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.
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. 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. This means that there are fewer collisions per unit of time, which results in lower pressure and an underinflated car tire.
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. 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. 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. 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.
Over four hundred years, scientists including Rudolf Clausius and James Clerk Maxwell developed the kinetic-molecular theory KMT of gases, which describes how molecule properties relate to the macroscopic behaviors of an ideal gas—a theoretical gas that always obeys the ideal gas equation. 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.
Kinetic-molecular theory states that molecules have an energy of motion kinetic energy that depends on temperature. Rudolf Clausius developed the kinetic theory of heat, which relates energy in the form of heat to the kinetic energy of molecules.
Over four hundred years, scientists have developed the kinetic-molecular theory of gases, which describes how molecule properties relate to the macroscopic behaviors of an ideal gas—a theoretical gas that always obeys the ideal gas equation. The molecules in an ideal gas are assumed to have no volume, and to experience no intermolecular forces of attraction or repulsion.
Figure 1 : A new, clear light bulb compared with a blackened one. Comprehension Checkpoint Like modern kinetic-molecular theory, Bernoulli theorized that a. Comprehension Checkpoint Clausius proposed that a.
This provided a solid theoretical grounding for the kinetic theory of gases that had never before been in place. He also explained that the viscosity of a gas should be independent of its density an unexpected result as common sense seemed to indicate the opposite. He had first calculated this in but it was difficult to believe until he and his wife, Katherine, proved it experimentally in Maxwell continued to work sporadically on the kinetic theory of gases for the rest of his career.
In he published his book The Theory of Heat in which he created Maxwell's Demon , a thought experiment designed to contradict the second law of thermodynamics. This problem took 70 years to solve and also stimulated research in information technology. He also wrote a paper On Boltzmann's Theorem on the Average Distribution of Energy in a System of Material Points in which he introduced the idea of ergodic hypothesis which says in Maxwell's words: The system left to itself in its actual state of motion will, sooner or later, pass through every actual phase which is consistent with the equation of energy [10 , p ] This was an excellent piece of work that laid the foundations of statistical mechanics.
He then wrote a paper On Stresses in Rarefied Gases Arising from Inequalities of Temperature in which he developed the theory of rarefied gases to account for observations. This began an entirely new branch of physics that eventually enabled us to understand the upper atmosphere and the fringes of space. So it is clear Maxwell's influence in this area was great. However, real samples of gases comprise molecules with an entire distribution of molecular speeds and trajectories. The distribution function for velocities in the x direction, known as the Maxwell-Boltzmann distribution , is given by:.
This function has two parts: a normalization constant and an exponential term. The normalization constant is derived by noting that. The Maxwell-Boltzmann distribution has to be normalized because it is a continuous probability distribution. As such, the sum of the probabilities for all possible values of v x must be unity. It is then more simply written. Calculating an Average from a Probability Distribution. Calculating an average for a finite set of data is fairly easy.
The average is calculated by. But how does one proceed when the set of data is infinite? Or how does one proceed when all one knows are the probabilities for each possible measured outcome? It turns out that that is fairly simple too! This can also be extended to problems where the measurable properties are not discrete like the numbers that result from rolling a pair of dice but rather come from a continuous parent population.
In this case, if the probability is of measuring a specific outcome, the average value can then be determined by. A value that is useful and will be used in further developments is the average velocity in the x direction. This can be derived using the probability distribution, as shown in the mathematical development box above. This integral will, by necessity, be zero. These motions will have to cancel. Since this cannot be negative, and given the symmetry of the distribution, the problem becomes.
In other words, we will consider only half of the distribution, and then double the result to account for the half we ignored. This expression indicates the average speed for motion of in one direction. However, real gas samples have molecules not only with a distribution of molecular speeds and but also a random distribution of directions. Using normal vector magnitude properties or simply using the Pythagorean Theorem , it can be seen that.
Since the direction of travel is random, the velocity can have any component in x, y, or z directions with equal probability.
0コメント