}^2 = NkT$
4. Solving: $v_{rms} = \sqrt{\frac{3kT}{m}}$

Alternative forms:

• $v_{rms} = \sqrt{\frac{3RT}{M}}$ where $M$ is molecular weight
• $v_{rms} = \sqrt{\frac{3p}{\rho}}$ in terms of gas density

For different gases:

• At same temperature, $v_{rms} \propto \frac{1}{\sqrt{M}}$
• Lighter gases have higher rms speeds
• Example: Hydrogen molecules move approximately 4 times faster than oxygen molecules at the same temperature

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.

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.

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

• Saturation vapor pressure increases exponentially with temperature for liquids • The relationship is non-linear, with the rate of increase accelerating at higher temperatures • Different substances have different SVP curves at the same temperature • For example, methyl alcohol has a significantly higher SVP than water at the same temperature • This exponential relationship can be described by the Clausius-Clapeyron equation: ln(P₂/P₁) = (ΔH_vap/R)×(1/T₁ - 1/T₂) where ΔH_vap is the enthalpy of vaporization and R is the gas constant

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.

The boiling point of a substance is determined by the temperature at which its saturation vapor pressure equals the external pressure. On a phase diagram:

• Boiling occurs along the liquid-vapor phase boundary (curve PA in phase diagrams)
• Each point on this curve represents a different boiling point at a specific pressure
• Increasing external pressure raises the boiling point
• Decreasing external pressure lowers the boiling point

For water at standard pressure (1 atm), the boiling point is 100°C, but in a pressure cooker (higher pressure), water boils at higher temperatures (enabling faster cooking).

The liquid-vapor boundary terminates at the critical point, beyond which the distinction between liquid and gas phases disappears, forming a supercritical fluid.

Maxwell's Speed Distribution Law describes the statistical distribution of molecular speeds in a gas at thermal equilibrium:

Mathematical Formulation

$\frac{dN}{dv} = 4\pi N\left(\frac{m}{2\pi kT}\right)^{3/2}v^2e^{-mv^2/2kT}$

Where:

• $dN/dv$ = number of molecules with speeds between $v$ and $v+dv$
• $N$ = total number of molecules
• $m$ = molecular mass
• $k$ = Boltzmann constant
• $T$ = absolute temperature

Key Features

1. Distribution Shape:

   • Asymmetric (skewed right)
   • Starts at zero (no molecules with zero speed)
   • Rises to a peak, then falls with a long tail at higher speeds

2. Characteristic Speeds:

   • Most probable speed: $v_p = \sqrt{\frac{2kT}{m}}$
   • Average speed: $\bar{v} = \sqrt{\frac{8kT}{\pi m}}$
   • Root mean square speed: $v_{rms} = \sqrt{\frac{3kT}{m}}$
   • Relationship: $v_p < \bar{v} < v_{rms}$

3. Temperature Effects:

   • Distribution broadens at higher temperatures
   • Peak shifts to higher speeds as temperature increases
   • Area under the curve always equals total number of molecules

Example: In hydrogen gas at room temperature (~300K), the average speed is ~1800 m/s, but the distribution shows significant numbers of molecules moving both much faster and slower than this average.