13.3 The Ideal Gas Law

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, ${\text{N}}_2$, and oxygen, ${\text{O}}_2$, 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 $\beta$. 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.

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 $\text{PV}$ is a constant). In the ideal gas model, the volume occupied by its atoms and molecules is a negligible fraction of $V$. The ideal gas law describes the behavior of real gases under most conditions. (Note, for example, that $N$ 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 $P$ is essentially equal to atmospheric pressure, and the volume $V$ increases in direct proportion to the number of atoms and molecules $N$ put into the tire. Once the volume of the tire is constant, the equation $\text{PV}=\text{NkT}$ 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$(N_{\text{A}})$, 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

$$N_{\text{A}}=6\text{.}\text{02}\times {\text{10}}^{\text{23}}{\text{mol}}^{-1}\text{.}$$

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Avogadro’s Number

One mole always contains $6\text{.}\text{02}\times {\text{10}}^{\text{23}}$ particles (atoms or molecules), independent of the element or substance. A mole of any substance has a mass in grams equal to its molecular mass, which can be calculated from the atomic masses given in the periodic table of elements.

$$N_{\text{A}}=6\text{.}\text{02}\times {\text{10}}^{\text{23}}{\text{mol}}^{-1}$$

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Figure 13.19 How big is a mole? On a macroscopic level, one mole of table tennis balls would cover the Earth to a depth of about 40 km.

Check Your Understanding

The active ingredient in a Tylenol pill is 325 mg of acetaminophen $({\text{C}}_8{\text{H}}_9{\text{NO}}_2)$. Find the number of active molecules of acetaminophen in a single pill.

Solution

We first need to calculate the molar mass (the mass of one mole) of acetaminophen. To do this, we need to multiply the number of atoms of each element by the element’s atomic mass.

$$\begin{array}{}(\text{8 moles of carbon})(\text{12 grams/mole})+(\text{9 moles hydrogen})(\text{1 gram/mole})\\ +(\text{1 mole nitrogen})(\text{14 grams/mole})+(\text{2 moles oxygen})(\text{16 grams/mole})=\text{151 g}\end{array}$$

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Then we need to calculate the number of moles in 325 mg.

$$\left(\frac{\text{325 mg}}{\text{151 grams/mole}}\right)\left(\frac{\mathrm{1\; gram}}{\text{1000 mg}}\right)=2.15\times {\text{10}}^{-3}\text{moles}$$

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Then use Avogadro’s number to calculate the number of molecules.

$$N=\left(\text{2.15}\times {\text{10}}^{-3}\text{moles}\right)\left(\text{6.02}\times {\text{10}}^{\text{23}}\text{molecules/mole}\right)=\text{1.30}\times {\text{10}}^{\text{21}}\text{molecules}$$

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Example 13.8

Calculating Moles per Cubic Meter and Liters per Mole

Calculate: (a) the number of moles in $1\text{.}\text{00}{\text{m}}^3$ of gas at STP, and (b) the number of liters of gas per mole.

Strategy and Solution

(a) We are asked to find the number of moles per cubic meter, and we know from Example 13.7 that the number of molecules per cubic meter at STP is $2\text{.}\text{68}\times {\text{10}}^{\text{25}}$. The number of moles can be found by dividing the number of molecules by Avogadro’s number. We let $n$ stand for the number of moles,

$$n{\text{mol/m}}^3=\frac{N{\text{molecules/m}}^3}{6\text{.}\text{02}\times {\text{10}}^{\text{23}}\text{molecules/mol}}=\frac{2\text{.}\text{68}\times {\text{10}}^{\text{25}}{\text{molecules/m}}^3}{6\text{.}\text{02}\times {\text{10}}^{\text{23}}\text{molecules/mol}}=\text{44}\text{.}5{\text{mol/m}}^3\text{.}$$

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(b) Using the value obtained for the number of moles in a cubic meter, and converting cubic meters to liters, we obtain

$$\frac{\left({\text{10}}^3{\text{L/m}}^3\right)}{44\text{.}5{\text{mol/m}}^3}=\text{22}\text{.}5\text{L/mol}\text{.}$$

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Discussion

This value is very close to the accepted value of 22.4 L/mol. The slight difference is due to rounding errors caused by using three-digit input. Again this number is the same for all gases. In other words, it is independent of the gas.

The (average) molar weight of air (approximately 80% ${\text{N}}_2$ and 20% ${\text{O}}_2$ is $M=\text{28}\text{.}8\text{g}\text{.}$ Thus the mass of one cubic meter of air is 1.28 kg. If a living room has dimensions $5\text{m}\times \text{5 m}\times \text{3 m,}$ the mass of air inside the room is 96 kg, which is the typical mass of a human.

Check Your Understanding

The density of air at standard conditions $(P=1\text{atm}$ and $T=\text{20}\text{º}\text{C})$ is $1\text{.}\text{28}{\text{kg/m}}^3$. At what pressure is the density $0\text{.}{\text{64 kg/m}}^3$ if the temperature and number of molecules are kept constant?

Solution

The best way to approach this question is to think about what is happening. If the density drops to half its original value and no molecules are lost, then the volume must double. If we look at the equation $\text{PV}=\text{NkT}$, we see that when the temperature is constant, the pressure is inversely proportional to volume. Therefore, if the volume doubles, the pressure must drop to half its original value, and $P_{\text{f}}=0\text{.}\text{50}\text{atm}\text{.}$

The Ideal Gas Law Restated Using Moles

A very common expression of the ideal gas law uses the number of moles, $n$, rather than the number of atoms and molecules, $N$. We start from the ideal gas law,

$$\text{PV}=\text{NkT,}$$

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and multiply and divide the equation by Avogadro’s number $N_{\text{A}}$. This gives

$$\text{PV}=\frac{N}{N_{\text{A}}}{N}_{\text{A}}\text{kT}\text{.}$$

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Note that $n=N/N_{\text{A}}$ is the number of moles. We define the universal gas constant $R=N_{\text{A}}k$, and obtain the ideal gas law in terms of moles.

Ideal Gas Law (in terms of moles)

The ideal gas law (in terms of moles) is

$$\text{PV}=\text{nRT}.$$

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The numerical value of $R$ in SI units is

$$R=N_{\text{A}}k=\left(6\text{.}\text{02}\times {\text{10}}^{\text{23}}{\text{mol}}^{-1}\right)\left(1\text{.}\text{38}\times {\text{10}}^{-\text{23}}\text{J/K}\right)=8\text{.}\text{31}\text{J}/\text{mol}\cdot \text{K}.$$

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In other units,

$$\begin{array}{lll}R& =& 1\text{.}\text{99}\text{cal/mol}\cdot \text{K}\\ R& =& \text{0}\text{.}\text{0821 L}\cdot \text{atm/mol}\cdot \text{K}\text{.}\end{array}$$

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You can use whichever value of $R$ is most convenient for a particular problem.

Example 13.9

Calculating Number of Moles: Gas in a Bike Tire

How many moles of gas are in a bike tire with a volume of $2\text{.}\text{00}\times {\text{10}}^{–3}{\text{m}}^3(2\text{.}\text{00 L}),$ a pressure of $7\text{.}\text{00}\times {\text{10}}^5\text{Pa}$ (a gauge pressure of just under $\text{90}\text{.}0{\text{lb/in}}^2$), and at a temperature of $\text{18}\text{.}0\text{º}\text{C}$?

Strategy

Identify the knowns and unknowns, and choose an equation to solve for the unknown. In this case, we solve the ideal gas law, $\text{PV}=\text{nRT}$, for the number of moles $n$.

Solution

1. Identify the knowns.

$$\begin{array}{lll}P& =& 7\text{.}\text{00}\times {\text{10}}^5\text{Pa}\\ V& =& 2\text{.}\text{00}\times {\text{10}}^{-3}{\text{m}}^3\\ T& =& \text{18}\text{.}0\text{º}\text{C}=\text{291 K}\\ R& =& 8\text{.}\text{31}\text{J/mol}\cdot \text{K}\end{array}$$

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2. Rearrange the equation to solve for $n$ and substitute known values.

$$\begin{array}{lll}n& =& \frac{\text{PV}}{\text{RT}}=\frac{\left(7\text{.}\text{00}\times {\text{10}}^5\text{Pa}\right)\left(2\text{.}00\times {\text{10}}^{-3}{\text{m}}^3\right)}{\left(8\text{.}\text{31}\text{J/mol}\cdot \text{K}\right)\left(\text{291}\text{K}\right)}\\ & =& \text{0}\text{.}\text{579}\text{mol}\end{array}$$

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Discussion

The most convenient choice for $R$ in this case is $8\text{.}\text{31}\text{J/mol}\cdot \text{K,}$ because our known quantities are in SI units. The pressure and temperature are obtained from the initial conditions in Example 13.6, but we would get the same answer if we used the final values.

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 $\text{PV}=\text{NkT}$ is $\text{NkT}$. This term is roughly the amount of translational kinetic energy of $N$ atoms or molecules at an absolute temperature $T$, 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 $\text{PV}$, 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.

Problem-Solving Strategy: The Ideal Gas Law

Step 1 Examine the situation to determine that an ideal gas is involved. Most gases are nearly ideal.

Step 2 Make a list of what quantities are given, or can be inferred from the problem as stated (identify the known quantities). Convert known values into proper SI units (K for temperature, Pa for pressure, ${\text{m}}^3$ for volume, molecules for $N$, and moles for $n$).

Step 3 Identify exactly what needs to be determined in the problem (identify the unknown quantities). A written list is useful.

Step 4 Determine whether the number of molecules or the number of moles is known, in order to decide which form of the ideal gas law to use. The first form is $\text{PV}=\text{NkT}$ and involves $N$, the number of atoms or molecules. The second form is $\text{PV}=\text{nRT}$ and involves $n$, the number of moles.

Step 5 Solve the ideal gas law for the quantity to be determined (the unknown quantity). You may need to take a ratio of final states to initial states to eliminate the unknown quantities that are kept fixed.

Step 6 Substitute the known quantities, along with their units, into the appropriate equation, and obtain numerical solutions complete with units. Be certain to use absolute temperature and absolute pressure.

Step 7 Check the answer to see if it is reasonable: Does it make sense?

Check Your Understanding

Liquids and solids have densities about 1000 times greater than gases. Explain how this implies that the distances between atoms and molecules in gases are about 10 times greater than the size of their atoms and molecules.

Solution

Atoms and molecules are close together in solids and liquids. In gases they are separated by empty space. Thus gases have lower densities than liquids and solids. Density is mass per unit volume, and volume is related to the size of a body (such as a sphere) cubed. So if the distance between atoms and molecules increases by a factor of 10, then the volume occupied increases by a factor of 1000, and the density decreases by a factor of 1000.