# Temperature, Kinetic Theory, and the Gas Laws

## The Ideal Gas Law


### Learning Objectives

By the end of this section, you will be able to:
1. State the ideal gas law in terms of molecules and in terms of moles.
2. Use the ideal gas law to calculate pressure change, temperature change, volume change, or the number of molecules or moles in a given volume.
3. Use Avogadro’s number to convert between number of molecules and number of moles.
In this section, we continue to explore the thermal behavior of gases. In particular, we examine the characteristics of atoms and molecules that compose gases. (Most gases, for example nitrogen, , and oxygen, , are composed of two or more atoms. We will primarily use the term “molecule” in discussing a gas because the term can also be applied to monatomic gases, such as helium.)
Gases are easily compressed. We can see evidence of this in , where you will note that gases have the largest coefficients of volume expansion. The large coefficients mean that gases expand and contract very rapidly with temperature changes. In addition, you will note that most gases expand at the same rate, or have the same . This raises the question as to why gases should all act in nearly the same way, when liquids and solids have widely varying expansion rates.
The answer lies in the large separation of atoms and molecules in gases, compared to their sizes, as illustrated in . Because atoms and molecules have large separations, forces between them can be ignored, except when they collide with each other during collisions. The motion of atoms and molecules (at temperatures well above the boiling temperature) is fast, such that the gas occupies all of the accessible volume and the expansion of gases is rapid. In contrast, in liquids and solids, atoms and molecules are closer together and are quite sensitive to the forces between them.
To get some idea of how pressure, temperature, and volume of a gas are related to one another, consider what happens when you pump air into an initially deflated tire. The tire’s volume first increases in direct proportion to the amount of air injected, without much increase in the tire pressure. Once the tire has expanded to nearly its full size, the walls limit volume expansion. If we continue to pump air into it, the pressure increases. The pressure will further increase when the car is driven and the tires move. Most manufacturers specify optimal tire pressure for cold tires. (See .)
At room temperatures, collisions between atoms and molecules can be ignored. In this case, the gas is called an ideal gas, in which case the relationship between the pressure, volume, and temperature is given by the equation of state called the ideal gas law.
The ideal gas law can be derived from basic principles, but was originally deduced from experimental measurements of Charles’ law (that volume occupied by a gas is proportional to temperature at a fixed pressure) and from Boyle’s law (that for a fixed temperature, the product  is a constant). In the ideal gas model, the volume occupied by its atoms and molecules is a negligible fraction of . The ideal gas law describes the behavior of real gases under most conditions. (Note, for example, that  is the total number of atoms and molecules, independent of the type of gas.)
Let us see how the ideal gas law is consistent with the behavior of filling the tire when it is pumped slowly and the temperature is constant. At first, the pressure  is essentially equal to atmospheric pressure, and the volume  increases in direct proportion to the number of atoms and molecules  put into the tire. Once the volume of the tire is constant, the equation  predicts that the pressure should increase in proportion to the number N of atoms and molecules. 

### Moles and Avogadro’s Number

It is sometimes convenient to work with a unit other than molecules when measuring the amount of substance. A mole (abbreviated mol) is defined to be the amount of a substance that contains as many atoms or molecules as there are atoms in exactly 12 grams (0.012 kg) of carbon-12. The actual number of atoms or molecules in one mole is called Avogadro’s number, in recognition of Italian scientist Amedeo Avogadro (1776–1856). He developed the concept of the mole, based on the hypothesis that equal volumes of gas, at the same pressure and temperature, contain equal numbers of molecules. That is, the number is independent of the type of gas. This hypothesis has been confirmed, and the value of Avogadro’s number is 

### The Ideal Gas Law Restated Using Moles

A very common expression of the ideal gas law uses the number of moles, , rather than the number of atoms and molecules, . We start from the ideal gas law,
and multiply and divide the equation by Avogadro’s number . This gives
Note that  is the number of moles. We define the universal gas constant , and obtain the ideal gas law in terms of moles.
The ideal gas law can be considered to be another manifestation of the law of conservation of energy (see Conservation of Energy). Work done on a gas results in an increase in its energy, increasing pressure and/or temperature, or decreasing volume. This increased energy can also be viewed as increased internal kinetic energy, given the gas’s atoms and molecules.

### The Ideal Gas Law and Energy

Let us now examine the role of energy in the behavior of gases. When you inflate a bike tire by hand, you do work by repeatedly exerting a force through a distance. This energy goes into increasing the pressure of air inside the tire and increasing the temperature of the pump and the air.
The ideal gas law is closely related to energy: the units on both sides are joules. The right-hand side of the ideal gas law in  is . This term is roughly the amount of translational kinetic energy of  atoms or molecules at an absolute temperature , as we shall see formally in Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature. The left-hand side of the ideal gas law is , which also has the units of joules. We know from our study of fluids that pressure is one type of potential energy per unit volume, so pressure multiplied by volume is energy. The important point is that there is energy in a gas related to both its pressure and its volume. The energy can be changed when the gas is doing work as it expands—something we explore in Heat and Heat Transfer Methods—similar to what occurs in gasoline or steam engines and turbines.

### Test Prep for AP Courses


### Section Summary

1. The ideal gas law relates the pressure and volume of a gas to the number of gas molecules and the temperature of the gas.
2. The ideal gas law can be written in terms of the number of molecules of gas:
    
where  is pressure,  is volume,  is temperature,  is number of molecules, and  is the Boltzmann constant
    
3. A mole is the number of atoms in a 12-g sample of carbon-12. 
4. The number of molecules in a mole is called Avogadro’s number ,
    
5. A mole of any substance has a mass in grams equal to its molecular weight, which can be determined from the periodic table of elements.
6. The ideal gas law can also be written and solved in terms of the number of moles of gas:
    
where  is number of moles and  is the universal gas constant,
    
7. The ideal gas law is generally valid at temperatures well above the boiling temperature.

### Conceptual Questions


### Problems & Exercises
