Kinetic Theory of Gases: HRK Physics Chapter 23 Complete Notes & Solutions

House of Physics October 01, 2025
Kinetic Theory and The Ideal Gas: Complete Physics Guide | HRK Chapter 23
Mastering Thermodynamics, Gas Laws, Molecular Motion, and Real Gas Behavior
Kinetic Theory Ideal Gas Law Thermodynamics Molecular Motion RMS Speed Reading Time: 25 min

📋 Table of Contents

📜 Historical Background

The kinetic theory of gases developed over several centuries, with contributions from many scientists:

  • Robert Boyle (1662): Established Boyle's Law relating pressure and volume
  • Daniel Bernoulli (1738): First proposed kinetic theory, explaining pressure as molecular impacts
  • John Dalton (1803): Atomic theory provided foundation for molecular approach
  • James Joule (1840s): Established mechanical equivalent of heat
  • Rudolf Clausius (1850s): Developed kinetic theory mathematically
  • James Clerk Maxwell (1860s): Derived Maxwell-Boltzmann distribution of molecular speeds
  • Ludwig Boltzmann (1870s): Connected entropy with probability

These developments transformed our understanding of heat from a fluid ("caloric") to molecular motion.

Introduction to Kinetic Theory

🔬 What is Kinetic Theory?

Kinetic theory provides a molecular explanation for the behavior of gases, connecting microscopic molecular motion to macroscopic thermodynamic properties like pressure, temperature, and volume.

While thermodynamics deals with relationships between macroscopic properties, kinetic theory explains these relationships in terms of the motion and interactions of molecules.

📝 Two Approaches to Averaging

  • Kinetic Theory: Follows motion of representative particles and averages this behavior
  • Statistical Mechanics: Applies probability laws to statistical distributions of molecular properties

Both approaches yield the same results for systems with large numbers of particles.

Thermodynamics and Temperature

🔬 Thermodynamics Definition

Thermodynamics is the branch of physics that deals with the conversion of heat energy into mechanical energy and vice versa.

🌡️ Temperature Definition

Temperature is the degree of hotness or coldness of a body. More precisely, temperature is a physical property that determines the direction of heat flow when two bodies are in thermal contact.

Temperature Scales Conversion

Q #1: Temperature Conversion

One day when temperature is 15°F. What will be the temperature on Celsius and Kelvin scale?

Given: F = 15°F
To Find: C = ?, K = ?
Celsius scale conversion:
\[ C = \frac{5}{9} (F - 32) \]
\[ = \frac{5}{9} (15 - 32) \]
\[ = \frac{5}{9} \times (-17) \]
\[ = -9.4^\circ C \]
Kelvin scale conversion:
\[ K = C + 273 \]
\[ = -9.4 + 273 \]
\[ = 263.6 \, K \]

State Variables

📊 State Variables (Thermodynamic Coordinates)

State variables are physical quantities that describe the state of a system. Examples include:

  • Pressure (P)
  • Volume (V)
  • Temperature (T)
  • Internal Energy (U)
  • Entropy (S)

These variables completely define the thermodynamic state of a system.

Ideal Gas and Equation of State

🔬 Ideal Gas Definition

An ideal gas is one that obeys gas laws under all temperatures and pressures. In an ideal gas:

  • Molecules have no potential energy, only kinetic energy
  • Molecular size is negligible compared to intermolecular distances
  • Molecular collisions are perfectly elastic
  • Intermolecular forces are negligible except during collisions

Boyle's, Charles's, and Gay-Lussac's Laws

🧮 Gas Laws Foundation

Boyle's Law

At constant temperature, volume is inversely proportional to pressure:

\[ V \propto \frac{1}{P} \]

Charles's Law

At constant pressure, volume is directly proportional to absolute temperature:

\[ V \propto T \]

Gay-Lussac's Law

At constant pressure and temperature, volume is directly proportional to number of molecules:

\[ V \propto N \]

Ideal Gas Law Derivation

🧮 Derivation of PV = nRT

Step 1: Combine Gas Laws

\[ V \propto \frac{TN}{P} \]
\[ V = \text{constant} \frac{TN}{P} \]

Step 2: Introduce Boltzmann Constant

\[ PV = k_B NT \]

where \( k_B = 1.38066 \times 10^{-23} \, \text{J/K} \) is Boltzmann's constant

Step 3: Express in Terms of Moles

\[ N = nN_A \]
\[ PV = k_B (nN_A)T \]
\[ PV = n(N_A k_B)T \]

Step 4: Universal Gas Constant

\[ R = N_A k_B = 8.3145 \, \text{J/mol·K} \]
\[ PV = nRT \]
Q #2: Ideal Gas Law Application

A 2.5 liter container holds 0.5 moles of gas at 27°C. What is the pressure of the gas?

Given:
\[ V = 2.5 \, \text{L} = 2.5 \times 10^{-3} \, \text{m}^3 \]
\[ n = 0.5 \, \text{mol} \]
\[ T = 27^\circ C = 300 \, K \]
\[ R = 8.314 \, \text{J/mol·K} \]
Using ideal gas law:
\[ PV = nRT \]
\[ P = \frac{nRT}{V} \]
\[ = \frac{0.5 \times 8.314 \times 300}{2.5 \times 10^{-3}} \]
\[ = \frac{1247.1}{0.0025} \]
\[ = 498840 \, \text{Pa} \]
\[ = 4.99 \times 10^5 \, \text{Pa} \]

Kinetic Theory of Gases

🔍 Molecular Motion Visualization

Gas Molecule Behavior

[Molecular Diagram: Random motion of gas molecules in a container]

Molecular Characteristics

  • Molecules move randomly in all directions
  • Molecular speeds follow Maxwell-Boltzmann distribution
  • Collisions with walls create pressure
  • Average kinetic energy depends on temperature

Basic Assumptions

📝 Kinetic Theory Assumptions

  • A gas consists of a large number of molecules in random motion
  • Molecular size is negligible compared to intermolecular distances
  • Molecules exert no forces on each other except during collisions
  • Collisions between molecules and with walls are perfectly elastic
  • Molecular motion follows Newton's laws of motion
  • Time during collision is negligible compared to time between collisions

Pressure Derivation

🧮 Derivation of Pressure Formula

Step 1: Force on Wall from One Molecule

Consider a molecule with velocity vx hitting a wall perpendicular to x-axis:

\[ \text{Momentum change} = mv_x - (-mv_x) = 2mv_x \]

Step 2: Time Between Collisions

Time between successive collisions with same wall:

\[ \Delta t = \frac{2L}{v_x} \]

Step 3: Force from One Molecule

\[ F = \frac{\Delta p}{\Delta t} = \frac{2mv_x}{2L/v_x} = \frac{mv_x^2}{L} \]

Step 4: Total Force from All Molecules

\[ F_{\text{total}} = \sum \frac{mv_{xi}^2}{L} = \frac{m}{L} \sum v_{xi}^2 \]

Step 5: Average Square Velocity

\[ \langle v_x^2 \rangle = \frac{1}{N} \sum v_{xi}^2 \]
\[ \sum v_{xi}^2 = N \langle v_x^2 \rangle \]

Step 6: Pressure Calculation

\[ P = \frac{F}{A} = \frac{F}{L^2} = \frac{m}{L^3} N \langle v_x^2 \rangle \]
\[ = \frac{Nm}{V} \langle v_x^2 \rangle \]

Step 7: Three-Dimensional Motion

\[ \langle v^2 \rangle = \langle v_x^2 \rangle + \langle v_y^2 \rangle + \langle v_z^2 \rangle \]
\[ \langle v_x^2 \rangle = \langle v_y^2 \rangle = \langle v_z^2 \rangle = \frac{1}{3} \langle v^2 \rangle \]

Step 8: Final Pressure Formula

\[ P = \frac{1}{3} \frac{Nm}{V} \langle v^2 \rangle \]
\[ = \frac{1}{3} \rho \langle v^2 \rangle \]

where ρ = Nm/V is the gas density

Molecular Speeds and Temperature

🔬 Root Mean Square Speed

The root mean square (RMS) speed is defined as the square root of the average of the squares of molecular speeds:

\[ v_{\text{rms}} = \sqrt{\langle v^2 \rangle} \]

This is the most probable measure of molecular speed in a gas.

Root Mean Square Speed

🧮 Derivation of RMS Speed

Step 1: From Pressure Equation

\[ P = \frac{1}{3} \frac{Nm}{V} \langle v^2 \rangle \]

Step 2: Express in Terms of RMS Speed

\[ \langle v^2 \rangle = v_{\text{rms}}^2 \]
\[ P = \frac{1}{3} \frac{Nm}{V} v_{\text{rms}}^2 \]

Step 3: Relate to Ideal Gas Law

\[ PV = Nk_B T \]
\[ \frac{1}{3} \frac{Nm}{V} v_{\text{rms}}^2 \cdot V = Nk_B T \]
\[ \frac{1}{3} Nm v_{\text{rms}}^2 = Nk_B T \]

Step 4: Solve for RMS Speed

\[ v_{\text{rms}}^2 = \frac{3k_B T}{m} \]
\[ v_{\text{rms}} = \sqrt{\frac{3k_B T}{m}} \]

Step 5: Express in Terms of Molar Mass

\[ m = \frac{M}{N_A}, \quad k_B = \frac{R}{N_A} \]
\[ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]

where M is the molar mass of the gas

Kinetic Interpretation of Temperature

🧮 Temperature and Kinetic Energy

Step 1: From RMS Speed Equation

\[ \frac{1}{2} m v_{\text{rms}}^2 = \frac{3}{2} k_B T \]

Step 2: Average Kinetic Energy

\[ \langle K \rangle = \frac{1}{2} m \langle v^2 \rangle = \frac{1}{2} m v_{\text{rms}}^2 \]
\[ = \frac{3}{2} k_B T \]

Step 3: Important Conclusion

\[ \langle K \rangle = \frac{3}{2} k_B T \]

The average translational kinetic energy of gas molecules is directly proportional to the absolute temperature and is independent of the nature of the gas.

Q #3: RMS Speed Calculation

Calculate the RMS speed of oxygen molecules at 27°C. (Molar mass of oxygen = 32 g/mol)

Given:
\[ T = 27^\circ C = 300 \, K \]
\[ M = 32 \, \text{g/mol} = 0.032 \, \text{kg/mol} \]
\[ R = 8.314 \, \text{J/mol·K} \]
Using RMS speed formula:
\[ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]
\[ = \sqrt{\frac{3 \times 8.314 \times 300}{0.032}} \]
\[ = \sqrt{\frac{7482.6}{0.032}} \]
\[ = \sqrt{233831.25} \]
\[ = 483.56 \, \text{m/s} \]

📈 Maxwell-Boltzmann Speed Distribution

[Graph: Maxwell-Boltzmann distribution showing molecular speed distribution at different temperatures]

The graph shows how molecular speeds are distributed in a gas. As temperature increases, the distribution shifts to higher speeds and becomes broader.

Internal Energy of Gases

🔬 Internal Energy Definition

The internal energy of a gas is the sum of the kinetic and potential energies of all its molecules. For an ideal gas, where intermolecular forces are negligible, the internal energy consists only of the kinetic energy of molecular motion.

Monatomic vs Diatomic Gases

Gas Type Degrees of Freedom Internal Energy per Mole Specific Heats
Monatomic (He, Ne, Ar) 3 (translational) U = (3/2)RT CV = (3/2)R, CP = (5/2)R
Diatomic (O2, N2) 5 (3 trans + 2 rot) U = (5/2)RT CV = (5/2)R, CP = (7/2)R
Polyatomic (CO2, H2O) 6 (3 trans + 3 rot) U = 3RT CV = 3R, CP = 4R

Degrees of Freedom

📝 Equipartition Theorem

The equipartition theorem states that each degree of freedom contributes (1/2)kBT to the average energy per molecule, or (1/2)RT per mole.

Degrees of Freedom:

  • Translational: 3 (motion along x, y, z axes)
  • Rotational: 2 for diatomic, 3 for polyatomic molecules
  • Vibrational: Additional degrees at high temperatures
Q #4: Internal Energy Calculation

Calculate the internal energy of 2 moles of helium gas at 27°C.

Given:
\[ n = 2 \, \text{mol} \]
\[ T = 27^\circ C = 300 \, K \]
Helium is monatomic gas
For monatomic gas:
\[ U = \frac{3}{2} nRT \]
\[ = \frac{3}{2} \times 2 \times 8.314 \times 300 \]
\[ = 3 \times 8.314 \times 300 \]
\[ = 7482.6 \, \text{J} \]

Real Gases: Van der Waals Equation

🔬 Real Gas Behavior

Real gases deviate from ideal gas behavior at high pressures and low temperatures due to:

  • Finite molecular size (excluded volume)
  • Intermolecular attractive forces

🧮 Van der Waals Equation

Volume Correction

Available volume for molecular motion is less than container volume:

\[ V_{\text{available}} = V - nb \]

where b is the volume excluded by one mole of molecules

Pressure Correction

Intermolecular attractions reduce the pressure:

\[ P_{\text{actual}} = P + \frac{an^2}{V^2} \]

where a is a constant depending on intermolecular forces

Van der Waals Equation

\[ \left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT \]

📉 Real Gas vs Ideal Gas Behavior

[Graph: PV vs P curves showing deviation of real gases from ideal gas behavior]

The graph shows how real gases deviate from ideal gas behavior, especially at high pressures and low temperatures.

Work Done on Ideal Gas

🧮 Work in Thermodynamic Processes

Definition of Work

Work done on a gas during a volume change:

\[ dW = -P dV \]

The negative sign indicates that work is done on the gas when volume decreases.

Work in Different Processes

Isobaric Process (Constant Pressure):

\[ W = -P \Delta V \]

Isothermal Process (Constant Temperature):

\[ W = -nRT \ln\left(\frac{V_f}{V_i}\right) \]

Adiabatic Process (No Heat Exchange):

\[ W = \frac{P_i V_i - P_f V_f}{\gamma - 1} \]

where γ = CP/CV is the adiabatic index

Frequently Asked Questions

❓ Why do real gases deviate from ideal gas behavior?

Real gases deviate from ideal behavior because:

  • Molecular Volume: Gas molecules have finite size, reducing the available volume
  • Intermolecular Forces: Attractive forces between molecules reduce the pressure
  • These effects become significant at high pressures and low temperatures

The van der Waals equation accounts for these deviations.

❓ What is the physical significance of the RMS speed?

The root mean square (RMS) speed represents the square root of the average of the squares of molecular speeds. It's significant because:

  • It directly relates to the kinetic energy of gas molecules
  • It appears in the pressure equation derived from kinetic theory
  • It's proportional to the square root of absolute temperature
  • It's inversely proportional to the square root of molecular mass

For oxygen at room temperature, vrms ≈ 480 m/s, while for hydrogen it's about 4 times faster.

❓ How does temperature relate to molecular motion?
Temperature Molecular Motion Average Kinetic Energy
Absolute Zero (0 K) Theoretical minimum motion Minimum possible
Low Temperature Slow molecular motion Low kinetic energy
Room Temperature (300 K) Fast molecular motion ~6.21 × 10-21 J per molecule
High Temperature Very fast molecular motion High kinetic energy

Temperature is a measure of the average translational kinetic energy of molecules: ⟨K⟩ = (3/2)kBT

📚 Master Thermodynamics and Kinetic Theory

Understanding kinetic theory is fundamental to thermodynamics, statistical mechanics, and many engineering applications. Continue your journey into the fascinating world of molecular physics.

Read More: Physics HRK Notes of Thermodynamics

© House of Physics | HRK Physics Chapter 23: Kinetic Theory and The Ideal Gas

Based on Halliday, Resnick, and Krane's "Physics" with additional insights from university physics curriculum

House of Physics | Contact: aliphy2008@gmail.com

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