A. Boyle's Law
Boyle's Law states: If the temperature of a gas sample is kept constant, the volume of the sample will vary inversely as the pressure varies. This statement means that, if the pressure increases, the volume will decrease. If the pressure decreases, the volume will increase. This law can be expressed as an equation that relates the initial volume (V₁) and the initial pressure (P₁) to the final volume (V₂) and the final pressure (P₂). At constant temperature,

V1 V2

=

P2 P1

Rearranging this equation gives:

V1P1

=

V2P2

or

V2

=

V1

X

P1 P2

Boyle's Law is illustrated in Figure 9.8 which shows a sample of gas enclosed in a container with a movable piston. The container is kept at a constant temperature and subjected to a regularly increasing amount of pressure. When the piston is stationary, the pressure it exerts on the gas sample is equal to the pressure the gas exerts on it. When the pressure on the piston is doubled, it moves downward until the pressure exerted by the gas equals the pressure exerted by the piston. At this point the volume of the gas is halved. If the pressure on the piston is again doubled, the volume of gas decreases to one-fourth its original volume.

FIGURE 9.8 Boyle's Law: At constant temperature, the volume of a gas sample is inversely proportional to the pressure. The curve is a graph based on the data listed in the figure.

 

At the molecular level, the pressure of a gas depends on the number of collisions its molecules have with the walls of the container. If the pressure on the piston is doubled, the volume of the gas decreases by one-half. The gas molecules, now confined in a smaller volume, collide with the walls of the container twice as often and their pressure once again equals that of the piston.

How does Boyle's Law relate to the kinetic molecular theory? The first postulate of the theory states that a gas sample occupies a relatively enormous empty space containing molecules of negligible volume. Changing the pressure on the sample changes only the volume of that empty space - not the volume of the molecules.

B. Charles' Law
Charles' Law states: If the pressure of a gas sample is kept constant, the volume of the sample will vary directly with the temperature in Kelvin (Figure 9.9). As the temperature increases, so will the volume; if the temperature decreases, the volume will decrease. This relationship can be expressed by an equation relating the initial volume (V₁) and initial temperature (T₁ measured in K) to the final volume (V₂) and final temperature (T₂ measured in K). At constant pressure,

V1 V2

=

T1 T2

Rearranging this equation gives:

V2

=

V1

X

T2 T1

or

V2 T2

=

V1 T1

FIGURE 9.9 Charles' Law: At constant pressure, the volume of a gas sample is directly proportional to the temperature in degrees Kelvin.

How does Charles' Law relate to the postulates of the kinetic molecular theory? The theory states that the molecules in a gas sample are in constant, rapid, random motion. This motion allows the tiny molecules to effectively occupy the relatively large volume filled by the entire gas sample.

What is meant by "effectively occupy"? Consider a basketball game, with thirteen persons on the court during a game (ten players and three officials). Standing still, they occupy only a small fraction of the floor. During play they are in constant, rapid motion effectively occupying the entire court. You could not cross the floor without danger of collision. The behavior of the molecules in a gas sample is similar. Although the actual volume of the molecules is only a tiny fraction of the volume of the sample, the constant motion of the molecules allows them to effectively fill that space. As the temperature increases, so does the kinetic energy of the molecules. As they are all of the same mass, an increased kinetic energy must mean an increased velocity. This increased velocity allows the molecules to occupy or fill an increased volume, as do the basketball players in fast action. Similarly, with decreased temperature, the molecules move less rapidly and fill a smaller space.

The next example shows how Charles' Law can be used in calculations.

C. The Combined Gas Law
Frequently, a gas sample is subjected to changes in both temperature and pressure. In such cases, the Boyle's Law and Charles' Law equations can be combined into a single equation, representing the Combined Gas Law, which states: The volume of a gas sample changes inversely with its pressure and directly with its Kelvin temperature.

V2

=

V1

X

T2 T1

X

P1 P2

As before, V₁ , P₁ , and T₁ are the initial conditions, and V₂ , P₂, and T₂ are the final conditions. The Combined Gas Law equation can be rearranged to another frequently used form:

P1V1 T1

=

P2V2 T2

Example: A gas sample occupies a volme of 2.5 L at 10°C and 0.95 atm. What is its volume at 25°C and 0.75 atm? Solution   Initial Conditions Final Conditions volume V1 = 2.5 L V2 = ? pressure P1 = 0.95 atm P2 = 0.75 atm temperature T1 = 10°C = 283 K T2 =25°C = 298 K Check that P1 and P2 are measured in the same units and that both temperatures have been changed to Kelvin. Substitute in the equation: Solving this equation we get: This answer is reasonable. Both the pressure change (lower) and the temperature change (higher) would cause an increased volume.

Example: A gas sample originally ocupies a volume of 0.546 L at 745 mm Hg and 95 °C. What pressure will be needed to contain the sample in 155 mL at 25 °C? Solution   Initial Conditions Final Conditions volume V1 = 0.546 L V2 = 155 mL = 0.155 L pressure P1 = 745 mm Hg P2 = ? temperature T1 = 95°C = 368 K T2 =25°C = 298 K Notice that the units of each property are now the same in the initial and final state. Substituting into the equation:

D. Avogadro's Hypothesis and Molar Volume
Avogadro's Hypothesis states: At the same temperature and pressure, equal volumes of gases contain equal numbers of molecules (Figure 9.10). This statement means that, if one liter of nitrogen at a particular temperature and pressure contains 1.0 X 10²² molecules, then one liter of any other gas at the same temperature and pressure also contains 1.0 X 10²² molecules.

FIGURE 9.10 Avogadro's Hypothesis: At the same temperature and pressure, equal volumes of different gases contain the same number of molecules. Each balloon holds 1.0 L of gas at 20°C and 1 atm pressure. Each contains 0.045 mol or 2.69 X 1022 molecules of gas.

The reasoning behind Avogadro's Hypothesis is not always immediately apparent. But consider that the properties of a gas that relate its volume to its temperature and pressure have been described using the postulates of the kinetic molecular theory without mentioning the composition of the gas. One of the conclusions we drew from those postulates was that, at any pressure, the volume a gas sample occupies depends on the kinetic energy of its molecules and the average of those kinetic energies is dependent only on the temperature of the sample. Stated slightly differently, at a given temperature, all gas molecules, regardless of their chemical composition, have the same average kinetic energy and therefore occupy the same effective volume.

One corollary of Avogadro's Hypothesis is the concept of molar volume. The molar volume (the volume occupied by one mole) of a gas under 1.0 atm pressure and at 0°C (273.15 K) (STP or standard conditions) is, to three significant figures, 22.4 L. Molar volume can be used to calculate gas densities, d_gas, under standard conditions. The equation for this calculation is:

At STP, dgas

=

formula or molecular weight in grams 22.4 liters per mole

Example: Calculate the density of nitrogen under standard conditions (STP) Solution The mole weight of nitrogen is (2 x 14.0) or 28.9 g/mol. The molar volume is 22.4 L. Density is the ratio of mass to volume (mass/volume). Therefore:

A second corollary of Avogadro's Hypothesis is that, at constant temperature and pressure, the volume of a gas sample depends on the number of molecules (or moles) the sample contains. Stated a little differently, if the pressure and temperature are constant, the ratio between the volume of a gas sample and the number of molecules the sample contains is a constant. Stating this ratio as an equation,

Volume of sample 1 Volume of sample 2

=

Number of molecules in sample 1 Number of molecules in sample 2

Example: A gas sample containing 5.02x1023 molecules has a volume of 19.6 L. At the same temperature and pressure, how many molecules will be contained in 7.9 L of the gas? Solution If the temperature and pressure are kept constant, the volume of a gas is directly proportional to the number of molecules it contains. Substituting values in the equation: Rearranging and solving:

E. The Ideal Gas Equation
The various statements relating the pressure, volume, temperature, and number of moles of a gas sample can be combined into one statement: The volume (V) occupied by a gas is directly proportional to its Kelvin temperature (T) and the number of moles (n) of gas in the sample, and it is inversely proportional to its pressure (P). In mathematical form, this statement becomes:

V

=

nRT P

where V = volume, n = moles of sample, P = pressure, T = temperature in K, and R = a proportionality constant known as the gas constant. This equation, called the ideal gas equation, is often seen in the form

PV

=

nRT

The term ideal gas means a gas that obeys exactly the gas laws. Real gases, those gases whose molecules do not follow exactly the postulates of the kinetic molecular theory, exhibit minor variations in behavior from those predicted by the gas laws.

The value of the gas constant R can be determined by substituting into the equation the known values for one mole of gas at standard conditions.

R

=

PV nT

=

1 atm X 22.4 L 1 mol X 273 K

=

0.0821

L-atm mol-K

Table 9.3 shows the value of the gas constant R when the units are different from those shown here.

Example: What volume is occupied by 5.50 g of carbon dioxide at 25°C and 742 torr? Solution 1. Identify the variables in the equation, and convert the units to match those of the gas constant. We will use the gas constant 0.082 L-atm/mol-K. This value establishes the units of volume (L), of pressure (atm), of moles, and temperature (K) to be used in solving the problem. 2. Substituting these values into the ideal gas equation: The units cancel; the answer is reasonable. The amount of carbon dioxide is about one-eight mole. The conditions are not far from STO. The answer (3.13 L) is about one-eight of the molar volume (22.4 L).

Example: Laughing gas is dinitrogen oxide, N2O. What is the density of laughing gas at 30 °C and 745 torr? Wanted: Density (that is mass/volume) of N2O at 30°C and 745 torr. Strategy: The mass of one mole at STP is known. Using the ideal gas equation, we can calculate the volume of one mole at the given conditions. The density at the given conditions can be calculated. Data: Substituting into the ideal gas equation, Calculating the density:

Molar volume is often used to determine the molecular mass of a low-boiling liquid. The compound becomes gaseous at a measured temperature and pressure, and the mass of a measured volume of the vapor is determined. Example 9.10 illustrates this process.