Kinetic Theory of an Ideal Gas: Class 11 Physics Notes with Pressure and Temperature Derivations

The kinetic theory of an ideal gas is a theoretical and mathematical framework that provides a molecular interpretation for the macroscopic properties of gases. That includes observable characteristics, including pressure and temperature. 

This theory makes several assumptions about how an ideal gas behaves. You can get a broader context as you review our exam-focused overview of the Kinetic Theory of Gases. But to leave no stone unturned, go revise what kinetic energy is.  

One critical assumption of the Kinetic Theory of an Ideal Gas is that, at ordinary temperature or pressure, the average distance between identical molecules is significantly larger. It’s also greater than the size of the molecule. That leads to negligible interactions between molecules, except during collisions. 

So, using Newton’s First Law of Motion, the movement of these molecules in a gas is in straight lines. But when these molecules do collide or interact with each other, considering within a container’s walls, the collisions are elastic. Both kinetic energy and momentum are conserved during a collision. The time spent in collisions is assumed to be negligible in comparison to the time spent between collisions. That’s how the energy is conserved. 

Because of these assumptions, outlined in the Kinetic Theory of an Ideal Gas, we can derive equations for macroscopic properties, such as pressure and temperature. Put, with them, we can draw the link between these macroscopic properties (pressure and temperature) based on the microscopic behaviour of how molecules behave in an ideal gas. 

Pressure arises from molecular collisions with container walls.

Let’s look at the step-by-step derivation of the pressure of an ideal gas. 

Consider $N$ molecules, each of mass $m$ , in a cube of side $L$ (volume $V=L^3$ ):

• A molecule with velocity $\vec{v}=\left(v_x,v_y,v_z\right)$ hits a wall perpendicular to the x-axis, changing momentum by $\mathrm{\Delta }p=-2m{v}_x$
• Time between collisions with the same wall: $\mathrm{\Delta }t=\frac{2L}{v_x}$
• Collisions per second: $\frac{v_x}{2L}$
• Force by one molecule: $F=\frac{\mathrm{\Delta }p}{\mathrm{\Delta }t}=\frac{m{v}_x^2}{L}$
• Total force on the wall: $F_x=\frac{m}{L}\sum v_x^2$
• Pressure: $P=\frac{F_x}{L^2}=\frac{m}{V}\sum v_x^2$
• For isotropic motion, we have the equation below. Isotropy means the motion of gas molecules is equally likely in all directions.
• $\sum v_x^2=\sum v_y^2=\sum v_z^2=\frac{1}{3}\sum v^2$ , so: $P=\frac{1}{3}\frac{mN}{V}{v}_{\mathrm{r}\mathrm{m}\mathrm{s}}^2=\frac{1}{3}\rho v_{\mathrm{r}\mathrm{m}\mathrm{s}}^2$ where $v_{\text{rms}}=\sqrt[]{\frac{\sum v^2}{N}}$ is the root-mean-square speed, and $\rho =\frac{mN}{V}$ is the density.

This equation illustrates that pressure is directly proportional to molecular speed and density. You can start applying this equation more by practising the pressure equation with our NCERT Solutions for the Kinetic Theory chapter.

Role of Pressure on an Ideal Gas

• Gas pressure results from molecules colliding with the walls. The faster they move (higher vrms) and the more molecules there are in a given volume (higher density), the greater the pressure.
• This equation indicates that pressure is dependent on density and the average molecular speed. That’s why we used the root-mean-square velocity.
• The gas is ideal with no intermolecular forces. These molecules undergo perfect elastic collisions as the system (or the container that holds the gas) is in thermal equilibrium. 

Here is a step-by-step derivation of the kinetic interpretation of temperature. 

Combining the pressure equation with the Ideal Gas Law

$PV=\mu RT=NkT$ : $PV=\frac{1}{3}Nm{v}_{\mathrm{r}\mathrm{m}\mathrm{s}}^2$

Equating: $\begin{array}{c}\frac{1}{3}Nm{v}_{\mathrm{r}\mathrm{m}\mathrm{s}}^2=NkT\\ \frac{1}{2}m{v}_{\mathrm{r}\mathrm{m}\mathrm{s}}^2=\frac{3}{2}kT\end{array}$

The average translational kinetic energy per molecule is: $E=\frac{3}{2}kT$

For one mole ( $\mu =1,N=N_A$ ): $E=\frac{3}{2}RT$

Temperature measures the average molecular kinetic energy of an ideal gas, independent of pressure or volume.

Role of Temperature on Ideal Gas

• This equation helps in relating the temperature (T) to the average kinetic energy (E) of gas molecules.
• The Boltzmann constant 𝑘 connects microscopic energy per molecule to macroscopic temperature.
• It comes from equating the kinetic theory pressure formula and the ideal gas law. That illustrates how microscopic motion is connected to macroscopic temperature.
• The final equation applies when a gas behaves ideally. The pressure is low, the temperature is high, and intermolecular forces are negligible.
• By learning this equation, you can show that temperature is a direct measure of molecular motion. It explains why hot gases have faster-moving molecules and how temperature affects pressure and energy. 

The kinetic theory of an ideal gas is a small section in your CBSE Physics textbook in Class 11. It does not go into real gas scenarios until later. But forms the primary foundation for understanding how molecular behaviour can affect pressure and temperature with respect to the ideal gas equation. You could, however, have a look at our other comprehensive guide on the Behaviour of Gases.

If you are sitting for JEE Mains next year, here are some categories of questions that you should not ignore. 

Category	Core Concepts Tested
Assumptions of Kinetic Theory	Negligible intermolecular forces, Elastic collisions, Random motion, Point particles
Molecular Motion & Pressure	Derivation of pressure from collisions, Use of isotropy, Density and pressure link
Temperature & Kinetic Energy	Kinetic interpretation of temperature
Ideal Gas Law Integration	Connection to PV = μRT = NkT, Linking microscopic & macroscopic quantities
Theory-Based Reasoning	Limitations of ideal gas assumptions, Relation of temperature to motion, Isotropy & elastic collision implications

You can also refer to previous JEE Main papers to gain an idea. 

Here are some essential concepts that NCERT briefly mentions while explaining the Kinetic Theory of an Ideal Gas. 

Concept	Description	Connection to Kinetic Theory of an Ideal Gas
Conservation of Momentum	In gas collisions, momentum remains unchanged before and after contact.	Helps explain how molecule-wall impacts generate pressure, as force arises from repeated momentum changes.
Boltzmann Constant (𝑘)	Serves as a connection between microscopic molecular motions and the visible temperature effects.	Used to connect the average energy per molecule with temperature. That shows how individual motion affects thermal properties.
Consistency with the Ideal Gas Law	Kinetic theory explains the relationship between gas pressure, volume, and temperature.	The theory explains how pressure and temperature arise from molecular behaviour, thereby validating the assumptions of an ideal gas.
Dalton’s Law of Partial Pressures	Each type of gas contributes to the total pressure independently.	Since ideal gas molecules don't interact, each type behaves independently, matching Dalton’s idea of pressure being additive.

Go through these notes on Physics chapters in Class 11. Also, look out for the topic explanations for each of these chapters, with in-depth and expert-reviewed guides. 

Units and Measurements Class 11 Notes	Mechanical Properties of Solids Class 11 Notes
Motion in a Straight Line Class 11 Notes	Mechanical Properties of Fluids Class 11 Notes
NCERT Class 11 Notes for Motion in a Plane	Thermal Properties of Matter Class 11 Notes
Laws of Motion Class 11 Notes	Thermodynamics Class 11 Notes
Work, Energy, and Power Class 11 Notes	Kinetic Theory of Gas Class 11 Notes
System of Particles and Rotational Motion Class 11 Notes	Oscillations Class 11 Notes
Gravitation Class 11 Notes	Waves Class 11 Notes

Look for more curriculum notes across all subjects in Class 11 Science. 

NCERT Class 11 Notes for PCM

NCERT Class 11 Physics Notes

 

Struggling with conceptually based numerical problems in Physics before exams? Reduce some pressure with expert-written  NCERT Solutions for Physics Class 11. Find for all chapters below.  

Units and Measurements Class 11 NCERT Solutions	Mechanical Properties of Solids Class 11 NCERT Solutions
Motion in a Straight Line Class 11 NCERT Solutions	Mechanical Properties of Fluids Class 11 NCERT Solutions
Motion in a Plane Class 11 NCERT Solutions	Thermal Properties of Matter Class 11 NCERT Solutions
Laws of Motion Class 11 NCERT Solutions	Thermodynamics Class 11 NCERT Solutions
Work, Energy, and Power Class 11 NCERT Solutions	Kinetic Theory Class 11 NCERT Solutions
System of Particles and Rotational Motion Class 11 NCERT Solutions	Oscillations Class 11 NCERT Solutions
Gravitation Class 11 NCERT Solutions	Waves Class 11 NCERT Solutions