When the number of molecules of a gas is increased what happened to the volume of its container?

Learning Objectives

• Use the kinetic-molecular theory’s postulates to explain the gas laws

The gas laws that we have seen to this point, as well as the ideal gas equation, are empirical, that is, they have been derived from experimental observations. The mathematical forms of these laws closely describe the macroscopic behavior of most gases at pressures less than about 1 or 2 atm. Although the gas laws describe relationships that have been verified by many experiments, they do not tell us why gases follow these relationships.

The test of the KMT and its postulates is its ability to explain and describe the behavior of a gas. The various gas laws can be derived from the assumptions of the KMT, which have led chemists to believe that the assumptions of the theory accurately represent the properties of gas molecules. We will first look at the individual gas laws (Boyle’s, Charles’s, Amontons’s, Avogadro’s, and Dalton’s laws) conceptually to see how the KMT explains them. Then, we will more carefully consider the relationships between molecular masses, speeds, and kinetic energies with temperature, and explain Graham’s law.

Recalling that gas pressure is exerted by rapidly moving gas molecules and depends directly on the number of molecules hitting a unit area of the wall per unit of time, we see that the KMT conceptually explains the behavior of a gas as follows:

• Amontons’s law. If the temperature is increased, the average speed and kinetic energy of the gas molecules increase. If the volume is held constant, the increased speed of the gas molecules results in more frequent and more forceful collisions with the walls of the container, therefore increasing the pressure (Figure \(\PageIndex{1a}\)).
• Charles’s law. If the temperature of a gas is increased, a constant pressure may be maintained only if the volume occupied by the gas increases. This will result in greater average distances traveled by the molecules to reach the container walls, as well as increased wall surface area. These conditions will decrease both the frequency of molecule-wall collisions and the number of collisions per unit area, the combined effects of which balance the effect of increased collision forces due to the greater kinetic energy at the higher temperature.
• Boyle’s law. If the gas volume is decreased, the container wall area decreases and the molecule-wall collision frequency increases, both of which increase the pressure exerted by the gas (Figure \(\PageIndex{1b}\)).
• Avogadro’s law. At constant pressure and temperature, the frequency and force of molecule-wall collisions are constant. Under such conditions, increasing the number of gaseous molecules will require a proportional increase in the container volume in order to yield a decrease in the number of collisions per unit area to compensate for the increased frequency of collisions (Figure \(\PageIndex{1c}\)).
• Dalton’s Law. Because of the large distances between them, the molecules of one gas in a mixture bombard the container walls with the same frequency whether other gases are present or not, and the total pressure of a gas mixture equals the sum of the (partial) pressures of the individual gases.

According to Graham’s law, the molecules of a gas are in rapid motion and the molecules themselves are small. The average distance between the molecules of a gas is large compared to the size of the molecules. As a consequence, gas molecules can move past each other easily and diffuse at relatively fast rates.

The rate of effusion of a gas depends directly on the (average) speed of its molecules:

\[\textrm{effusion rate} ∝ u_\ce{rms}\]

Using this relation, and the equation relating molecular speed to mass, Graham’s law may be easily derived as shown here:

\[u_\ce{rms}=\sqrt{\dfrac{3RT}{m}}\]

\[m=\dfrac{3RT}{u^2_\ce{rms}}=\dfrac{3RT}{\overline{u}^2}\]

\[\mathrm{\dfrac{effusion\: rate\: A}{effusion\: rate\: B}}=\dfrac{u_\mathrm{rms\:A}}{u_\mathrm{rms\:B}}=\dfrac{\sqrt{\dfrac{3RT}{m_\ce{A}}}}{\sqrt{\dfrac{3RT}{m_\ce{B}}}}=\sqrt{\dfrac{m_\ce{B}}{m_\ce{A}}}\]

The ratio of the rates of effusion is thus derived to be inversely proportional to the ratio of the square roots of their masses. This is the same relation observed experimentally and expressed as Graham’s law.

The kinetic molecular theory is a simple but very effective model that effectively explains ideal gas behavior. The theory assumes that gases consist of widely separated molecules of negligible volume that are in constant motion, colliding elastically with one another and the walls of their container with average velocities determined by their absolute temperatures. The individual molecules of a gas exhibit a range of velocities, the distribution of these velocities being dependent on the temperature of the gas and the mass of its molecules.

kinetic molecular theory theory based on simple principles and assumptions that effectively explains ideal gas behavior root mean square velocity (urms) measure of average velocity for a group of particles calculated as the square root of the average squared velocity

Contributors and Attributions

Knowing how much gas is available for a dive is crucial to a diver's survival. The tank on the diver's back is equipped with gauges to indicate how much gas is present and what the pressure is. A basic knowledge of gas behavior allows the diver to assess how long they can stay underwater without developing problems.

Volume is a third way to measure the amount of matter, after item count and mass. With liquids and solids, volume varies greatly depending on the density of the substance. This is because solid and liquid particles are packed close together with very little space in between the particles. However, gases are largely composed of empty spaces between the actual gas particles (see figure below).

In 1811, Amadeo Avogadro explained that the volumes of all gases can be easily determined. Avogadro's hypothesis states that equal volumes of all gases at the same temperature and pressure contain equal numbers of particles. Since the total volume that a gas occupies is made up primarily of the empty space between the particles, the actual size of the particles themselves is nearly negligible. A given volume of a gas with small light particles, such as hydrogen \(\left( \ce{H_2} \right)\), contains the same number of particles as the same volume of a heavy gas with large particles, such as sulfur hexafluoride, \(\ce{SF_6}\).

Gases are compressible, meaning that when put under high pressure, the particles are forced closer to one another. This decreases the amount of empty space and reduces the volume of the gas. Gas volume is also affected by temperature. When a gas is heated, its molecules move faster and the gas expands. Because of the variation in gas volume due to pressure and temperature changes, the comparison of gas volumes must be done at standard temperature and pressure. Standard temperature and pressure (STP) is defined as \(0^\text{o} \text{C}\) \(\left( 273.15 \: \text{K} \right)\) and \(1 \: \text{atm}\) pressure. The molar volume of a gas is the volume of one mole of a gas at STP. At STP, one mole (\( 6.02 \times 10^{23}\) representative particles) of any gas occupies a volume of \(22.4 \: \text{L}\) (figure below).

The figure below illustrates how molar volume can be seen when comparing different gases. Samples of helium \(\left( \ce{He} \right)\), nitrogen \(\left( \ce{N_2} \right)\), and methane \(\left( \ce{CH_4} \right)\) are at STP. Each contains 1 mole or \(6.02 \times 10^{23}\) particles. However, the mass of each gas is different and corresponds to the molar mass of that gas: \(4.00 \: \text{g/mol}\) for \(\ce{He}\), \(28.0 \: \text{g/mol}\) for \(\ce{N_2}\), and \(16.0 \: \text{g/mol}\) for \(\ce{CH_4}\).

Summary

• Equal volumes of gases at the same conditions contain the same number of particles.
• Standard temperature and pressure is abbreviated (STP).
• Standard temperature is 0°C (273.15 K) and standard pressure is 1 atm. 
• At STP, one mole of any gas occupies a volume of 22.4 L

Review

• A container is filled with gas, what do we know about the space actually taken up by a gas?
• Why do we need to do all our comparisons at the same temperature and pressure?
• At standard temperature and pressure, 1 mole of gas is always equal to how many liters?

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