Chemistry 351

The Kinetic Molecular Theory of Gases

The Ideal Gas Law was originally developed from experimental observations, but it can also be understood from a microscopic picture of matter. The Kinetic Molecular Theory (KMT) models a gas as a collection of molecules moving continuously and colliding with one another and with the walls of their container.

In its modern form, the Kinetic Molecular Theory is based on five fundamental postulates:

1. Gas particles obey Newton's laws of motion and travel in straight lines between collisions.
2. Gas particles are very small compared to the average distance separating them.
3. Molecular collisions are perfectly elastic, so kinetic energy is conserved.
4. Gas particles do not exert attractive or repulsive forces on one another except during collisions.
5. The average molecular kinetic energy is proportional to the temperature.

These assumptions describe an ideal gas. Real gases deviate from this behavior because molecules have finite size and experience intermolecular forces.

Pressure from Molecular Collisions

According to the Kinetic Molecular Theory, gas pressure arises from collisions of molecules with the walls of a container. Each collision transfers momentum to the wall, producing a force. The greater the number of collisions and the larger the momentum change per collision, the greater the pressure.

A detailed analysis of molecular collisions leads to the expression

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

where

• \(N\) is the number of molecules,
• \(m\) is the mass of a single molecule,
• \(\langle v^2\rangle\) is the average value of \(v^2\), and
• \(V\) is the volume of the container.

This equation provides a direct connection between a macroscopic observable property (pressure) and microscopic molecular motion.

The Maxwell-Boltzmann Distribution and RMS Speed

Real gas samples contain molecules with a distribution of speeds described by the Maxwell-Boltzmann distribution. From this distribution, the root-mean-square (RMS) speed is found to be

\[ v_{rms} = \sqrt{ \frac{3RT} {MW} } \]

Since the RMS speed is defined by

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

it follows that

\[ \langle v^2\rangle = \frac{3RT} {MW} \]

This result links molecular speed directly to temperature.

Consistency with the Ideal Gas Law

Substituting the RMS-speed expression into the Kinetic Molecular Theory expression for pressure gives

\[ p = \frac{Nm} {3V} \left( \frac{3RT} {MW} \right) \]

Recognizing that

\[ MW = mN_A \]

and

\[ N=nN_A \]

leads to

\[ p = \frac{nRT} {V} \]

or

\[ PV=nRT \]

which is exactly the Ideal Gas Law.

Big picture: The Kinetic Molecular Theory provides a microscopic explanation for the Ideal Gas Law. Pressure arises from molecular collisions, temperature reflects molecular kinetic energy, and together these ideas naturally lead to the equation \(PV=nRT\).

Practice

Question 1
According to the Kinetic Molecular Theory, what is the origin of gas pressure?
A. Intermolecular attractions	B. Molecular collisions with container walls	C. Molecular collisions with each other	D. Gravitational attraction between molecules

Question 2
Which of the following is NOT a postulate of the Kinetic Molecular Theory?
A. Molecular collisions are elastic	B. Molecules travel in straight lines between collisions	C. Molecules exert strong attractive forces on one another	D. Average kinetic energy depends on temperature

Question 3
If the temperature of an ideal gas is increased while the volume remains constant, what happens to the average molecular kinetic energy?
A. It decreases	B. It remains constant	C. It increases	D. It becomes zero

Question 4
According to the Kinetic Molecular Theory, why do real gases deviate from ideal behavior?
A. Molecules have finite size and intermolecular forces	B. Molecules violate Newton's laws	C. Molecules stop moving at room temperature	D. Pressure is not caused by collisions

Question 5
The Kinetic Molecular Theory predicts that the pressure of a gas should be proportional to which combination of variables?
A. \(nVT\)	B. \(nT/V\)	C. \(V/(nT)\)	D. \(nV/T\)

Key points (one glance)

• The Kinetic Molecular Theory (KMT) models a gas as a collection of molecules moving continuously and colliding with one another and with the walls of their container.
• The five postulates of KMT assume that:
  • Molecules obey Newton's laws of motion.
  • Molecules are very small compared to the distance between them.
  • Collisions are perfectly elastic.
  • Molecules do not interact except during collisions.
  • Average molecular kinetic energy is proportional to temperature.
• Gas pressure arises from molecular collisions with the walls of a container.
• The KMT expression for pressure is \[ p = \frac{Nm\langle v^2\rangle} {3V} \]
• The Maxwell-Boltzmann distribution provides the statistical description of molecular speeds in a thermalized gas sample.
• The root-mean-square speed is \[ v_{rms} = \sqrt{ \frac{3RT} {MW} } \]
• The average molecular kinetic energy depends only on temperature.
• The KMT predicts that \[ p \propto n \] \[ p \propto \frac{1}{V} \] \[ p \propto T \]
• Combining these relationships yields \[ p \propto \frac{nT}{V} \] which has exactly the same form as the Ideal Gas Law.
• Substituting the Maxwell-Boltzmann prediction for molecular speeds into the KMT pressure expression leads directly to \[ PV=nRT \]
• Real gases deviate from the KMT assumptions because molecules have finite size and experience intermolecular attractions and repulsions.

Big picture: The Kinetic Molecular Theory provides a microscopic explanation for the Ideal Gas Law. By modeling gases as collections of rapidly moving molecules, the theory explains the origin of pressure, the role of temperature, and why the empirical gas laws are so successful.