Kinetic-Molecular Theory: _{Molecule collisions, the mean free path, and modern KMT}

Bernoulli proposes that molecule collisions cause air pressure

In the 17th century, the Italian mathematician Evangelista Torricelli built the first mercury barometer by filling a glass tube sealed at one end with mercury and then inverting the open end into a tub full of the liquid metal. To the surprise of his contemporaries, the tube remained partially filled—almost as if something was pushing down on the mercury in the tub, and forcing the liquid metal up the tube (Figure 2). Most significantly, the level at which mercury rose in the tube changed from day to day, challenging scientists to explain how mercury was forced up the closed glass tube.

The British scientists Robert Boyle and Robert Hooke devised an experiment to figure out what was pressing down on the mercury. Working with a candy cane-shaped glass tube that had its short leg sealed off, Boyle poured in just enough mercury to fill the tube’s curve and trap air inside the short leg. When he poured in still more mercury, Boyle saw that the while the volume of trapped air shrank, the air somehow pushed against the mercury and forced it back up the long leg. Boyle reasoned that air must also weigh down on the mercury in the Torricelli’s tub and exert the force driving mercury partially up the tube (to learn more about Boyle’s experiment, see our module Properties of Gases).

But how could air—which many scientists had regarded as an indivisible element—weigh down on mercury? By the early 18th century, scientists realized that air was made up of tiny particles. However, these same scientists couldn’t imagine air particles just floating in space. They assumed that the particles vibrated and spun while being held in place by an invisible substance called ether.

The Swiss mathematician Daniel Bernoulli had a different idea about how particles could be suspended in air. In his 1738 book Hydrodynamica, Bernoulli sketched out a thought experiment illustrating how the linear motion of air particles could exert pressure. Bernoulli first asked his readers to imagine a cylinder fitted with a movable piston. Next, he directed the reader to picture the air particles as tiny spheres that zipped around in all directions, colliding with each other and the piston. These numerous, constant collisions would “kick” the piston aloft. Furthermore, Bernoulli suggested that if the air was heated, the particles would zoom faster, striking the piston more often and kicking the piston still higher in the cylinder.

With his thought experiment, Bernoulli was the first to develop several assumptions about molecules and heat that are integral to modern KMT. Like modern KMT, Bernoulli assumed that molecules behave like tiny spheres in constant linear motion. Working from this assumption, he reasoned that the molecules would constantly collide with each other and the walls of a container, thereby exerting pressure on these walls. Importantly, he also assumed that heat affects the movement of molecules.

Though largely correct, Bernoulli’s ideas were mostly dismissed by contemporary scientists. He didn’t offer any experimental data to support his ideas. Furthermore, accepting his ideas required a literal belief in atoms, which many scientists were skeptical existed up to the 19th century. Equating motion with temperature implied that there must be an absolute minimum temperature at which point all motion ceased.

And finally, Bernoulli’s kinetic theory competed with caloric theory, a prominent idea at the time. According to caloric theory, caloric was a “heat substance” that engulfed gas molecules, causing them to repel each other so forcefully they would shoot into the walls of a container. This competing idea about what caused air pressure was championed by influential chemists such as Antoine Lavoisier and John Dalton, while Bernoulli’s idea was largely ignored until the 19th century. Unfortunately, it is not uncommon for good ideas to take time to be accepted in science as previously accepted theories are disproven. However, over time, scientific progress assures that theories which better explain the data collected take hold, as Bernoulli’s did.

Clausius incorporates energy into kinetic theory

Unlike Lavoisier and Dalton, the 19th century German physicist Rudolf Clausius rejected caloric theory. Instead of regarding heat as a substance that surrounds molecules, Clausius proposed that heat is a form of energy that affects the temperature of matter by changing the motion of molecules in matter. This kinetic theory of heat enabled Clausius to study and predict the flow of heat—a field we now call thermodynamics (for more information, see our module Thermodynamics I).

In his 1857 paper, “On the nature of the motion which we call heat,” Clausius speculated on how heat energy, temperature, and molecule motion could explain gas behavior. In doing this, he proposed several ideas about the molecules of gases. These ideas have come to be accepted for ideal gases—theoretical gases that perfectly obey the ideal gas equation (for more information, see our Properties of Gases module). Clausius proposed that the space taken up by ideal gas molecules should be regarded as infinitesimal when compared to the space occupied by the whole gas – in other words, a gas consists mostly of empty space. Second, he suggested that the intermolecular forces between molecules should be treated as infinitesimal.

A key part of Clausius’s ideas was his work on the mathematical relationship between heat, temperature, molecule motion, and kinetic energy—the energy of motion. He proposed that the net kinetic energy of the molecules in an ideal gas is directly proportional to the gas’s absolute temperature, T. Its kinetic energy, E_k, is therefore determined by the number of gas molecules, n, which each have a molecule mass of m, and are moving with velocity u, as shown below:

With this equation and observational data on the weights and volumes of gases at specific temperatures, Clausius was able to calculate the average speeds of gas molecules such as oxygen (an astonishing 461 m/s!). However, the Dutch meteorologist Christoph Hendrik Diederik Buys-Ballot quickly pointed out a problem with these speed calculations. If gas molecules moved hundreds of meters per second, then an odorous gas (like in perfume) should spread across a room almost instantly. Instead, perfumes and other scents usually took several minutes to reach people across a room. This suggested that either Clausius’s mathematical relationship was wrong, or that something more complicated was happening with real gas molecules.

Clausius proposes the idea of mean free path between molecule collisions

Buys-Ballot’s objection forced Clausius to re-think his ideas about gas molecules. If a gas molecule could travel at 461 m/s, but still take minutes to cross a room, it must be encountering lots of obstacles—such as other gas molecules. Clausius realized then that one of his core ideas about ideal gas molecules had to change.

In 1859, Clausius published a paper proposing that, instead of being infinitesimally small, gas molecules had to be big enough that it could collide with another molecule, and there had to be so many fast-moving gas molecules present that it couldn’t travel far before doing so. The average distance that a molecule travels between collisions has come to be known as its mean free path. Clausius realized that while the mean free path must be very big compared to the actual size of the molecules, it would still have to be small enough that a fast-moving molecule would collide with other molecules many times each second (Figure 3).

Thus, gas molecules are constantly colliding and changing directions. While it’s tempting to picture molecules colliding every few seconds like bumper cars at an amusement park, the collisions are so much more frequent. For example, at room temperature, one oxygen molecule travels an average distance of 67 nm (almost 1,500 times narrower than the width of a human hair) before colliding with another molecule. And this single molecule collides with others 7.2 billion times per second! This astounding frequency of collisions explains how gas molecules can zip along at hundreds of meters per second, but still take minutes to cross a room.

Clausius’s idea about mean free path was vital to how Langmuir solved the blackening light bulb problem. Thanks to Clausius, Langmuir understood that he needed to decrease the mean free path for the tungsten atoms sublimating off the filament. In a vacuum, the mean free path was very long, and the tungsten atoms quickly make their way from the filament to the inside of the bulb. By adding an inert gas like argon, Langmuir increased the number of molecules in the bulb and the frequency of collisions—thereby decreasing the mean free path, and increasing the bulb’s life.

Clausius’s ideas about mean free path, molecule motion, and kinetic energy intrigued James Clerk Maxwell, a contemporary Scottish physicist. In 1860, Maxwell published his first kinetic theory paper expanding on Clausius’s work. Building on Clausius’s calculations for the average speed of oxygen and other molecules, Maxwell further developed the idea that the gas molecules in a sample of gas are moving at different speeds. Within this range of possible speeds, some molecules are slower than the average and some faster (Figure 4); furthermore, a molecule’s speed can change when it collides with another molecule. Only a tiny number of the gas molecules are actually moving at the slowest and fastest speeds possible—but we know now that this small number of speedy molecules are especially important, because they are the most likely molecules to undergo a chemical reaction.

Along with these ideas, Maxwell proposed that gas particles should be treated mathematically as spheres that undergo perfectly elastic collisions. This means that the net kinetic energy of the spheres is the same before and after they collide, even if their velocities change. Together, Clausius and Maxwell developed several key assumptions that are vital to modern KMT.

Modern KMT

A major use of modern KMT is as a framework for understanding gases and predicting their behavior. KMT links the microscopic behaviors of ideal gas molecules to the macroscopic properties of gases. In its current form, KMT makes five assumptions about ideal gas molecules:

1. Gases consist of many molecules in constant, random, linear motion.
2. The volume of all the molecules is negligible compared to the gas’s total volume.
3. Intermolecular forces are negligible.
4. The average kinetic energy of all molecules does not change, so long as the gas’s temperature is constant. In other words, collisions between molecules are perfectly elastic.
5. The average kinetic energy of all molecules is proportional to the absolute temperature of the gas. This means that, at any temperature, gas molecules in equilibrium have the same average kinetic energy (but NOT the same velocity and mass).

With KMT’s assumptions, scientists are able to describe on a molecular level the behaviors of gases. These behaviors are common to all gases because of the relationships between gas pressure, volume, temperature, and amount, which are described and predicted by the gas laws (for more on the gas laws, please see our Properties of Gases module). But KMT and the gas laws are useful for understanding more than abstract ideas about chemistry. With KMT and the gas laws, we can better understand the behaviors of real gases, such as the air we use to inflate tires, as we’ll explore more below.

KMT and Boyle’s law

Boyle’s law describes how, for a fixed amount of gas, its volume is inversely proportional to its pressure (Figure 5). This means that if you took all the air from a fully inflated bike tire and put the air inside a much larger, empty car tire, the air would not be able to exert enough pressure to inflate the car tire. While this example about the relationship between gas volume and pressure may seem intuitive, KMT can help us understand the relationship on a molecular level. According to KMT, air pressure depends on how often and how forcefully air molecules collide with tire walls. So when the volume of the container increases (like when we transfer air from the bike tire to the car tire), the air molecules have to travel farther before they can collide with the tire’s walls. This means that there are fewer collisions per unit of time, which results in lower pressure (and an underinflated car tire).

KMT and Charles’s law

Charles’s law describes how a gas’s volume is directly proportional to its absolute temperature (Figure 6)—and also why your car tire pressure increases the longer you drive (and thus why you should always measure tire pressure after your car has been parked for a long period). Using KMT, we can understand that as the friction between the tires and road raises the air temperature inside the tires, the air molecules’ kinetic energy and speed are also increasing. Because the molecules are zipping around faster, they collide more frequently and more forcefully with the tire’s walls, thereby increasing the pressure. Both Charles’s law and KMT also explain why tire pressure decreases after you park. As the tires cool down, the air molecules move more slowly and collide less with the tire’s walls, thereby exerting less pressure.

KMT's limitations

While KMT is a useful tool for understanding the linked behaviors of molecules and matter, particularly gases, KMT does have limitations related to how its theoretical assumptions differ from the behavior of real matter. In particular, KMT’s assumptions that intermolecular forces are negligible, and the volume of molecules is negligible, aren’t always valid. Real gas molecules do experience intermolecular forces. As pressure on a real gas increases and forces its molecules closer together, the molecules can attract one another. This attraction slows down the molecules just a little bit before they slam into one another or the walls of a container, so that the pressure inside a container of real gas molecules is slightly lower than we would expect based on KMT. These intermolecular forces are particularly influential when gas molecules are moving more slowly, such as at low temperature.

While growing pressure on a real gas initially allows its intermolecular forces to have more influence, a different factor gains more influence as the pressure continues to grow. While KMT assumes that gas molecules have no volume, real gas molecules do have volume. This gives a real gas greater volume at high pressure than would be predicted from KMT. Furthermore, as a real gas is compressed, the mean free path of its molecules decreases and the molecules collide more often—thereby increasing the pressure exerted by a real gas compared to KMT’s prediction.

Ultimately, KMT is most useful and accurate when gases are under conditions that cause molecules to behave consistently with KMT’s assumptions. These conditions often happen at low pressure, where molecules have lots of empty space to move in, and the molecule volumes are very small compared to the total volume. And the conditions often occur at high temperature, when the molecules possess a high kinetic energy and fast speed, which lets them overcome the attractive forces between molecules.

Ultimately, KMT provides assumptions about molecule behavior that can be used both as the basis for other theories about molecules, and to solve real-world problems. Clausius’ concept of a molecule’s mean free path underlies our modern ideas of diffusion and Brownian motion, which can explain why scents from perfume or baking cookies take so long to cross a room. And while Langmuir used KMT’s assumptions to develop a longer-lasting light bulb, still other scientists are deploying their knowledge of KMT in more dramatic and controversial ways. By understanding how real gas molecules behave and move, scientists are able to separate gas molecules from each other based on tiny differences in mass—a key principle behind, for example, how uranium isotopes are enriched for use in nuclear weapons.