Unit 1 Kinetic Theory of Gases

KINETIC THEORY OF MATTER

Overview

• Presented by Mr. W. Williams.

STATES OF MATTER

COMPARING STATES

Properties of Different States

PROPERTIES OF GAS

Describing Gases

• Gases are characterized based on four properties:

  • Pressure (P)

  • Definition: The force exerted by gas against the walls of its container.

  • Units: atmosphere (atm), mm Hg, torr, pascal.

  • Volume (V)

  • Definition: The space occupied by the gas.

  • Units: liter (L), milliliter (mL).

  • Temperature (T)

  • Definition: Determines the kinetic energy and rate of motion of gas particles.

  • Units: Celsius (°C), Kelvin (K).

  • Amount (n)

  • Definition: The quantity of gas present in a container.

  • Units: grams (g), moles (n, required in calculations).

GAS LAWS

Boyle’s Law (BeT)

• Formula: At a fixed mass of gas at a constant temperature, the pressure (P) is inversely proportional to the volume (V).

  • Equation: $P imes V = ext{constant}$

  • Derived formula: $P_1V_1 = P_2V_2$

Examples Illustrating Boyle's Law:

1. A sample of oxygen gas has:

   • Volume, $V_1 = 225 ext{ mL}$ at pressure, $P_1 = 1.12 ext{ atm}$.

   • To find volume at pressure $P_2 = 0.98 ext{ atm}$:

     • Use: $P_1V_1 = P_2V_2$.

     • Calculation: $V_2 = rac{P_1V_1}{P_2} = rac{1.12 imes 225}{0.98} = 257.14 ext{ mL}$.

2. A gas with:

   • Volume, $V_1 = 1 ext{ L}$ and pressure, $P_1 = 400 ext{ kPa}$ is transferred to a volume, $V_2 = 3 ext{ L}$:

     • Calculation: $P_2 = rac{P_1V_1}{V_2} = rac{400 imes 1}{3} = 133.33 ext{ kPa}$.

Charles’ Law (CeP)

• Formula: At a fixed mass of gas at a constant pressure, the volume (V) is directly proportional to the temperature (T).

  • Equation: $rac{V_1}{T_1} = rac{V_2}{T_2}$.

  • Remember to convert temperatures to Kelvin:

  • $T(K) = T(°C) + 273$.

Examples Illustrating Charles’ Law:

1. Volume of gas at 20 °C (293 K) is 600 mL, find volume at 60 °C (333 K):

   • Calculation: $V_2 = rac{V_1 imes T_2}{T_1} = rac{600 imes 333}{293} = 681.9 ext{ mL}$.

2. Weather balloon:

   • Initial conditions: $V_1 = 5000 ext{ cm}^3$, $T_1 = 32°C (305 K)$; find temperature when volume is $V_2 = 4400 ext{ cm}^3$:

   • Calculate: $T_2 = rac{V_2 imes T_1}{V_1} = rac{4400 imes 305}{5000} = 268.4 K (-4.6 °C)$.

Pressure Law

• Formula: At a constant volume, the pressure (P) is directly proportional to the temperature (T).

Combined Gas Law

• Formulation: $rac{P_1V_1}{T_1} = rac{P_2V_2}{T_2}$.

Examples Illustrating Combined Gas Law:

1. A gas has volume of 800.0 mL at -23.0 °C and pressure of 300.0 torr; find its volume at 227.0 °C and 600.0 torr:

   • Use: $V_2 = rac{P_1V_1T_2}{P_2T_1}$.

   • Calculation results: Should lead to recalculating volume.

2. Balloon initially in a freezer:

   • Initial volume 9.40 L, pressure 0.939 atm, need to find initial temperature for volume 10.0 L at pressure 1.00 atm:

   • $T_1 = rac{P_2V_2T_1}{P_1V_1}$

   • Results yield: $T_1 = 263 K$.

IDEAL GASES

Assumptions of Ideal Gases (VMIKE)

1. Volume: The volume of individual gas molecules is negligible compared to the total volume.

2. Motion: Molecules are in constant, random motion.

3. Interatomic Forces: Negligible forces between molecules except during collisions.

4. Kinetic Energy: Average kinetic energy depends only on temperature.

5. Elastic Collisions: Collisions of gas particles with container walls are perfectly elastic.

Real Gases vs. Ideal Gases

Conditions Where Real Gases Behave Like Ideal

• High Temperatures and Low Pressures:

  • At high temperatures, particles possess more kinetic energy and move faster.

  • At low pressures, particles are further apart, weakening interatomic forces.

Conditions Where Real Gases Deviate From Ideal

• Low Temperatures and High Pressures:

  • At low temperatures, particles have less kinetic energy, moving slower.

  • At high pressures, particles are closer together, strengthening interatomic forces.

Ideal Gas Equation

• Formulation: $PV = nRT$

  • Where:

  • $P$ = pressure (Pa)

  • $V$ = volume (m³)

  • $n$ = number of moles

  • $R$ = gas constant (8.31 J K⁻¹ mol⁻¹)

  • $T$ = temperature (K)

Conversions and Applications

• Moles: $n = rac{m}{M_r}$

• Substituting into ideal gas equation:

  • $PV = rac{mRT}{M_r}$

• For pressure in kPa, volume must be in