
You have started studying Kinetic Theory again in Class 11 for your annuals, perhaps, brushing through the molecular nature of matter and the behaviour of gases. You may still feel clear about the kinetic interpretation of macroscopic properties. But, the equations and derivations in the NCERT Solutions for Kinetic Theory are becoming a bit hazy. And when you shift gears to preparing for JEE Mains, the struggle intensifies more, seeing that mounting pile of study material at your desk.
You aren’t alone in this. Every student finds it challenging right before exams to visualise atoms and molecules moving in perpetual motion, as well as understanding when and why ideal gases deviate.
Let’s simplify those complex concepts and derivations in the Kinetic Theory of Gases with this overview-style guide, prepared for a quick brushup before you begin to move on to the nitty-gritty of this chapter and tackle JEE and any other competitive test with confidence.
- Kinetic Theory of Gases - An Overview
- Eight (8) Important Assumptions of Kinetic Theory
- Pressure of an Ideal Gas
- Ideal Gas Equation for Kinetic Theory
- Vander Waal's Gas Equation for Kinetic Theory of Non-Ideal Gas
- Molecular Speeds
- Kinetic Energy of Gas Molecules
- Degrees of Freedom
- Law of Equipartition of Energy
- Specific Heat of Gases
- Mean Free Path
- Common Mistakes in Kinetic Theory
- Sample Problems
Kinetic Theory of Gases - An Overview
The kinetic theory of gases (KTG) is a model to help us understand macroscopic properties of gases (pressure, temperature, volume) by showing how molecules behave at the microscopic level. It assumes gases consist of numerous tiny particles in constant random motion, undergoing elastic collisions with each other and container walls. This theory connects molecular motion to observable gas properties.
Important Backgrounds to the Kinetic Theory
- Your previous chapter, Thermal Properties of Matter contains the ideal gas equation section, which you should refer to when trying to draw the connection with Gas Laws by Boyle, Charles, and Lussac. Do practice the NCERT Solutions for Physics Chapter 10 to draw connections on how ideal gas behaviour deviates.
- The Kinetic Theory is applied universally to all gaseous systems where interatomic forces are negligible. Scientists, including Boyle, Newton, Boltzmann, and Maxwell developed this during the 16th-17th centuries, which in turn is a result of centuries-old observations of atoms by Kanada from India and Democritus from Greece - later concretised by Dalton.
- The macroscopic properties of gases, such as pressure, temperature, and volume, are known as thermodynamic variables.
Eight (8) Important Assumptions of Kinetic Theory
The kinetic theory relies on these assumptions for an ideal gas. This section should accompany you with the Kinetic Theory of An Ideal Gas section of your Physics textbook from NCERT.
We elaborate on the reasons for these assumptions as well, because learning about them helps us draw the link between the microscopic properties of gas molecules and their macroscopic properties, including pressure, temperature, and volume.
- Gases consist of many identical molecules, treated as point masses with negligible size . In physics, a point mass is a small object that has no measurable size or internal structure.
-
Molecules move randomly in all directions with a range of velocities. This ensures that gas properties, such as pressure, are uniform with no preferred direction of molecular motion.
- Collisions between molecules and with container walls are perfectly elastic, conserving momentum and kinetic energy. Meaning, the molecules can rebound without any loss of total kinetic energy or momentum.
- No intermolecular forces exist except during collisions. This is because gas molecules are considered to move freely and independently between brief, instantaneous interactions.
- Molecular volume is negligible compared to the gas volume. For instance, it is 0.014% for oxygen at STP, i.e., standard temperature 273 K and pressure 1 atm.
- Collision time is negligible compared to the time between collisions. The reason for this is that the duration of an actual collision between molecules (when they are interacting) is extremely short compared to the much more extended periods they spend travelling freely through space without any interaction.
- Molecules follow Newtonian mechanics, moving in straight lines between collisions. This is a simplification of classical physics, or, more accurately, a framework to simplify the complex behaviour of molecules when they aren’t colliding with each other.
Pressure of an Ideal Gas
We know that pressure results from molecular collisions with container walls.
From a physics perspective, we need to understand how a macroscopic property, such as pressure, arises from the random motion of individual molecules within a container.
For a gas with molecules, each of mass , in a cube of side :
- A molecule with velocity hits a wall perpendicular to the x-axis, changing momentum by .
- Time between collisions with the same wall: .
- Collisions per second: .
- Force by one molecule: .
- Total force: .
- Pressure: .
- For isotropic motion, , so: where and .
Key Implications
- Pressure
.
Pressure is directly proportional to the mean squared speed of the gas molecules.
- For constant
and
.
For a gas kept at a constant volume (V) and temperature (T), the pressure is proportional to the total mass of the gas (mN).
- For constant and
This is precisely Boyle's Law, which states that for a fixed amount of gas at constant temperature, pressure and volume are inversely related.
- For constant and .
-
This directly aligns with the ideal gas law, where if N and V are constant, P is directly proportional to T. It’s also Charles’ Law, when extended to constant pressure.
Ideal Gas Equation for Kinetic Theory
The NCERT textbook mentions that the “kinetic theory of an ideal gas is completely consistent with the ideal gas equation and the various gas laws based on it.”
As you know, an ideal gas obeys
where is moles, , and .
For density : where is the molar mass.
Vander Waal's Gas Equation for Kinetic Theory of Non-Ideal Gas
So far, we have considered kinetic theory in terms of an ideal gas, which is a theoretical model. In reality, however, there is no ‘ideal gas’, a concept corroborated by your NCERT book as well. That’s why we need to learn about the Van der Waals equation. It is a modification of the ideal gas equation, which takes into account the intermolecular forces between gas molecules and the finite volume of the gas molecules themselves.
Real gases deviate from ideal behaviour due to molecular size and forces,
where corrects for attractions, for molecular volume.
Molecular Speeds
Molecules have a distribution of speeds, where macroscopic properties, such as temperature and pressure, play a crucial role. In kinetic theory, the molecules of a gas move at various speeds when looked at from the perspective of Newtonian mechanics. This distribution of gas is described by the Maxwell-Boltzmann distribution.
The main macroscopic properties—temperature and pressure—are directly related to this distribution, and it applies to real gases. Temperature is a measure of the average kinetic energy of the molecules, and pressure results from collisions of these molecules with the walls of the container.
Root Mean Square Speed
This is the square root of the average of the squares of the molecular speeds.
Here, R is the universal gas constant, k is Boltzmann’s constant, T is the absolute temperature, M is the molar mass, and m is the molecular mass.
Most Probable Speed
This is the speed at which the majority of molecules are moving. It shows the peak of the Maxwell-Boltzmann distribution.
Average Speed
This is the arithmetic mean of the speeds of all molecules.
Now, all three of the above depend on temperature and molecular mass. But, they are not equal. Their ratio reflects the different ways in which speed is averaged.
Relation: .
Kinetic Energy of Gas Molecules
So far, we have seen how molecular speeds are distributed in a gas. But how does such motion convert into kinetic energy, especially when we have to quantify it?
We have taken into account how temperature is linked to the movement of molecules at measurable speeds. Now we need to determine how much kinetic energy they possess.
Average translational kinetic energy per molecule
E is the average kinetic energy per molecule
k is the Boltzmann constant
T is the absolute temperature
For one mole or Avogadro’s number of molecules
Here, R is the Universal Gas Constant.
This energy also connects to pressure through the kinetic theory equation.
Degrees of Freedom
Law of Equipartition of Energy
Specific Heat of Gases
Mean Free Path
Common Mistakes in Kinetic Theory
Sample Problems
Physics Kinetic Theory Exam