Class 11 Physics Ideal Gas Notes

Chapter Overview

An ideal gas is a theoretical gas whose molecules are treated as point particles with negligible molecular volume and no intermolecular attraction except during collisions. Real gases approach ideal-gas behavior most closely at low pressure and high temperature.

This chapter connects the experimental gas laws with the microscopic kinetic model of matter. The central ideas are the ideal-gas equation, absolute temperature, molecular motion, pressure due to molecular collisions, translational kinetic energy, the Boltzmann constant, root mean square speed, and heat capacities.

Macroscopic View

Describes a gas by measurable quantities such as pressure P, volume V, temperature T, and amount n.

Microscopic View

Explains gas behavior using molecules, molecular mass, random motion, collisions, momentum, and kinetic energy.

1. Gas Laws and the Ideal-Gas Equation

1.1 Boyle’s Law

For a fixed mass of gas at constant temperature, pressure is inversely proportional to volume.

P ∝ 1/V ⇒ PV = constant ⇒ P₁V₁ = P₂V₂

Diagram 1 — Boyle’s Law: Pressure–Volume Graph

At constant temperature, increasing volume lowers pressure; the P–V curve is a rectangular hyperbola.

1.2 Charles’s Law

For a fixed mass of gas at constant pressure, volume is directly proportional to its absolute temperature.

V ∝ T ⇒ V/T = constant ⇒ V₁/T₁ = V₂/T₂

If temperature is measured in degree Celsius and the graph is extrapolated, the volume tends toward zero near −273.15 °C. This defines absolute zero, or 0 K.

Diagram 2 — Charles’s Law and Absolute Zero

Extrapolation of V against Celsius temperature gives zero volume at about −273.15 °C.

1.3 Pressure Law and Equality of Gas Coefficients

At constant volume, the pressure of a fixed mass of gas is directly proportional to absolute temperature:

P ∝ T ⇒ P/T = constant

For an ideal gas near 0 °C, the volume coefficient at constant pressure and pressure coefficient at constant volume both have the limiting value approximately:

α_V = α_P ≈ 1/273.15 K⁻¹

Diagram 3 — Pressure–Temperature Graph

At constant volume, pressure increases linearly with absolute temperature.

1.4 Combined Gas Equation and Ideal-Gas Equation

Combining Boyle’s law and Charles’s law for a fixed amount of gas gives:

PV/T = constant

For n moles of an ideal gas, the constant becomes nR:

Ideal-gas equation
PV = nRT

Using the number of molecules N instead of moles:

PV = Nk_BT

Symbol	Meaning	Common SI unit / value
P	Absolute pressure	pascal (Pa)
V	Gas volume	m³
n	Amount of substance	mol
T	Absolute temperature	kelvin (K)
R	Universal gas constant	8.314 J mol⁻¹ K⁻¹
N	Number of molecules	dimensionless count
k_B	Boltzmann constant	1.380649 × 10⁻²³ J K⁻¹

Exam tip: Always convert Celsius temperature to kelvin before using the ideal-gas equation: T(K) = t(°C) + 273.15.

2. Molecular Properties of Matter

2.1 Molecules and Intermolecular Forces

Matter is composed of atoms and molecules. Molecules in a real gas have finite size and experience weak attractive or repulsive forces. In the ideal-gas model, these effects are neglected except for the impulsive force during collision.

2.2 Mole and Avogadro Constant

One mole contains exactly 6.02214076 × 10²³ specified particles. This number is the Avogadro constant, N_A.

n = N/N_A and R = N_Ak_B

Diagram 4 — Microscopic Model of a Gas

Gas pressure is a macroscopic result of enormous numbers of microscopic molecular collisions.

3. Kinetic-Molecular Model of an Ideal Gas

The kinetic theory uses a simplified microscopic model. Its main assumptions are:

A gas contains a very large number of identical molecules in continuous random motion.
The molecular size is negligible compared with the separation between molecules and the container volume.
Molecules obey Newton’s laws of motion.
No intermolecular force acts except during collisions.
Collisions between molecules and with the walls are perfectly elastic.
The duration of a collision is negligible compared with the time between collisions.
Molecular motion is isotropic: no direction is preferred.

Idealization: No real gas satisfies all assumptions perfectly. The model works best for dilute gases, especially at low pressure and sufficiently high temperature.

4. Derivation of Pressure Exerted by an Ideal Gas

Consider a cubical container of side l and volume V = l³. Let one molecule of mass m have velocity components c_x, c_y, c_z.

Diagram 5 — Molecular Collision with a Wall

In an elastic collision with the wall, the x-component reverses while its magnitude is unchanged.

Step 1: Change of Momentum

Before collision, x-momentum is mc_x; after collision it is −mc_x.

Δp_x = (−mc_x) − (mc_x) = −2mc_x

The impulse delivered to the wall has magnitude 2mc_x.

Step 2: Time Between Successive Collisions with the Same Wall

The molecule travels a round-trip distance 2l.

Δt = 2l/c_x

Step 3: Average Force Due to One Molecule

F = Δp/Δt = (2mc_x)/(2l/c_x) = mc_x²/l

Step 4: Sum Over N Molecules

F = (m/l) Σ c_xi²

Since wall area A = l² and V = l³:

P = F/A = (m/V) Σ c_xi²

Step 5: Use Isotropy of Molecular Motion

Σc_x² = Σc_y² = Σc_z² = (1/3)Σc²

Therefore, for N molecules:

P = (1/3)(Nm/V) c̄² = (1/3)ρc_rms²

where ρ = Nm/V is gas density and c_rms = √c̄².

Core derivation to memorize: P = (1/3)ρc_rms². In a long-answer question, clearly state the cube, momentum change, collision time, force, isotropy, and final result.

5. Pressure and Translational Kinetic Energy

Starting from:

P = (1/3)(Nm/V)c̄²

Multiply by volume:

PV = (1/3)Nm c̄² = (2/3)[(1/2)Nm c̄²]

The bracketed term is the total translational kinetic energy K of all molecules.

PV = (2/3)K ⇒ P = (2/3)(K/V)

Thus, pressure equals two-thirds of the translational kinetic-energy density.

Diagram 6 — Link Between Pressure and Molecular Kinetic Energy

At fixed volume, hotter molecules collide more energetically with the walls and produce greater pressure.

6. Average Translational Kinetic Energy and Boltzmann Constant

For an ideal gas:

PV = Nk_BT

But from kinetic theory:

PV = (2/3)K

Therefore:

K = (3/2)Nk_BT

Hence, average translational kinetic energy per molecule is:

K̄ = (3/2)k_BT

For one mole:

K_mole = (3/2)RT

Physical meaning: The average translational kinetic energy of an ideal-gas molecule depends only on absolute temperature, not on pressure, volume, or the chemical identity of the gas.

7. Root Mean Square (RMS) Speed

Molecular speeds are not all equal, so kinetic theory uses the root mean square speed:

c_rms = √(c̄²)

From the pressure equation and ideal-gas equation:

c_rms = √(3RT/M)

where M is molar mass in kg mol⁻¹. For a single molecule of mass m:

c_rms = √(3k_BT/m)

Dependence on Temperature and Molecular Mass

c_rms ∝ √T and c_rms ∝ 1/√M

At higher temperature, gas molecules have greater RMS speed.
At the same temperature, lighter gases have greater RMS speed than heavier gases.

Diagram 7 — RMS Speed: Temperature and Molar Mass

RMS speed rises with √T and falls with √M.

8. Heat Capacities of Gases and Solids

8.1 Heat Capacity and Specific Heat Capacity

Heat capacity is the heat required to raise the temperature of a body by 1 K:

C = Q/ΔT

Specific heat capacity is heat required per unit mass per kelvin:

c = Q/(mΔT)

8.2 Molar Heat Capacities of an Ideal Gas

A gas can be heated at constant volume or at constant pressure, giving two molar heat capacities: C_V and C_P.

At constant volume, no expansion work is done, so supplied heat increases internal energy.
At constant pressure, part of the supplied heat increases internal energy and part does expansion work.
Therefore, C_P > C_V.

C_P − C_V = R

The heat-capacity ratio is:

γ = C_P/C_V

8.3 Simple Equipartition Results

In the classical model, each independent quadratic degree of freedom contributes (1/2)k_BT per molecule to average energy.

Idealized substance	Active degrees of freedom	C_V	C_P	γ
Monatomic ideal gas	3 translational	3R/2	5R/2	5/3 ≈ 1.67
Diatomic ideal gas (ordinary temperatures, simple model)	3 translational + 2 rotational	5R/2	7R/2	7/5 = 1.40
Classical crystalline solid (Dulong–Petit limit)	Lattice vibrations	Molar heat capacity ≈ 3R		—

Diagram 8 — Constant-Volume vs Constant-Pressure Heating

Because a gas can expand at constant pressure, more heat is needed for the same temperature rise than at constant volume.

9. Formula Summary

Concept	Formula	Condition / Meaning
Boyle’s law	PV = constant	Fixed mass, constant T
Charles’s law	V/T = constant	Fixed mass, constant P
Pressure law	P/T = constant	Fixed mass, constant V
Combined gas equation	PV/T = constant	Fixed amount of gas
Ideal-gas equation	PV = nRT = Nk_BT	Ideal gas
Boltzmann relation	R = N_Ak_B	Links molar and molecular scales
Pressure from kinetic theory	P = (1/3)ρc_rms²	Ideal gas
Pressure–energy relation	PV = (2/3)K	K = total translational KE
Average KE per molecule	K̄ = (3/2)k_BT	Translational KE
KE per mole	K = (3/2)RT	One mole
RMS speed	c_rms = √(3RT/M)	M in kg mol⁻¹
Mayer’s relation	C_P − C_V = R	Ideal gas, molar capacities
Heat-capacity ratio	γ = C_P/C_V	Ideal gas

10. Solved Numerical Examples

Example 1 — Ideal-Gas Equation

Question: Find the volume occupied by 2.0 mol of an ideal gas at 300 K and 1.00 × 10⁵ Pa.

V = nRT/P = (2.0)(8.314)(300)/(1.00 × 10⁵) ≈ 4.99 × 10⁻² m³

Answer: V ≈ 0.0499 m³ ≈ 49.9 L.

Example 2 — RMS Speed

Question: Estimate the RMS speed of nitrogen gas at 300 K. Take M = 28 × 10⁻³ kg mol⁻¹.

c_rms = √(3RT/M) = √[(3)(8.314)(300)/(28 × 10⁻³)] ≈ 517 m s⁻¹

Answer: c_rms ≈ 5.17 × 10² m s⁻¹.

Example 3 — Average Molecular Kinetic Energy

Question: Find the average translational kinetic energy of one ideal-gas molecule at 300 K.

K̄ = (3/2)k_BT = (3/2)(1.380649 × 10⁻²³)(300) ≈ 6.21 × 10⁻²¹ J

Answer: K̄ ≈ 6.21 × 10⁻²¹ J per molecule.

Example 4 — Molecular Speed and Temperature

Question: If the absolute temperature becomes four times as large, by what factor does RMS speed change?

c_rms ∝ √T ⇒ c′/c = √(4T/T) = 2

Answer: The RMS speed doubles.

Example 5 — Pressure from Density and RMS Speed

Question: A gas has density 1.20 kg m⁻³ and RMS speed 500 m s⁻¹. Find its pressure.

P = (1/3)ρc_rms² = (1/3)(1.20)(500)² = 1.00 × 10⁵ Pa

Answer: P = 1.00 × 10⁵ Pa.

Important Exam Questions

Short-Answer Questions

Define an ideal gas. Under what conditions do real gases approximately behave ideally?
State Boyle’s law and Charles’s law.
What is absolute zero? How is it obtained from a V–t or P–t graph?
Define mole and Avogadro constant.
State the principal assumptions of the kinetic-molecular model of an ideal gas.
Define RMS speed. How does it depend on absolute temperature and molar mass?
Write the relation between gas pressure and translational kinetic-energy density.
What is Boltzmann constant? Write its relation with R and N_A.
Why is C_P greater than C_V for a gas?
State Mayer’s relation for an ideal gas.

Long-Answer Questions

Explain the experimental gas laws and combine them to obtain the ideal-gas equation.
Explain absolute zero with the help of volume–temperature and pressure–temperature graphs.
State and explain the assumptions of kinetic theory of gases.
Derive the expression for pressure exerted by an ideal gas on the walls of a container.
Show that the pressure of an ideal gas is two-thirds of its translational kinetic-energy density.
Derive the average translational kinetic energy per molecule and per mole.
Derive the expression c_rms = √(3RT/M) and discuss its dependence on T and M.
Explain C_V, C_P, γ, and the heat capacities of simple gases and solids.

Derivations to Practice

PV = nRT from the gas laws.
P = (1/3)ρc_rms² from molecular collisions.
PV = (2/3)K.
K̄ = (3/2)k_BT.
c_rms = √(3RT/M).

Numerical Questions

A gas occupies 2.0 L at 1.2 × 10⁵ Pa. Find its volume at 0.8 × 10⁵ Pa if temperature is constant. Answer: 3.0 L.
A fixed amount of gas occupies 300 cm³ at 300 K. What volume will it occupy at 450 K at constant pressure? Answer: 450 cm³.
Find the pressure of 1.5 mol of an ideal gas occupying 0.030 m³ at 320 K. Answer: ≈ 1.33 × 10⁵ Pa.
Find the average translational kinetic energy per molecule at 400 K. Answer: ≈ 8.28 × 10⁻²¹ J.
Calculate the RMS speed of oxygen at 300 K, taking M = 32 × 10⁻³ kg mol⁻¹. Answer: ≈ 484 m s⁻¹.

Diagram Questions

Draw and label the Boyle’s-law P–V graph.
Draw a V–t graph and indicate absolute zero.
Draw a P–t graph and indicate absolute zero.
Draw a cubical container and show the collision used in the kinetic-theory pressure derivation.
Draw a simple comparison of constant-volume and constant-pressure heating.

One-Minute Revision

An ideal gas obeys PV = nRT and is best approximated by real gases at low pressure and high temperature.
Boyle’s law: PV = constant at constant T.
Charles’s law: V/T = constant at constant P.
At constant V, P/T = constant.
Absolute zero is 0 K = −273.15 °C.
One mole contains N_A = 6.02214076 × 10²³ particles.
R = N_Ak_B.
Kinetic theory assumes random motion, negligible molecular size and perfectly elastic collisions.
Pressure from kinetic theory: P = (1/3)ρc_rms².
Pressure–energy relation: PV = (2/3)K.
Average translational KE per molecule = (3/2)k_BT.
RMS speed: c_rms = √(3RT/M).
RMS speed increases as √T and decreases as 1/√M.
For an ideal gas, C_P − C_V = R and C_P > C_V.
A classical crystalline solid approaches molar heat capacity ≈ 3R at sufficiently high temperature.

Diagram Practice

Redraw the Boyle’s-law rectangular hyperbola and mark two states (P₁,V₁) and (P₂,V₂).
Redraw the V–t graph and extend it to −273.15 °C.
Redraw the P–t graph and extend it to −273.15 °C.
Draw molecules moving randomly inside a container and label the walls and molecular velocity.
Draw the wall-collision diagram and label +c_x, −c_x, side length l, and the wall normal to the x-axis.
Create a concept map linking temperature → average kinetic energy → collision effect → pressure.
Draw qualitative curves showing c_rms ∝ √T and c_rms ∝ 1/√M.
Sketch constant-volume and constant-pressure heating arrangements and state why C_P > C_V.

Syllabus Coverage Checklist

NEB/CDC Chapter 13 scope	Covered here
13.1 Ideal gas equation; gas laws; pressure/volume coefficients; absolute zero	Yes
13.2 Molecular properties of matter	Yes
13.3 Kinetic-molecular model of an ideal gas	Yes
13.4 Derivation of pressure exerted by gas	Yes
13.5 Average translational kinetic energy of gas molecule	Yes
13.6 Boltzmann constant and root mean square speed	Yes
13.7 Heat capacities: gases and solids	Yes
Related mathematical problems	Yes — solved and practice numericals included