Learning Objectives The ideal gas equation \[PV = nRT\] describes how gases behave, e.g.:
However, the ideal gas law (nor any of the constituent gas laws) does not explain why gases behave this way? What happens to gas particles when conditions such as pressure and temperature change? This is addressed via Kinetic Molecule Theory.
The Fundamentals of Kinetic Molecular Theory (KMT)The molecules of a gas are in a state of perpetual motion in which the velocity (that is, the speed and direction) of each molecule is completely random and independent of that of the other molecules. This fundamental assumption of the kinetic-molecular model helps us understand a wide range of commonly-observed phenomena. According to this model, most of the volume occupied by a gas is empty space; this is the main feature that distinguishes gases from condensed states of matter (liquids and solids) in which neighboring molecules are constantly in contact. Gas molecules are in rapid and continuous motion; at ordinary temperatures and pressures their velocities are of the order of 0.1-1 km/sec and each molecule experiences approximately 1010collisions with other molecules every second. The five basic tenets of the kinetic-molecular theory are as follows:
Note The Kinetic-Molecular Theory is "the theory of moving molecules." -Rudolf Clausius, 1857
If gases do in fact consist of widely-separated particles, then the observable properties of gases must be explainable in terms of the simple mechanics that govern the motions of the individual molecules. The kinetic molecular theory makes it easy to see why a gas should exert a pressure on the walls of a container. Any surface in contact with the gas is constantly bombarded by the molecules. Figure 5.6.1: Pressure arises from the force due to the acceleration of molecules as they bound off a container's wallsAt each collision, a molecule moving with momentum mv strikes the surface. Since the collisions are elastic, the molecule bounces back with the same velocity in the opposite direction. This change in velocity ΔV is equivalent to an acceleration \(a\); according to Newton's second law, \[F = ma\] with a force, \(F\), that is exerted on the surface of area \(A\) exerting a pressure \[P = \dfrac{f}{A}\]
According to the kinetic molecular theory, the average kinetic energy of an ideal gas is directly proportional to the absolute temperature. Kinetic energy is the energy a body has by virtue of its motion: \[ KE = \dfrac{1}{2}m v^2\] with
As the temperature of a gas rises, the average velocity of the molecules will increase; a doubling of the temperature will increase this velocity by a factor of four. Collisions with the walls of the container will transfer more momentum, and thus more kinetic energy, to the walls. Figure 5.6.2: A microscopic picture of the molecules in a gas (balls) as a specific time. The magnitude of the velocity of each molecule is indicated by the length of the arrow.If the walls are cooler than the gas, they will get warmer, returning less kinetic energy to the gas, and causing it to cool until thermal equilibrium is reached. Because temperature depends on the average kinetic energy, the concept of temperature only applies to a statistically meaningful sample of molecules. We will have more to say about molecular velocities and kinetic energies farther on.
Although the molecules in a sample of gas have an average kinetic energy (and therefore an average speed) the individual molecules move at various speeds. Some are moving fast, others relatively slowly Figure 5.6.3: The distribution of speeds of nitrogen molecules at 0 °C and 100 °C.At higher temperatures at greater fraction of the molecules are moving at higher speeds (Figure 3). What is the speed (velocity) of a molecule possessing average kinetic energy? KMT theory shows the the average kinetic energy (KE) is related to the root mean square (rms) speed \(u\) \[KE = \dfrac{1}{2} m u^2\] This is different from the typical definition of an average speed \( \langle v \rangle\) as demonstrated in Example 5.6.1.
Example 5.6.1 Suppose a gas consists of four molecules with speeds of 3.0, 4.5, 5.2 and 8.3 m/s. What is the difference between the average speed and root mean square speed of this gas? Solution The average speed is: \[ \langle v \rangle = \dfrac{3.0 + 4.5 + 5.2 + 8.3}{4}=5.25\; m/s\] The root mean square speed is: \[u= \sqrt{\dfrac{3.0^2+ 4.5^2+5.2^2+8.3^2}{4}}= 5.59 \; m/s\]
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