,
molecules, or
ions) move around freely in the absence of an applied
electric field.|184x184pxIf one could observe a gas under a powerful microscope, one would see a collection of particles without any definite shape or volume that are in more or less random motion. These gas particles only change direction when they collide with another particle or with the sides of the container. This
microscopic view of gas is well-described by
statistical mechanics, but it can be described by many different theories. The
kinetic theory of gases, which makes the assumption that these collisions are perfectly
elastic, does not account for intermolecular forces of attraction and repulsion.
Kinetic theory of gases Kinetic theory provides insight into the macroscopic properties of gases by considering their molecular composition and motion. Starting with the definitions of
momentum and
kinetic energy, one can use the
conservation of momentum and geometric relationships of a cube to relate macroscopic system properties of temperature and pressure to the microscopic property of kinetic energy per molecule. The theory provides averaged values for these two properties. The
kinetic theory of gases can help explain how the system (the collection of gas particles being considered) responds to changes in temperature, with a corresponding change in
kinetic energy. For example: Imagine you have a sealed container of a fixed-size (a
constant volume), containing a fixed-number of gas particles; starting from
absolute zero (the theoretical temperature at which atoms or molecules have no thermal energy, i.e. are not moving or vibrating), you begin to add energy to the system by heating the container, so that energy transfers to the particles inside. Once their
internal energy is above
zero-point energy, meaning their
kinetic energy (also known as
thermal energy) is non-zero, the gas particles will begin to move around the container. As the box is further heated (as more energy is added), the individual particles increase their average speed as the system's total internal energy increases. The higher average-speed of all the particles leads to a greater
rate at which
collisions happen (i.e. greater number of collisions per unit of time), between particles and the container, as well as between the particles themselves. The
macroscopic, measurable quantity of
pressure, is the direct result of these
microscopic particle collisions with the surface, over which, individual molecules exert a small force, each contributing to the total force applied within a specific area. (
Read .) Likewise, the macroscopically measurable quantity of
temperature, is a quantification of the overall amount of
motion, or kinetic energy that the particles exhibit. (
Read .)
Thermal motion and statistical mechanics In the
kinetic theory of gases, kinetic energy is assumed to purely consist of linear translations according to a
speed distribution of
particles in the system. However, in
real gases and other real substances, the motions which define the kinetic energy of a system (which collectively determine the temperature), are much more complex than simple linear
translation due to the more complex structure of molecules, compared to single atoms which act similarly to
point-masses. In real thermodynamic systems, quantum phenomena play a large role in determining thermal motions. The random, thermal motions (kinetic energy) in molecules is a combination of a finite set of possible motions including translation, rotation, and
vibration. This finite range of possible motions, along with the finite set of molecules in the system, leads to a finite number of
microstates within the system; we call the set of all microstates an
ensemble. Specific to atomic or molecular systems, we could potentially have three different kinds of ensemble, depending on the situation:
microcanonical ensemble,
canonical ensemble, or
grand canonical ensemble. Specific combinations of microstates within an ensemble are how we truly define
macrostate of the system (temperature, pressure, energy, etc.). In order to do that, we must first count all microstates though use of a
partition function. The use of statistical mechanics and the partition function is an important tool throughout all of physical chemistry, because it is the key to connection between the microscopic states of a system and the macroscopic variables which we can measure, such as temperature, pressure, heat capacity, internal energy, enthalpy, and entropy, just to name a few. (
Read:
Partition function Meaning and significance) Using the partition function to find the energy of a molecule, or system of molecules, can sometimes be approximated by the
Equipartition theorem, which greatly-simplifies calculation. However, this method assumes all molecular
degrees of freedom are equally populated, and therefore equally utilized for storing energy within the molecule. It would imply that internal energy changes linearly with temperature, which is not the case. This ignores the fact that
heat capacity changes with temperature, due to certain degrees of freedom being unreachable (a.k.a. "frozen out") at lower temperatures. As internal energy of molecules increases, so does the ability to store energy within additional degrees of freedom. As more degrees of freedom become available to hold energy, this causes the molar heat capacity of the substance to increase..
Brownian motion Brownian motion is the mathematical model used to describe the random movement of particles suspended in a fluid. The gas particle animation, using pink and green particles, illustrates how this behavior results in the spreading out of gases (
entropy). These events are also described by
particle theory. Since it is at the limit of (or beyond) current technology to observe individual gas particles (atoms or molecules), only theoretical calculations give suggestions about how they move, but their motion is different from Brownian motion because Brownian motion involves a smooth drag due to the frictional force of many gas molecules, punctuated by violent collisions of an individual (or several) gas molecule(s) with the particle. The particle (generally consisting of millions or billions of atoms) thus moves in a jagged course, yet not so jagged as would be expected if an individual gas molecule were examined.
Intermolecular forces - the primary difference between Real and Ideal gases Forces between two or more molecules or atoms, either attractive or repulsive, are called
intermolecular forces. Intermolecular forces are experienced by molecules when they are within physical proximity of one another. These forces are very important for properly modeling molecular systems, as to accurately predict the microscopic behavior of molecules in
any system, and therefore, are necessary for accurately predicting the physical properties of gases (and liquids) across wide variations in physical conditions. Arising from the study of
physical chemistry, one of the most prominent intermolecular forces throughout physics, are
van der Waals forces. Van der Waals forces play a key role in determining nearly all
physical properties of fluids such as
viscosity,
flow rate, and
gas dynamics (see physical characteristics section). The van der Waals interactions between gas molecules, is the reason why modeling a "real gas" is more mathematically difficult than an "
ideal gas". Ignoring these proximity-dependent forces allows a
real gas to be treated like an
ideal gas, which greatly simplifies calculation. ) which occur as the pressure is varied. The compressibility factor Z, is equal to the ratio Z = PV/nRT. An ideal gas, with compressibility factor Z = 1, is described by the horizontal line where the y-axis is equal to 1. Non-ideality can be described as the deviation of a gas above or below Z = 1. The intermolecular attractions and repulsions between two gas molecules depend on the distance between them. The combined attractions and repulsions are well-modelled by the
Lennard-Jones potential, which is one of the most extensively studied of all
interatomic potentials describing the
potential energy of molecular systems. Due to the general applicability and importance, the Lennard-Jones model system is often referred to as 'Lennard-Jonesium'. The Lennard-Jones potential between molecules can be broken down into two separate components: a long-distance attraction due to the
London dispersion force, and a short-range repulsion due to electron-electron
exchange interaction (which is related to the
Pauli exclusion principle). When two molecules are relatively distant (meaning they have a high
potential energy), they experience a weak attracting force, causing them to move toward each other, lowering their potential energy. However, if the molecules are
too far away, then they would not experience attractive force of any significance. Additionally, if the molecules get
too close then they will collide, and experience a
very high repulsive force (modelled by
Hard spheres) which is a
much stronger force than the attractions, so that any attraction due to proximity is disregarded. As two molecules approach each other, from a distance that is
neither too-far,
nor too-close, their attraction increases as the magnitude of their potential energy increases (becoming more negative), and lowers their total internal energy. The attraction causing the molecules to get closer, can only happen if the molecules remain in proximity for the duration of time it takes to physically
move closer. Therefore, the attractive forces are strongest when the molecules move at
low speeds. This means that the attraction between molecules is
significant when gas temperatures is
low. However, if you were to isothermally compress this cold gas into a small volume,
forcing the molecules into close proximity, and raising the pressure, the repulsions will begin to dominate over the attractions, as the rate at which collisions are happening will increase significantly. Therefore, at low temperatures, and low pressures,
attraction is the dominant intermolecular interaction. If two molecules are moving at high speeds, in arbitrary directions, along non-intersecting paths, then they will not spend enough time in proximity to be affected by the attractive London-dispersion force. If the two molecules collide, they are moving too fast and their kinetic energy will be much greater than any attractive potential energy, so they will only experience repulsion upon colliding. Thus, attractions between molecules can be neglected at
high temperatures due to high speeds. At high temperatures, and high pressures,
repulsion is the dominant intermolecular interaction. Accounting for the above stated effects which cause these attractions and repulsions,
real gases, delineate from the
ideal gas model by the following generalization: • At low temperatures, and low pressures, the volume occupied by a real gas, is
less than the volume predicted by the ideal gas law. • At high temperatures, and high pressures, the volume occupied by a real gas, is
greater than the volume predicted by the ideal gas law. ==Mathematical models==