2025 9 Kinetic Molecular Theory

Kinetic Molecular Theory

• Describes the behavior of gases

• Key assumptions include:

  • Gas molecules have tiny volumes compared to the collective volume they occupy.

  • They move constantly and randomly.

  • Average kinetic energy is proportional to absolute temperature; at a given temperature, all gases have the same average kinetic energy.

  • Gas molecules engage in elastic collisions with the walls of their container and with each other.

  • They act independently of each other.

Pressure

Definition

• Pressure (P) is defined as the ratio of force (F) exerted on a surface to the area (A) of that surface:[ P = \frac{F}{A} ]

• Atmospheric pressure is the force exerted by gases surrounding the Earth on its surface.

Units of Pressure

• SI units: (\text{Newton/meter}^2 = 1 \text{ Pascal (Pa)})

• Common conversions:

  • 1 atm = 101,325 Pa

  • 1 atm = 760 mmHg = 760 torr

Converting Units of Pressure

Table of Units

Boyle’s Law: Relating Pressure and Volume

Principle

• Pressure increases with the rate of molecular collisions.

• Gases are compressible: volume decreases as pressure increases.

• Boyle's Law states that for a constant temperature and number of moles, (PV = \text{constant}).

  • The value of the constant depends on the amount of gas and temperature.

Application

• The product (PV) remains unchanged for a given mass of gas at constant temperature:[ P_1V_1 = P_2V_2 ]

• Example: Used in gas cylinders for scuba diving.

Practice: Boyle’s Law

Calculating Volume Change

• Given:

  • Initial volume (V_1 = 5.00 L)

  • Initial pressure (P_1 = 1.00 atm)

  • Final pressure (P_2 = 0.83 atm)

• Find final volume (V_2) using Boyle's Law:[ V_2 = \frac{P_1V_1}{P_2} = \frac{(1.00)(5.00)}{0.83} = 6.0 L ]

Charles’s Law: Relating Volume and Temperature

Principle

• Charles's Law states that for a fixed quantity of gas at constant pressure, the volume is directly proportional to its temperature (in Kelvin):[ V \propto T ]

Practice: Charles's Law

Example Problem

• Initial volume (V_1 = 2.00 L), increased to (V_2 = 3.00 L) at initial temperature (T_1 = 15°C = 288 K).

• Solve for final temperature (T_2) using:[ T_2 = \frac{V_2 T_1}{V_1} = \frac{(3.00)(288)}{2.00} = 432 K]

• Convert (T_2) to Celsius: (T_2 = 432 - 273 = 159°C]

Avogadro’s Law: Relating Volume and Moles (n)

• States that volume is directly proportional to the number of moles of gas:[ V \propto n ]

• Increase in moles results in an increase in volume, demonstrating the expansion of gas.

Amonton’s Law: Relating Pressure and Temperature

• Demonstrates that pressure is proportional to temperature at constant volume:[ P \propto T ]

• Higher temperatures increase the average velocity of gas molecules, increasing pressure.

Ideal Gas Law

Equation

• The ideal gas law combines all the above relationships into:[ PV = nRT ]

  • Where:

    • (R = 0.08206 , L , atm/(K , mol))

    • (n = \text{moles of gas})

    • (T = \text{temperature in Kelvin})

Practice: Ideal Gas Law

Example Problem

• Calculate pressure of 1.2 mol of methane in a 3.3 L container at 25°C (298 K).[ P = \frac{(1.2)(0.08206)(298)}{3.3} = 8.9 , atm ]

Combined Gas Law

• Combines various gas laws:

  • [ \frac{PV}{T} = \text{constant} ]

• Represents the relationships between pressure, volume, and temperature of gases.

Reference Points for Gases

Standard Temperature and Pressure (STP)

• Defined as:

  • P = 1 atm; T = 273 K (0.0 °C)

• Molar volume of one mole of an ideal gas at STP is 22.4 L.

Practice: Stoichiometry Calculations

Example Problem

• Determine grams of NaClO3 needed to produce 125 L of O2 gas at 20.0 °C and 1.00 atm.

• Using ideal gas equation and stoichiometry,

  • Required (5.20 , mol O2) leads to (553 , g NaClO3).

Graham’s Law of Effusion and Diffusion

Definitions

• Effusion: Gas escaping through a tiny hole.

• Diffusion: Spread of one substance through another.

• Faster moving molecules diffuse and effuse more quickly.

Application of Graham's Law

• The diffusion rate of a gas is inversely proportional to the square root of its molar mass:[ r \propto \frac{1}{\sqrt{M}} ]

Real vs. Ideal Gases

• Ideal gas assumptions break down at high pressures and low volumes, where attractions between molecules become significant.

Summary

Practice Problems

• Helium balloon in hot trunk

• Balloon expands due to increased temperature (Charles’s Law).

• Helium tank pressure increases with temperature (Amonton’s Law).

• Kinetic energy of gas atoms increases with temperature; they move faster and collide more with the tank.