# Temperature, Kinetic Theory, and the Gas Laws

## Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature


### Learning Objectives

By the end of this section, you will be able to:
1. Express the ideal gas law in terms of molecular mass and velocity.
2. Define thermal energy.
3. Calculate the kinetic energy of a gas molecule, given its temperature.
4. Describe the relationship between the temperature of a gas and the kinetic energy of atoms and molecules.
5. Describe the distribution of speeds of molecules in a gas.
We have developed macroscopic definitions of pressure and temperature. Pressure is the force divided by the area on which the force is exerted, and temperature is measured with a thermometer. We gain a better understanding of pressure and temperature from the kinetic theory of gases, which assumes that atoms and molecules are in continuous random motion.
 shows an elastic collision of a gas molecule with the wall of a container, so that it exerts a force on the wall (by Newton’s third law). Because a huge number of molecules will collide with the wall in a short time, we observe an average force per unit area. These collisions are the source of pressure in a gas. As the number of molecules increases, the number of collisions and thus the pressure increase. Similarly, the gas pressure is higher if the average velocity of molecules is higher. The actual relationship is derived in the Things Great and Small feature below. The following relationship is found:
where  is the pressure (average force per unit area),  is the volume of gas in the container,  is the number of molecules in the container,  is the mass of a molecule, and  is the average of the molecular speed squared.
What can we learn from this atomic and molecular version of the ideal gas law? We can derive a relationship between temperature and the average translational kinetic energy of molecules in a gas. Recall the previous expression of the ideal gas law:
Equating the right-hand side of this equation with the right-hand side of  gives
We can get the average kinetic energy of a molecule, , from the right-hand side of the equation by canceling  and multiplying by 3/2. This calculation produces the result that the average kinetic energy of a molecule is directly related to absolute temperature.
The average translational kinetic energy of a molecule, , is called thermal energy. The equation  is a molecular interpretation of temperature, and it has been found to be valid for gases and reasonably accurate in liquids and solids. It is another definition of temperature based on an expression of the molecular energy.
It is sometimes useful to rearrange , and solve for the average speed of molecules in a gas in terms of temperature, 
where  stands for root-mean-square (rms) speed. 

### Distribution of Molecular Speeds

The motion of molecules in a gas is random in magnitude and direction for individual molecules, but a gas of many molecules has a predictable distribution of molecular speeds. This distribution is called the Maxwell-Boltzmann distribution, after its originators, who calculated it based on kinetic theory, and has since been confirmed experimentally. (See .) The distribution has a long tail, because a few molecules may go several times the rms speed. The most probable speed  is less than the rms speed .  shows that the curve is shifted to higher speeds at higher temperatures, with a broader range of speeds.
The distribution of thermal speeds depends strongly on temperature. As temperature increases, the speeds are shifted to higher values and the distribution is broadened.
What is the implication of the change in distribution with temperature shown in  for humans? All other things being equal, if a person has a fever, they are likely to lose more water molecules, particularly from linings along moist cavities such as the lungs and mouth, creating a dry sensation in the mouth.

### Test Prep for AP Courses


### Section Summary

1. Kinetic theory is the atomistic description of gases as well as liquids and solids.
2. Kinetic theory models the properties of matter in terms of continuous random motion of atoms and molecules.
3. The ideal gas law can also be expressed as
    
      
where  is the pressure (average force per unit area),  is the volume of gas in the container,  is the number of molecules in the container,  is the mass of a molecule, and  is the average of the molecular speed squared.
    
4. Thermal energy is defined to be the average translational kinetic energy  of an atom or molecule.
5. The temperature of gases is proportional to the average translational kinetic energy of atoms and molecules.
    
      
    or
    
      

6. The motion of individual molecules in a gas is random in magnitude and direction. However, a gas of many molecules has a predictable distribution of molecular speeds, known as the Maxwell-Boltzmann distribution. 

### Conceptual Questions


### Problems & Exercises
