## Kinetic Molecular Theory and Gas Laws

Kinetic Molecular Theory explains the macroscopic properties of gases and can be used to understand and explain the gas laws.

### Learning Objectives

Express the five basic assumptions of the Kinetic Molecular Theory of Gases.

### Key Takeaways

#### Key Points

- Kinetic Molecular Theory states that gas particles are in constant motion and exhibit perfectly elastic collisions.
- Kinetic Molecular Theory can be used to explain both Charles’ and Boyle’s Laws.
- The average kinetic energy of a collection of gas particles is directly proportional to absolute temperature only.

#### Key Terms

**ideal gas**: a hypothetical gas whose molecules exhibit no interaction and undergo elastic collision with each other and the walls of the container**macroscopic properties**: properties that can be visualized or measured by the naked eye; examples include pressure, temperature, and volume

### Basic Assumptions of the Kinetic Molecular Theory

By the late 19th century, scientists had begun accepting the atomic theory of matter started relating it to individual molecules. The Kinetic Molecular Theory of Gases comes from observations that scientists made about gases to explain their macroscopic properties. The following are the basic assumptions of the Kinetic Molecular Theory:

- The volume occupied by the individual particles of a gas is negligible compared to the volume of the gas itself.
- The particles of an ideal gas exert no attractive forces on each other or on their surroundings.
- Gas particles are in a constant state of random motion and move in straight lines until they collide with another body.
- The collisions exhibited by gas particles are completely elastic; when two molecules collide, total kinetic energy is conserved.
- The average kinetic energy of gas molecules is directly proportional to absolute temperature only; this implies that all molecular motion ceases if the temperature is reduced to absolute zero.

### Applying Kinetic Theory to Gas Laws

Charles’ Law states that at constant pressure, the volume of a gas increases or decreases by the same factor as its temperature. This can be written as:

[latex]\frac{V_1}{T_1}=\frac{V_2}{T_2}[/latex]

According to Kinetic Molecular Theory, an increase in temperature will increase the average kinetic energy of the molecules. As the particles move faster, they will likely hit the edge of the container more often. If the reaction is kept at constant pressure, they must stay farther apart, and an increase in volume will compensate for the increase in particle collision with the surface of the container.

Boyle’s Law states that at constant temperature, the absolute pressure and volume of a given mass of confined gas are inversely proportional. This relationship is shown by the following equation:

[latex]P_1V_1=P_2V_2[/latex]

At a given temperature, the pressure of a container is determined by the number of times gas molecules strike the container walls. If the gas is compressed to a smaller volume, then the same number of molecules will strike against a smaller surface area; the number of collisions against the container will increase, and, by extension, the pressure will increase as well. Increasing the kinetic energy of the particles will increase the pressure of the gas.

## Distribution of Molecular Speeds and Collision Frequency

The Maxwell-Boltzmann Distribution describes the average molecular speeds for a collection of gas particles at a given temperature.

### Learning Objectives

Identify the relationship between velocity distributions and temperature and molecular weight of a gas.

### Key Takeaways

#### Key Points

- Gaseous particles move at random speeds and in random directions.
- The Maxwell-Boltzmann Distribution describes the average speeds of a collection gaseous particles at a given temperature.
- Temperature and molecular weight can affect the shape of Boltzmann Distributions.
- Average velocities of gases are often expressed as root-mean-square averages.

#### Key Terms

**velocity**: a vector quantity that denotes the rate of change of position with respect to time or a speed with a directional component**quanta**: the smallest possible packet of energy that can be transferred or absorbed

According to the Kinetic Molecular Theory, all gaseous particles are in constant random motion at temperatures above absolute zero. The movement of gaseous particles is characterized by straight-line trajectories interrupted by collisions with other particles or with a physical boundary. Depending on the nature of the particles’ relative kinetic energies, a collision causes a transfer of kinetic energy as well as a change in direction.

### Root-Mean-Square Velocities of Gaseous Particles

Measuring the velocities of particles at a given time results in a large distribution of values; some particles may move very slowly, others very quickly, and because they are constantly moving in different directions, the velocity could equal zero. (Velocity is a vector quantity, equal to the speed and direction of a particle) To properly assess the average velocity, average the *squares* of the velocities and take the square root of that value. This is known as the root-mean-square (RMS) velocity, and it is represented as follows:

[latex]\bar{v}=v_{rms}=\sqrt{\frac{3RT}{M_m}}[/latex]

[latex]KE=\frac{1}{2}mv^2[/latex]

[latex]KE=\frac{1}{2}mv^2[/latex]

In the above formula, R is the gas constant, T is absolute temperature, and M_{m} is the molar mass of the gas particles in kg/mol.

### Energy Distribution and Probability

Consider a closed system of gaseous particles with a fixed amount of energy. With no external forces (e.g. a change in temperature) acting on the system, the total energy remains unchanged. In theory, this energy can be distributed among the gaseous particles in many ways, and the distribution constantly changes as the particles collide with each other and with their boundaries. Given the constant changes, it is difficult to gauge the particles’ velocities at any given time. By understanding the nature of the particle movement, however, we can predict the probability that a particle will have a certain velocity at a given temperature.

Kinetic energy can be distributed only in discrete amounts known as quanta, so we can assume that any one time, each gaseous particle has a certain amount of quanta of kinetic energy. These quanta can be distributed among the three directions of motions in various ways, resulting in a velocity state for the molecule; therefore, the more kinetic energy, or quanta, a particle has, the more velocity states it has as well.

If we assume that all velocity states are equally probable, higher velocity states are favorable because there are greater in quantity. Although higher velocity states are favored statistically, however, lower energy states are more likely to be occupied because of the limited kinetic energy available to a particle; a collision may result in a particle with greater kinetic energy, so it must also result in a particle with *less* kinetic energy than before.

**Interactive: Diffusion & Molecular Mass**: Explore the role of molecular mass on the rate of diffusion. Select the mass of the molecules behind the barrier. Remove the barrier, and measure the amount of time it takes the molecules to reach the gas sensor. When the gas sensor has detected three molecules, it will stop the experiment. Compare the diffusion rates of the lightest, heavier and heaviest molecules. Trace an individual molecule to see the path it takes.

### Maxwell-Boltzmann Distributions

Using the above logic, we can hypothesize the velocity distribution for a given group of particles by plotting the number of molecules whose velocities fall within a series of narrow ranges. This results in an asymmetric curve, known as the Maxwell-Boltzmann distribution. The peak of the curve represents the most probable velocity among a collection of gas particles.

Velocity distributions are dependent on the temperature and mass of the particles. As the temperature increases, the particles acquire more kinetic energy. When we plot this, we see that an increase in temperature causes the Boltzmann plot to spread out, with the relative maximum shifting to the right.

Larger molecular weights narrow the velocity distribution because all particles have the same kinetic energy at the same temperature. Therefore, by the equation [latex]KE=\frac{1}{2}mv^2[/latex], the fraction of particles with higher velocities will increase as the molecular weight decreases.

## Root-Mean-Square Speed

The root-mean-square speed measures the average speed of particles in a gas, defined as [latex]v_{rms}=\sqrt{\frac{3RT}{M}}[/latex].

### Learning Objectives

Recall the mathematical formulation of the root-mean-square velocity for a gas.

### Key Takeaways

#### Key Points

- All gas particles move with random speed and direction.
- Solving for the average velocity of gas particles gives us the average velocity of zero, assuming that all particles are moving equally in different directions.
- You can acquire the average speed of gaseous particles by taking the root of the square of the average velocities.
- The root-mean-square speed takes into account both molecular weight and temperature, two factors that directly affect a material’s kinetic energy.

#### Key Terms

**velocity**: a vector quantity that denotes the rate of change of position, with respect to time or a speed with a directional component

### Kinetic Molecular Theory and Root-Mean-Square Speed

According to Kinetic Molecular Theory, gaseous particles are in a state of constant random motion; individual particles move at different speeds, constantly colliding and changing directions. We use velocity to describe the movement of gas particles, thereby taking into account both speed and direction.

Although the velocity of gaseous particles is constantly changing, the distribution of velocities does not change. We cannot gauge the velocity of each individual particle, so we often reason in terms of the particles’ average behavior. Particles moving in opposite directions have velocities of opposite signs. Since a gas’ particles are in random motion, it is plausible that there will be about as many moving in one direction as in the opposite direction, meaning that the average velocity for a collection of gas particles equals zero; as this value is unhelpful, the average of velocities can be determined using an alternative method.

By squaring the velocities and taking the square root, we overcome the “directional” component of velocity and simultaneously acquire the particles’ average velocity. Since the value excludes the particles’ direction, we now refer to the value as the average speed. The root-mean-square speed is the measure of the speed of particles in a gas, defined as the square root of the average velocity-squared of the molecules in a gas.

It is represented by the equation: [latex]v_{rms}=\sqrt{\frac{3RT}{M}}[/latex], where v_{rms} is the root-mean-square of the velocity, M_{m} is the molar mass of the gas in *kilograms per * *mole*, R is the molar gas constant, and T is the temperature in Kelvin.

The root-mean-square speed takes into account both molecular weight and temperature, two factors that directly affect the kinetic energy of a material.

### Example

- What is the root-mean-square speed for a sample of oxygen gas at 298 K?

[latex]v_{rms}=\sqrt{\frac{3RT}{M_m}}=\sqrt{\frac{3(8.3145\frac{J}{K*mol})(298\;K)}{32\times10^{-3}\frac{kg}{mol}}}=482\;m/s[/latex]