Learning Objectives
By the end of this section, you will be able to do the following:
- State the ideal gas law in terms of molecules and in terms of moles
- Use the ideal gas law to calculate pressure change, temperature change, volume change, or the number of molecules or moles in a given volume
- Use Avogadro’s number to convert between the number of molecules and the 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 Table 13.2, 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 Figure 13.17. 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 Figure 13.18).
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.
Ideal Gas Law
The ideal gas law states that
where is the absolute pressure of a gas, is the volume it occupies, is the number of atoms and molecules in the gas, and is its absolute temperature. The constant is called the Boltzmann constant in honor of Austrian physicist Ludwig Boltzmann (1844–1906) and has the value
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.
Example 13.6 Calculating Pressure Changes Due to Temperature Changes: Tire Pressure
Suppose your bicycle tire is fully inflated, with an absolute pressure of —a gauge pressure of just under —at a temperature of What is the pressure after its temperature has risen to ? Assume that there are no appreciable leaks or changes in volume.
Strategy
The pressure in the tire is changing only because of changes in temperature. First, we need to identify what we know and what we want to know, and then identify an equation to solve for the unknown.
We know the initial pressure , the initial temperature , and the final temperature . We must find the final pressure . How can we use the equation ? At first, it may seem that not enough information is given, because the volume and number of atoms are not specified. What we can do is use the equation twice: and . If we divide by , we can come up with an equation that allows us to solve for .
Since the volume is constant, and are the same and they cancel out. The same is true for and , and , which is a constant. Therefore,
We can then rearrange this to solve for
where the temperature must be in units of kelvins, because and are absolute temperatures.
Solution
1. Convert temperatures from Celsius to Kelvin.
2. Substitute the known values into the equation.
Discussion
The final temperature is about 6 percent greater than the original temperature, so the final pressure is about 6 percent greater as well. Note that absolute pressure and absolute temperature must be used in the ideal gas law.
Making Connections: Take-Home Experiment—Refrigerating a Balloon
Inflate a balloon at room temperature. Leave the inflated balloon in the refrigerator overnight. What happens to the balloon, and why?
Example 13.7 Calculating the Number of Molecules in a Cubic Meter of Gas
How many molecules are in a typical object, such as gas in a tire or water in a drink? We can use the ideal gas law to give us an idea of how large typically is.
Calculate the number of molecules in a cubic meter of gas at standard temperature and pressure (STP), which is defined to be and atmospheric pressure.
Strategy
Because pressure, volume, and temperature are all specified, we can use the ideal gas law , to find .
Solution
1. Identify the knowns.
2. Identify the unknown: number of molecules, .
3. Rearrange the ideal gas law to solve for .
4. Substitute the known values into the equation and solve for .
Discussion
This number is undeniably large, considering that a gas is mostly empty space. is huge, even in small volumes. For example, of a gas at STP has molecules in it. Once again, note that is the same for all types or mixtures of gases.