Gas

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https://en.wikipedia.org/wiki/Gas

Drifting smoke particles indicate the movement of the surrounding gas.

Gas is one of the four fundamental states of matter. The others are solid, liquid, and plasma. A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or compound molecules made from a variety of atoms (e.g. carbon dioxide). A gas mixture, such as air, contains a variety of pure gases. What distinguishes gases from liquids and solids is the vast separation of the individual gas particles. This separation usually makes a colorless gas invisible to the human observer.

The gaseous state of matter occurs between the liquid and plasma states, the latter of which provides the upper-temperature boundary for gases. Bounding the lower end of the temperature scale lie degenerative quantum gases which are gaining increasing attention. High-density atomic gases super-cooled to very low temperatures are classified by their statistical behavior as either Bose gases or Fermi gases. For a comprehensive listing of these exotic states of matter, see list of states of matter.

Elemental gases

The only chemical elements that are stable diatomic homonuclear molecular gases at STP are hydrogen (H₂), nitrogen (N₂), oxygen (O₂), and two halogens: fluorine (F₂) and chlorine (Cl₂). When grouped with the monatomic noble gases – helium (He), neon (Ne), argon (Ar), krypton (Kr), xenon (Xe), and radon (Rn) – these gases are referred to as "elemental gases".

Etymology

The word gas was first used by the early 17th-century Flemish chemist Jan Baptist van Helmont. He identified carbon dioxide, the first known gas other than air. Van Helmont's word appears to have been simply a phonetic transcription of the Ancient Greek word χάος 'chaos' – the g in Dutch being pronounced like ch in "loch" (voiceless velar fricative, /x/) – in which case Van Helmont simply was following the established alchemical usage first attested in the works of Paracelsus. According to Paracelsus's terminology, chaos meant something like 'ultra-rarefied water'.

An alternative story is that Van Helmont's term was derived from "gahst (or geist), which signifies a ghost or spirit". That story is given no credence by the editors of the Oxford English Dictionary. In contrast, the French-American historian Jacques Barzun speculated that Van Helmont had borrowed the word from the German Gäscht, meaning the froth resulting from fermentation.

Physical characteristics

Because most gases are difficult to observe directly, they are described through the use of four physical properties or macroscopic characteristics: pressure, volume, number of particles (chemists group them by moles) and temperature. These four characteristics were repeatedly observed by scientists such as Robert Boyle, Jacques Charles, John Dalton, Joseph Gay-Lussac and Amedeo Avogadro for a variety of gases in various settings. Their detailed studies ultimately led to a mathematical relationship among these properties expressed by the ideal gas law (see § Ideal and perfect gas section below).

Gas particles are widely separated from one another, and consequently, have weaker intermolecular bonds than liquids or solids. These intermolecular forces result from electrostatic interactions between gas particles. Like-charged areas of different gas particles repel, while oppositely charged regions of different gas particles attract one another; gases that contain permanently charged ions are known as plasmas. Gaseous compounds with polar covalent bonds contain permanent charge imbalances and so experience relatively strong intermolecular forces, although the compound's net charge remains neutral. Transient, randomly induced charges exist across non-polar covalent bonds of molecules and electrostatic interactions caused by them are referred to as Van der Waals forces. The interaction of these intermolecular forces varies within a substance which determines many of the physical properties unique to each gas. A comparison of boiling points for compounds formed by ionic and covalent bonds leads us to this conclusion.

Compared to the other states of matter, gases have low density and viscosity. Pressure and temperature influence the particles within a certain volume. This variation in particle separation and speed is referred to as compressibility. This particle separation and size influences optical properties of gases as can be found in the following list of refractive indices. Finally, gas particles spread apart or diffuse in order to homogeneously distribute themselves throughout any container.

Macroscopic view of gases

See also: Gas kinetics

When observing gas, it is typical to specify a frame of reference or length scale. A larger length scale corresponds to a macroscopic or global point of view of the gas. This region (referred to as a volume) must be sufficient in size to contain a large sampling of gas particles. The resulting statistical analysis of this sample size produces the "average" behavior (i.e. velocity, temperature or pressure) of all the gas particles within the region. In contrast, a smaller length scale corresponds to a microscopic or particle point of view.

Macroscopically, the gas characteristics measured are either in terms of the gas particles themselves (velocity, pressure, or temperature) or their surroundings (volume). For example, Robert Boyle studied pneumatic chemistry for a small portion of his career. One of his experiments related the macroscopic properties of pressure and volume of a gas. His experiment used a J-tube manometer which looks like a test tube in the shape of the letter J. Boyle trapped an inert gas in the closed end of the test tube with a column of mercury, thereby making the number of particles and the temperature constant. He observed that when the pressure was increased in the gas, by adding more mercury to the column, the trapped gas' volume decreased (this is known as an inverse relationship). Furthermore, when Boyle multiplied the pressure and volume of each observation, the product was constant. This relationship held for every gas that Boyle observed leading to the law, (PV=k), named to honor his work in this field.

Shuttle imagery of re-entry phase

There are many mathematical tools available for analyzing gas properties. Boyle's lab equipment allowed the use of just a simple calculation to obtain his analytical results. His results were possible because he was studying gases in relatively low pressure situations where they behaved in an "ideal" manner. These ideal relationships apply to safety calculations for a variety of flight conditions on the materials in use. However, the high technology equipment in use today was designed to help us safely explore the more exotic operating environments where the gases no longer behave in an "ideal" manner. As gases are subjected to extreme conditions, tools to interpret them become more complex, from the Euler equations for inviscid flow to the Navier–Stokes equations that fully account for viscous effects. This advanced math, including statistics and multivariable calculus, adapted to the conditions of the gas system in question, makes it possible to solve such complex dynamic situations as space vehicle reentry. An example is the analysis of the space shuttle reentry pictured to ensure the material properties under this loading condition are appropriate. In this flight situation, the gas is no longer behaving ideally.

Pressure

Main article: Pressure

The symbol used to represent pressure in equations is "p" or "P" with SI units of pascals.

Diagram showing particles' movement as lines.

When describing a container of gas, the term pressure (or absolute pressure) refers to the average force per unit area that the gas exerts on the surface of the container. Within this volume, it is sometimes easier to visualize the gas particles moving in straight lines until they collide with the container (see diagram at top). The force imparted by a gas particle into the container during this collision is the change in momentum of the particle. During a collision only the normal component of velocity changes. A particle traveling parallel to the wall does not change its momentum. Therefore, the average force on a surface must be the average change in linear momentum from all of these gas particle collisions.

Pressure is the sum of all the normal components of force exerted by the particles impacting the walls of the container divided by the surface area of the wall.

Temperature

Main article: Thermodynamic temperature

The symbol used to represent temperature in equations is T with SI units of kelvins.

The speed of a gas particle is proportional to its absolute temperature. The volume of the balloon in the video shrinks when the trapped gas particles slow down with the addition of extremely cold nitrogen. The temperature of any physical system is related to the motions of the particles (molecules and atoms) which make up the [gas] system. In statistical mechanics, temperature is the measure of the average kinetic energy stored in a molecule (also known as the thermal energy). The methods of storing this energy are dictated by the degrees of freedom of the molecule itself (energy modes). Thermal (kinetic) energy added to a gas or liquid (an endothermic process) produces translational, rotational, and vibrational motion. In contrast, a solid can only increase its internal energy by exciting additional vibrational modes, as the crystal lattice structure prevents both translational and rotational motion. These heated gas molecules have a greater speed range (wider distribution of speeds) with a higher average or mean speed. The variance of this distribution is due to the speeds of individual particles constantly varying, due to repeated collisions with other particles. The speed range can be described by the Maxwell–Boltzmann distribution. Use of this distribution implies ideal gases near thermodynamic equilibrium for the system of particles being considered.

Specific volume

Main article: Specific volume

The symbol used to represent specific volume in equations is "v" with SI units of cubic meters per kilogram.

See also: Volume (thermodynamics)

The symbol used to represent volume in equations is "V" with SI units of cubic meters.

When performing a thermodynamic analysis, it is typical to speak of intensive and extensive properties. Properties which depend on the amount of gas (either by mass or volume) are called extensive properties, while properties that do not depend on the amount of gas are called intensive properties. Specific volume is an example of an intensive property because it is the ratio of volume occupied by a unit of mass of a gas that is identical throughout a system at equilibrium. 1000 atoms a gas occupy the same space as any other 1000 atoms for any given temperature and pressure. This concept is easier to visualize for solids such as iron which are incompressible compared to gases. However, volume itself --- not specific --- is an extensive property.

Density

Main article: Density

The symbol used to represent density in equations is ρ (rho) with SI units of kilograms per cubic meter. This term is the reciprocal of specific volume.

Since gas molecules can move freely within a container, their mass is normally characterized by density. Density is the amount of mass per unit volume of a substance, or the inverse of specific volume. For gases, the density can vary over a wide range because the particles are free to move closer together when constrained by pressure or volume. This variation of density is referred to as compressibility. Like pressure and temperature, density is a state variable of a gas and the change in density during any process is governed by the laws of thermodynamics. For a static gas, the density is the same throughout the entire container. Density is therefore a scalar quantity. It can be shown by kinetic theory that the density is inversely proportional to the size of the container in which a fixed mass of gas is confined. In this case of a fixed mass, the density decreases as the volume increases.

Microscopic view of gases

Gas-phase particles (atoms, molecules, or ions) move around freely in the absence of an applied electric field.

If 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

Main articles: Kinetic theory of gases and Maxwell–Boltzmann distribution

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 § Pressure.)

Likewise, the macroscopically measurable quantity of temperature, is a quantification of the overall amount of motion, or kinetic energy that the particles exhibit. (Read § Temperature.)

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.

Random motion of gas particles results in diffusion.

Brownian motion

Main article: 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

Main articles: van der Waals force, Intermolecular force, and Lennard-Jones potential

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.

Isothermal curves depicting the non-ideality of a real gas. The changes in volume (depicted by Z, compressibility factor) 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

Main article: Equation of state

An equation of state (for gases) is a mathematical model used to roughly describe or predict the state properties of a gas. At present, there is no single equation of state that accurately predicts the properties of all gases under all conditions. Therefore, a number of much more accurate equations of state have been developed for gases in specific temperature and pressure ranges. The "gas models" that are most widely discussed are "perfect gas", "ideal gas" and "real gas". Each of these models has its own set of assumptions to facilitate the analysis of a given thermodynamic system. Each successive model expands the temperature range of coverage to which it applies.

Ideal and perfect gas

Main article: Perfect gas

The equation of state for an ideal or perfect gas is the ideal gas law and reads

${\displaystyle PV=nRT,}$

where P is the pressure, V is the volume, n is amount of gas (in mol units), R is the universal gas constant, 8.314 J/(mol K), and T is the temperature. Written this way, it is sometimes called the "chemist's version", since it emphasizes the number of molecules n. It can also be written as

${\displaystyle P=\rho R_{s}T,}$

where ${\displaystyle R_s}$ is the specific gas constant for a particular gas, in units J/(kg K), and ρ = m/V is density. This notation is the "gas dynamicist's" version, which is more practical in modeling of gas flows involving acceleration without chemical reactions.

The ideal gas law does not make an assumption about the heat capacity of a gas. In the most general case, the specific heat is a function of both temperature and pressure. If the pressure-dependence is neglected (and possibly the temperature-dependence as well) in a particular application, sometimes the gas is said to be a perfect gas, although the exact assumptions may vary depending on the author and/or field of science.

For an ideal gas, the ideal gas law applies without restrictions on the specific heat. An ideal gas is a simplified "real gas" with the assumption that the compressibility factor Z is set to 1 meaning that this pneumatic ratio remains constant. A compressibility factor of one also requires the four state variables to follow the ideal gas law.

This approximation is more suitable for applications in engineering although simpler models can be used to produce a "ball-park" range as to where the real solution should lie. An example where the "ideal gas approximation" would be suitable would be inside a combustion chamber of a jet engine. It may also be useful to keep the elementary reactions and chemical dissociations for calculating emissions.

Real gas

21 April 1990 eruption of Mount Redoubt, Alaska, illustrating real gases not in thermodynamic equilibrium.

Main article: Real gas

Each one of the assumptions listed below adds to the complexity of the problem's solution. As the density of a gas increases with rising pressure, the intermolecular forces play a more substantial role in gas behavior which results in the ideal gas law no longer providing "reasonable" results. At the upper end of the engine temperature ranges (e.g. combustor sections – 1300 K), the complex fuel particles absorb internal energy by means of rotations and vibrations that cause their specific heats to vary from those of diatomic molecules and noble gases. At more than double that temperature, electronic excitation and dissociation of the gas particles begins to occur causing the pressure to adjust to a greater number of particles (transition from gas to plasma). Finally, all of the thermodynamic processes were presumed to describe uniform gases whose velocities varied according to a fixed distribution. Using a non-equilibrium situation implies the flow field must be characterized in some manner to enable a solution. One of the first attempts to expand the boundaries of the ideal gas law was to include coverage for different thermodynamic processes by adjusting the equation to read pVⁿ = constant and then varying the n through different values such as the specific heat ratio, γ.

Real gas effects include those adjustments made to account for a greater range of gas behavior:

• Compressibility effects (Z allowed to vary from 1.0)
• Variable heat capacity (specific heats vary with temperature)
• Van der Waals forces (related to compressibility, can substitute other equations of state)
• Non-equilibrium thermodynamic effects
• Issues with molecular dissociation and elementary reactions with variable composition.

For most applications, such a detailed analysis is excessive. Examples where real gas effects would have a significant impact would be on the Space Shuttle re-entry where extremely high temperatures and pressures were present or the gases produced during geological events as in the image of the 1990 eruption of Mount Redoubt.

Permanent gas

Permanent gas is a term used for a gas which has a critical temperature below the range of normal human-habitable temperatures and therefore cannot be liquefied by pressure within this range. Historically such gases were thought to be impossible to liquefy and would therefore permanently remain in the gaseous state. The term is relevant to ambient temperature storage and transport of gases at high pressure.

Historical research

See also: Gas laws and Timeline of fluid and continuum mechanics

Boyle's law

Boyle's equipment

Main article: Boyle's law

Boyle's law was perhaps the first expression of an equation of state. In 1662 Robert Boyle performed a series of experiments employing a J-shaped glass tube, which was sealed on one end. Mercury was added to the tube, trapping a fixed quantity of air in the short, sealed end of the tube. Then the volume of gas was carefully measured as additional mercury was added to the tube. The pressure of the gas could be determined by the difference between the mercury level in the short end of the tube and that in the long, open end. The image of Boyle's equipment shows some of the exotic tools used by Boyle during his study of gases.

Through these experiments, Boyle noted that the pressure exerted by a gas held at a constant temperature varies inversely with the volume of the gas. For example, if the volume is halved, the pressure is doubled; and if the volume is doubled, the pressure is halved. Given the inverse relationship between pressure and volume, the product of pressure (P) and volume (V) is a constant (k) for a given mass of confined gas as long as the temperature is constant. Stated as a formula, thus is:

${\displaystyle PV=k}$

Because the before and after volumes and pressures of the fixed amount of gas, where the before and after temperatures are the same both equal the constant k, they can be related by the equation:

${\displaystyle P_1{V}_1=P_2{V}_2.}$

Charles's law

Main article: Charles's law

In 1787, the French physicist and balloon pioneer, Jacques Charles, found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to the same extent over the same 80 kelvin interval. He noted that, for an ideal gas at constant pressure, the volume is directly proportional to its temperature:

${\displaystyle \frac{V_1}{T_1}=\frac{V_2}{T_2}}$

Gay-Lussac's law

Main article: Gay-Lussac's law

In 1802, Joseph Louis Gay-Lussac published results of similar, though more extensive experiments. Gay-Lussac credited Charles' earlier work by naming the law in his honor. Gay-Lussac himself is credited with the law describing pressure, which he found in 1809. It states that the pressure exerted on a container's sides by an ideal gas is proportional to its temperature.

${\displaystyle \frac{P_1}{T_1}=\frac{P_2}{T_2}}$

Avogadro's law

Main article: Avogadro's law

In 1811, Amedeo Avogadro verified that equal volumes of pure gases contain the same number of particles. His theory was not generally accepted until 1858 when another Italian chemist Stanislao Cannizzaro was able to explain non-ideal exceptions. For his work with gases a century prior, the physical constant that bears his name (the Avogadro constant) is the number of atoms per mole of elemental carbon-12 (6.022×10²³ mol⁻¹). This specific number of gas particles, at standard temperature and pressure (ideal gas law) occupies 22.40 liters, which is referred to as the molar volume.

Avogadro's law states that the volume occupied by an ideal gas is proportional to the amount of substance in the volume. This gives rise to the molar volume of a gas, which at STP is 22.4 dm³/mol (liters per mole). The relation is given by $${\displaystyle \frac{V_1}{n_1}=\frac{V_2}{n_2},}$$ where n is the amount of substance of gas (the number of molecules divided by the Avogadro constant).

Dalton's law

Dalton's notation.

Main article: Dalton's law

In 1801, John Dalton published the law of partial pressures from his work with ideal gas law relationship: The pressure of a mixture of non reactive gases is equal to the sum of the pressures of all of the constituent gases alone. Mathematically, this can be represented for n species as:

Pressure_total = Pressure₁ + Pressure₂ + ... + Pressure_n

The image of Dalton's journal depicts symbology he used as shorthand to record the path he followed. Among his key journal observations upon mixing unreactive "elastic fluids" (gases) were the following:

• Unlike liquids, heavier gases did not drift to the bottom upon mixing.
• Gas particle identity played no role in determining final pressure (they behaved as if their size was negligible).

Special topics

Compressibility

Compressibility factors for air.

Main article: Compressibility factor

Thermodynamicists use this factor (Z) to alter the ideal gas equation to account for compressibility effects of real gases. This factor represents the ratio of actual to ideal specific volumes. It is sometimes referred to as a "fudge-factor" or correction to expand the useful range of the ideal gas law for design purposes. Usually this Z value is very close to unity. The compressibility factor image illustrates how Z varies over a range of very cold temperatures.

Boundary layer

Delta wing in wind tunnel. The shadows form as the indices of refraction change within the gas as it compresses on the leading edge of this wing.

Main article: Boundary layer

Particles will, in effect, "stick" to the surface of an object moving through it. This layer of particles is called the boundary layer. At the surface of the object, it is essentially static due to the friction of the surface. The object, with its boundary layer is effectively the new shape of the object that the rest of the molecules "see" as the object approaches. This boundary layer can separate from the surface, essentially creating a new surface and completely changing the flow path. The classical example of this is a stalling airfoil. The delta wing image clearly shows the boundary layer thickening as the gas flows from right to left along the leading edge.

Turbulence

Satellite view of weather pattern in vicinity of Robinson Crusoe Islands on 15 September 1999, shows a turbulent cloud pattern called a Kármán vortex street

Main article: Turbulence

In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic, stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time. The satellite view of weather around Robinson Crusoe Islands illustrates one example.

Viscosity

Main article: Viscosity

Viscosity, a physical property, is a measure of how well adjacent molecules stick to one another. A solid can withstand a shearing force due to the strength of these sticky intermolecular forces. A fluid will continuously deform when subjected to a similar load. While a gas has a lower value of viscosity than a liquid, it is still an observable property. If gases had no viscosity, then they would not stick to the surface of a wing and form a boundary layer. A study of the delta wing in the Schlieren image reveals that the gas particles stick to one another (see Boundary layer section).

Reynolds number

Main article: Reynolds number

In fluid mechanics, the Reynolds number is the ratio of inertial forces (v_sρ) which dominate a turbulent flow, to viscous forces (μ/L) which is proportional to viscosity. It is one of the most important dimensionless numbers in fluid dynamics and is used, usually along with other dimensionless numbers, to provide a criterion for determining dynamic similitude. As such, the Reynolds number provides the link between modeling results (design) and the full-scale actual conditions. It can also be used to characterize the flow.

Maximum entropy principle

Main article: Principle of maximum entropy

As the total number of degrees of freedom approaches infinity, the system will be found in the macrostate that corresponds to the highest multiplicity. In order to illustrate this principle, observe the skin temperature of a frozen metal bar. Using a thermal image of the skin temperature, note the temperature distribution on the surface. This initial observation of temperature represents a "microstate". At some future time, a second observation of the skin temperature produces a second microstate. By continuing this observation process, it is possible to produce a series of microstates that illustrate the thermal history of the bar's surface. Characterization of this historical series of microstates is possible by choosing the macrostate that successfully classifies them all into a single grouping.

Thermodynamic equilibrium

Main article: Thermodynamic equilibrium

When energy transfer ceases from a system, this condition is referred to as thermodynamic equilibrium. Usually, this condition implies the system and surroundings are at the same temperature so that heat no longer transfers between them. It also implies that external forces are balanced (volume does not change), and all chemical reactions within the system are complete. The timeline varies for these events depending on the system in question. A container of ice allowed to melt at room temperature takes hours, while in semiconductors the heat transfer that occurs in the device transition from an on to off state could be on the order of a few nanoseconds.

Ideal gas law

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Ideal_gas_law

Isotherms of an ideal gas for different temperatures. The curved lines are rectangular hyperbolae of the form y = a/x. They represent the relationship between pressure (on the vertical axis) and volume (on the horizontal axis) for an ideal gas at different temperatures: lines that are farther away from the origin (that is, lines that are nearer to the top right-hand corner of the diagram) correspond to higher temperatures.

The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stated by Benoît Paul Émile Clapeyron in 1834 as a combination of the empirical Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. The ideal gas law is often written in an empirical form:

$${\displaystyle pV=nRT}$$ where ${\displaystyle p}$, ${\displaystyle V}$ and ${\displaystyle T}$ are the pressure, volume and temperature respectively; ${\displaystyle n}$ is the amount of substance; and ${\displaystyle R}$ is the ideal gas constant. It can also be derived from the microscopic kinetic theory, as was achieved (apparently independently) by August Krönig in 1856 and Rudolf Clausius in 1857.

Equation

Molecular collisions within a closed container (a propane tank) are shown (right). The arrows represent the random motions and collisions of these molecules. The pressure and temperature of the gas are directly proportional: As temperature increases, the pressure of the propane gas increases by the same factor. A simple consequence of this proportionality is that on a hot summer day, the propane tank pressure will be elevated, and thus propane tanks must be rated to withstand such increases in pressure.

The state of an amount of gas is determined by its pressure, volume, and temperature. The modern form of the equation relates these simply in two main forms. The temperature used in the equation of state is an absolute temperature: the appropriate SI unit is the kelvin.

Common forms

The most frequently introduced forms are:$${\displaystyle pV=nRT=n{k}_{\text{B}}{N}_{\text{A}}T=N{k}_{\text{B}}T}$$where:

• ${\displaystyle p}$ is the absolute pressure of the gas,
• ${\displaystyle V}$ is the volume of the gas,
• ${\displaystyle n}$ is the amount of substance of gas (also known as number of moles),
• ${\displaystyle R}$ is the ideal, or universal, gas constant, equal to the product of the Boltzmann constant and the Avogadro constant,
• ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant,
• ${\displaystyle N_A}$ is the Avogadro constant,
• ${\displaystyle T}$ is the absolute temperature of the gas,
• ${\displaystyle N}$ is the number of particles (usually atoms or molecules) of the gas.

In SI units, p is measured in pascals, V is measured in cubic metres, n is measured in moles, and T in kelvins (the Kelvin scale is a shifted Celsius scale, where 0 K = −273.15 °C, the lowest possible temperature). R has for value 8.314 J/(mol·K) = 1.989 ≈ 2 cal/(mol·K), or 0.0821 L⋅atm/(mol⋅K).

Molar form

How much gas is present could be specified by giving the mass instead of the chemical amount of gas. Therefore, an alternative form of the ideal gas law may be useful. The chemical amount, n (in moles), is equal to total mass of the gas (m) (in kilograms) divided by the molar mass, M (in kilograms per mole):

${\displaystyle n=\frac{m}{M}.}$

By replacing n with m/M and subsequently introducing density ρ = m/V, we get:

${\displaystyle pV=\frac{m}{M}RT}$

${\displaystyle p=\frac{m}{V}\frac{RT}{M}}$

${\displaystyle p=\rho \frac{R}{M}T}$

Defining the specific gas constant R_specific as the ratio R/M,

${\displaystyle p=\rho R_{\text{specific}}T}$

This form of the ideal gas law is very useful because it links pressure, density, and temperature in a unique formula independent of the quantity of the considered gas. Alternatively, the law may be written in terms of the specific volume v, the reciprocal of density, as

${\displaystyle pv=R_{\text{specific}}T.}$

It is common, especially in engineering and meteorological applications, to represent the specific gas constant by the symbol R. In such cases, the universal gas constant is usually given a different symbol such as ${\displaystyle \overline{R}}$ or ${\displaystyle R^{\ast }}$ to distinguish it. In any case, the context and/or units of the gas constant should make it clear as to whether the universal or specific gas constant is being used.

Statistical mechanics

In statistical mechanics, the following molecular equation (i.e. the ideal gas law in its theoretical form) is derived from first principles:

${\displaystyle p=n{k}_{\text{B}}T,}$

where p is the absolute pressure of the gas, n is the number density of the molecules (given by the ratio n = N/V, in contrast to the previous formulation in which n is the number of moles), T is the absolute temperature, and k_B is the Boltzmann constant relating temperature and energy, given by:

${\displaystyle k_{\text{B}}=\frac{R}{N_{\text{A}}}}$

where N_A is the Avogadro constant. The form can be further simplified by defining the kinetic energy corresponding to the temperature:

${\displaystyle T:=k_{\text{B}}T,}$

so the ideal gas law is more simply expressed as:

${\displaystyle p=nT,}$

From this we notice that for a gas of mass m, with an average particle mass of μ times the atomic mass constant, m_u, (i.e., the mass is μ Da) the number of molecules will be given by

${\displaystyle N=\frac{m}{\mu m_{\text{u}}},}$

and since ρ = m/V = nμm_u, we find that the ideal gas law can be rewritten as

${\displaystyle p=\frac{1}{V}\frac{m}{\mu m_{\text{u}}}{k}_{\text{B}}T=\frac{k_{\text{B}}}{\mu m_{\text{u}}}\rho T.}$

In SI units, p is measured in pascals, V in cubic metres, T in kelvins, and k_B = 1.38×10⁻²³ J⋅K⁻¹ in SI units.

Combined gas law

Combining the laws of Charles, Boyle and Gay-Lussac gives the combined gas law, which takes the same functional form as the ideal gas law says that the number of moles is unspecified, and the ratio of ${\displaystyle PV}$ to ${\displaystyle T}$ is simply taken as a constant:

${\displaystyle \frac{PV}{T}=k,}$

where ${\displaystyle P}$ is the pressure of the gas, ${\displaystyle V}$ is the volume of the gas, ${\displaystyle T}$ is the absolute temperature of the gas, and ${\displaystyle k}$ is a constant. When comparing the same substance under two different sets of conditions, the law can be written as

${\displaystyle \frac{P_1{V}_1}{T_1}=\frac{P_2{V}_2}{T_2}.}$

Energy associated with a gas

According to the assumptions of the kinetic theory of ideal gases, one can consider that there are no intermolecular attractions between the molecules, or atoms, of an ideal gas. In other words, its potential energy is zero. Hence, all the energy possessed by the gas is the kinetic energy of the molecules, or atoms, of the gas.

${\displaystyle E=\frac{3}{2}nRT}$

This corresponds to the kinetic energy of n moles of a monoatomic gas having 3 degrees of freedom; x, y, z. The table here below gives this relationship for different amounts of a monoatomic gas.

Energy of a monoatomic gas	Mathematical expression
Energy associated with one mole	${\displaystyle E=\frac{3}{2}RT}$
Energy associated with one gram	${\displaystyle E=\frac{3}{2}rT}$
Energy associated with one atom	${\displaystyle E=\frac{3}{2}{k}_{\mathrm{B}}T}$

Applications to thermodynamic processes

The table below essentially simplifies the ideal gas equation for a particular process, making the equation easier to solve using numerical methods.

A thermodynamic process is defined as a system that moves from state 1 to state 2, where the state number is denoted by a subscript. As shown in the first column of the table, basic thermodynamic processes are defined such that one of the gas properties (P, V, T, S, or H) is constant throughout the process.

For a given thermodynamic process, in order to specify the extent of a particular process, one of the properties ratios (which are listed under the column labeled "known ratio") must be specified (either directly or indirectly). Also, the property for which the ratio is known must be distinct from the property held constant in the previous column (otherwise the ratio would be unity, and not enough information would be available to simplify the gas law equation).

In the final three columns, the properties (p, V, or T) at state 2 can be calculated from the properties at state 1 using the equations listed.

Process	Constant	Known ratio or delta	p2	V2	T2
Isobaric process	Pressure	V2/V1	p2 = p1	V2 = V1(V2/V1)	T2 = T1(V2/V1)
		T2/T1	p2 = p1	V2 = V1(T2/T1)	T2 = T1(T2/T1)
Isochoric process (Isovolumetric process) (Isometric process)	Volume	p2/p1	p2 = p1(p2/p1)	V2 = V1	T2 = T1(p2/p1)
		T2/T1	p2 = p1(T2/T1)	V2 = V1	T2 = T1(T2/T1)
Isothermal process	Temperature	p2/p1	p2 = p1(p2/p1)	V2 = V1(p1/p2)	T2 = T1
		V2/V1	p2 = p1(V1/V2)	V2 = V1(V2/V1)	T2 = T1
Isentropic process (Reversible adiabatic process)	Entropy[a]	p2/p1	p2 = p1(p2/p1)	V2 = V1(p2/p1)(−1/γ)	T2 = T1(p2/p1)(γ − 1)/γ
		V2/V1	p2 = p1(V2/V1)−γ	V2 = V1(V2/V1)	T2 = T1(V2/V1)(1 − γ)
		T2/T1	p2 = p1(T2/T1)γ/(γ − 1)	V2 = V1(T2/T1)1/(1 − γ)	T2 = T1(T2/T1)
Polytropic process	P Vn	p2/p1	p2 = p1(p2/p1)	V2 = V1(p2/p1)(−1/n)	T2 = T1(p2/p1)(n − 1)/n
		V2/V1	p2 = p1(V2/V1)−n	V2 = V1(V2/V1)	T2 = T1(V2/V1)(1 − n)
		T2/T1	p2 = p1(T2/T1)n/(n − 1)	V2 = V1(T2/T1)1/(1 − n)	T2 = T1(T2/T1)
Isenthalpic process (Irreversible adiabatic process)	Enthalpy[b]	p2 − p1	p2 = p1 + (p2 − p1)		T2 = T1 + μJT(p2 − p1)
		T2 − T1	p2 = p1 + (T2 − T1)/μJT		T2 = T1 + (T2 − T1)

^{^} a. In an isentropic process, system entropy (S) is constant. Under these conditions, p₁V₁^γ = p₂V₂^γ, where γ is defined as the heat capacity ratio, which is constant for a calorifically perfect gas. The value used for γ is typically 1.4 for diatomic gases like nitrogen (N₂) and oxygen (O₂), (and air, which is 99% diatomic). Also γ is typically 1.6 for mono atomic gases like the noble gases helium (He), and argon (Ar). In internal combustion engines γ varies between 1.35 and 1.15, depending on constitution gases and temperature.

^{^} b. In an isenthalpic process, system enthalpy (H) is constant. In the case of free expansion for an ideal gas, there are no molecular interactions, and the temperature remains constant. For real gasses, the molecules do interact via attraction or repulsion depending on temperature and pressure, and heating or cooling does occur. This is known as the Joule–Thomson effect. For reference, the Joule–Thomson coefficient μ_JT for air at room temperature and sea level is 0.22 °C/bar.

Deviations from ideal behavior of real gases

The equation of state given here (PV = nRT) applies only to an ideal gas, or as an approximation to a real gas that behaves sufficiently like an ideal gas. There are in fact many different forms of the equation of state. Since the ideal gas law neglects both molecular size and intermolecular attractions, it is most accurate for monatomic gases at high temperatures and low pressures. The neglect of molecular size becomes less important for lower densities, i.e. for larger volumes at lower pressures, because the average distance between adjacent molecules becomes much larger than the molecular size. The relative importance of intermolecular attractions diminishes with increasing thermal kinetic energy, i.e., with increasing temperatures. More detailed equations of state, such as the van der Waals equation, account for deviations from ideality caused by molecular size and intermolecular forces.

Derivations

Empirical

The empirical laws that led to the derivation of the ideal gas law were discovered with experiments that changed only 2 state variables of the gas and kept every other one constant.

All the possible gas laws that could have been discovered with this kind of setup are:

• Boyle's law (Equation 1) $${\displaystyle PV=C_1\text{or}{P}_1{V}_1=P_2{V}_2}$$
• Charles's law (Equation 2) $${\displaystyle \frac{V}{T}=C_2\text{or}\frac{V_1}{T_1}=\frac{V_2}{T_2}}$$
• Avogadro's law (Equation 3) $${\displaystyle \frac{V}{N}=C_3\text{or}\frac{V_1}{N_1}=\frac{V_2}{N_2}}$$
• Gay-Lussac's law (Equation 4) $${\displaystyle \frac{P}{T}=C_4\text{or}\frac{P_1}{T_1}=\frac{P_2}{T_2}}$$
• Equation 5 $${\displaystyle NT=C_5\text{or}{N}_1{T}_1=N_2{T}_2}$$
• Equation 6 $${\displaystyle \frac{P}{N}=C_6\text{or}\frac{P_1}{N_1}=\frac{P_2}{N_2}}$$

Relationships between Boyle's, Charles's, Gay-Lussac's, Avogadro's, combined and ideal gas laws, with the Boltzmann constant k = ⁠R/N_A⁠ = ⁠n R/N⁠ (in each law, properties circled are variable and properties not circled are held constant)

where P stands for pressure, V for volume, N for number of particles in the gas and T for temperature; where ${\displaystyle C_1,C_2,C_3,C_4,C_5,C_6}$ are constants in this context because of each equation requiring only the parameters explicitly noted in them changing.

To derive the ideal gas law one does not need to know all 6 formulas, one can just know 3 and with those derive the rest or just one more to be able to get the ideal gas law, which needs 4.

Since each formula only holds when only the state variables involved in said formula change while the others (which are a property of the gas but are not explicitly noted in said formula) remain constant, we cannot simply use algebra and directly combine them all. This is why: Boyle did his experiments while keeping N and T constant and this must be taken into account (in this same way, every experiment kept some parameter as constant and this must be taken into account for the derivation).

Keeping this in mind, to carry the derivation on correctly, one must imagine the gas being altered by one process at a time (as it was done in the experiments). The derivation using 4 formulas can look like this:

at first the gas has parameters ${\displaystyle P_1,V_1,N_1,T_1}$

Say, starting to change only pressure and volume, according to Boyle's law (Equation 1), then:

$${\displaystyle P_1{V}_1=P_2{V}_2}$$

7

After this process, the gas has parameters ${\displaystyle P_2,V_2,N_1,T_1}$

Using then equation (5) to change the number of particles in the gas and the temperature,

$${\displaystyle N_1{T}_1=N_2{T}_2}$$

8

After this process, the gas has parameters ${\displaystyle P_2,V_2,N_2,T_2}$

Using then equation (6) to change the pressure and the number of particles,

$${\displaystyle \frac{P_2}{N_2}=\frac{P_3}{N_3}}$$

9

After this process, the gas has parameters ${\displaystyle P_3,V_2,N_3,T_2}$

Using then Charles's law (equation 2) to change the volume and temperature of the gas,

$${\displaystyle \frac{V_2}{T_2}=\frac{V_3}{T_3}}$$

10

After this process, the gas has parameters ${\displaystyle P_3,V_3,N_3,T_3}$

Using simple algebra on equations (7), (8), (9) and (10) yields the result: $${\displaystyle \frac{P_1{V}_1}{N_1{T}_1}=\frac{P_3{V}_3}{N_3{T}_3}}$$ or $${\displaystyle \frac{PV}{NT}=k_{\text{B}},}$$ where ${\displaystyle k_{\text{B}}}$ stands for the Boltzmann constant.

Another equivalent result, using the fact that ${\displaystyle nR=N{k}_{\text{B}}}$, where n is the number of moles in the gas and R is the universal gas constant, is: $${\displaystyle PV=nRT,}$$ which is known as the ideal gas law.

If three of the six equations are known, it may be possible to derive the remaining three using the same method. However, because each formula has two variables, this is possible only for certain groups of three. For example, if you were to have equations (1), (2) and (4) you would not be able to get any more because combining any two of them will only give you the third. However, if you had equations (1), (2) and (3) you would be able to get all six equations because combining (1) and (2) will yield (4), then (1) and (3) will yield (6), then (4) and (6) will yield (5), as well as would the combination of (2) and (3) as is explained in the following visual relation:

Relationship between the six gas laws

where the numbers represent the gas laws numbered above.

If you were to use the same method used above on 2 of the 3 laws on the vertices of one triangle that has a "O" inside it, you would get the third.

For example:

Change only pressure and volume first:

$${\displaystyle P_1{V}_1=P_2{V}_2}$$

1'

then only volume and temperature:

$${\displaystyle \frac{V_2}{T_1}=\frac{V_3}{T_2}}$$

2'

then as we can choose any value for ${\displaystyle V_3}$, if we set ${\displaystyle V_1=V_3}$, equation (2') becomes:

$${\displaystyle \frac{V_2}{T_1}=\frac{V_1}{T_2}}$$

3'

combining equations (1') and (3') yields ${\displaystyle \frac{P_1}{T_1}=\frac{P_2}{T_2}}$, which is equation (4), of which we had no prior knowledge until this derivation.

Theoretical

Kinetic theory

Main article: Kinetic theory of gases

The ideal gas law can also be derived from first principles using the kinetic theory of gases, in which several simplifying assumptions are made, chief among which are that the molecules, or atoms, of the gas are point masses, possessing mass but no significant volume, and undergo only elastic collisions with each other and the sides of the container in which both linear momentum and kinetic energy are conserved.

First we show that the fundamental assumptions of the kinetic theory of gases imply that

${\displaystyle P=\frac{1}{3}nm{v}_{\text{rms}}^2.}$

Consider a container in the ${\displaystyle xyz}$ Cartesian coordinate system. For simplicity, we assume that a third of the molecules moves parallel to the ${\displaystyle x}$-axis, a third moves parallel to the ${\displaystyle y}$-axis and a third moves parallel to the ${\displaystyle z}$-axis. If all molecules move with the same velocity ${\displaystyle v}$, denote the corresponding pressure by ${\displaystyle P_0}$. We choose an area ${\displaystyle S}$ on a wall of the container, perpendicular to the ${\displaystyle x}$-axis. When time ${\displaystyle t}$ elapses, all molecules in the volume ${\displaystyle vtS}$ moving in the positive direction of the ${\displaystyle x}$-axis will hit the area. There are ${\displaystyle NvtS}$ molecules in a part of volume ${\displaystyle vtS}$ of the container, but only one sixth (i.e. a half of a third) of them moves in the positive direction of the ${\displaystyle x}$-axis. Therefore, the number of molecules ${\displaystyle N^{\prime }}$ that will hit the area ${\displaystyle S}$ when the time ${\displaystyle t}$ elapses is ${\displaystyle NvtS/6}$.

When a molecule bounces off the wall of the container, it changes its momentum ${\displaystyle {\mathbf{p}}_1}$ to ${\displaystyle {\mathbf{p}}_2=-{\mathbf{p}}_1}$. Hence the magnitude of change of the momentum of one molecule is ${\displaystyle |{\mathbf{p}}_2-{\mathbf{p}}_1|=2mv}$. The magnitude of the change of momentum of all molecules that bounce off the area ${\displaystyle S}$ when time ${\displaystyle t}$ elapses is then ${\displaystyle |\mathrm{\Delta }\mathbf{p}|=2mv{N}^{\prime }/V=NtSm{v}^2/(3V)=ntSm{v}^2/3}$. From ${\displaystyle F=|\mathrm{\Delta }\mathbf{p}|/t}$ and ${\displaystyle P_0=F/S}$ we get

${\displaystyle P_0=\frac{1}{3}nm{v}^2.}$

We considered a situation where all molecules move with the same velocity ${\displaystyle v}$. Now we consider a situation where they can move with different velocities, so we apply an "averaging transformation" to the above equation, effectively replacing ${\displaystyle P_0}$ by a new pressure ${\displaystyle P}$ and ${\displaystyle v^2}$ by the arithmetic mean of all squares of all velocities of the molecules, i.e. by ${\displaystyle v_{\text{rms}}^2.}$ Therefore

${\displaystyle P=\frac{1}{3}nm{v}_{\text{rms}}^2}$

which gives the desired formula.

Using the Maxwell–Boltzmann distribution, the fraction of molecules that have a speed in the range ${\displaystyle v}$ to ${\displaystyle v+dv}$ is ${\displaystyle f(v)dv}$, where

${\displaystyle f(v)=4\pi {\left(\frac{m}{2\pi k_{\mathrm{B}}T}\right)}^{\frac{3}{2}}{v}^2{e}^{-\frac{m{v}^2}{2k_{\mathrm{B}}T}}}$

and ${\displaystyle k}$ denotes the Boltzmann constant. The root-mean-square speed can be calculated by

${\displaystyle v_{\text{rms}}^2={\int }_0^{\mathrm{\infty }}{v}^{2}f(v)dv=4\pi {\left(\frac{m}{2\pi k_{\mathrm{B}}T}\right)}^{\frac{3}{2}}{\int }_0^{\mathrm{\infty }}{v}^4{e}^{-\frac{m{v}^2}{2k_{\mathrm{B}}T}}dv.}$

Using the integration formula

${\displaystyle {\int }_0^{\mathrm{\infty }}{x}^{2n}{e}^{-\frac{x^2}{a^2}}dx=\sqrt{\pi }\frac{(2n)!}{n!}{\left(\frac{a}{2}\right)}^{2n+1},n\in \mathbb{N},a\in {\mathbb{R}}^{+},}$

it follows that

${\displaystyle v_{\text{rms}}^2=4\pi {\left(\frac{m}{2\pi k_{\mathrm{B}}T}\right)}^{\frac{3}{2}}\sqrt{\pi }\frac{4!}{2!}{\left(\frac{\sqrt{\frac{2k_{\mathrm{B}}T}{m}}}{2}\right)}^5=\frac{3k_{\mathrm{B}}T}{m},}$

from which we get the ideal gas law:

${\displaystyle P=\frac{1}{3}nm\left(\frac{3k_{\mathrm{B}}T}{m}\right)=n{k}_{\mathrm{B}}T.}$

Statistical mechanics

Main article: Statistical mechanics

Let q = (q_x, q_y, q_z) and p = (p_x, p_y, p_z) denote the position vector and momentum vector of a particle of an ideal gas, respectively. Let F denote the net force on that particle. Then (two times) the time-averaged kinetic energy of the particle is:

${\displaystyle \begin{array}{rl}⟨\mathbf{q}\cdot \mathbf{F}⟩& =⟨q_x\frac{d{p}_x}{dt}⟩+⟨q_y\frac{d{p}_y}{dt}⟩+⟨q_z\frac{d{p}_z}{dt}⟩\\ & =-⟨q_x\frac{\mathrm{\partial }H}{\mathrm{\partial }{q}_x}⟩-⟨q_y\frac{\mathrm{\partial }H}{\mathrm{\partial }{q}_y}⟩-⟨q_z\frac{\mathrm{\partial }H}{\mathrm{\partial }{q}_z}⟩=-3k_{\text{B}}T,\end{array}}$

where the first equality is Newton's second law, and the second line uses Hamilton's equations and the equipartition theorem. Summing over a system of N particles yields

${\displaystyle 3N{k}_{\mathrm{B}}T=-⟨\sum _{k=1}^N{\mathbf{q}}_k\cdot {\mathbf{F}}_k⟩.}$

By Newton's third law and the ideal gas assumption, the net force of the system is the force applied by the walls of the container, and this force is given by the pressure P of the gas. Hence

${\displaystyle -⟨\sum _{k=1}^N{\mathbf{q}}_k\cdot {\mathbf{F}}_k⟩=P{\oint }_{\text{surface}}\mathbf{q}\cdot d\mathbf{S},}$

where dS is the infinitesimal area element along the walls of the container. Since the divergence of the position vector q is

${\displaystyle \mathrm{\nabla }\cdot \mathbf{q}=\frac{\mathrm{\partial }{q}_x}{\mathrm{\partial }{q}_x}+\frac{\mathrm{\partial }{q}_y}{\mathrm{\partial }{q}_y}+\frac{\mathrm{\partial }{q}_z}{\mathrm{\partial }{q}_z}=3,}$

the divergence theorem implies that

${\displaystyle P{\oint }_{\text{surface}}\mathbf{q}\cdot d\mathbf{S}=P{\int }_{\text{volume}}\left(\mathrm{\nabla }\cdot \mathbf{q}\right)dV=3PV,}$

where dV is an infinitesimal volume within the container and V is the total volume of the container.

Putting these equalities together yields

${\displaystyle 3N{k}_{\text{B}}T=-⟨\sum _{k=1}^N{\mathbf{q}}_k\cdot {\mathbf{F}}_k⟩=3PV,}$

which immediately implies the ideal gas law for N particles:

${\displaystyle PV=N{k}_{\mathrm{B}}T=nRT,}$

where n = N/N_A is the number of moles of gas and R = N_Ak_B is the gas constant.

Other dimensions

For a d-dimensional system, the ideal gas pressure is:

${\displaystyle P^{(d)}=\frac{N{k}_{\mathrm{B}}T}{L^d},}$

where ${\displaystyle L^d}$ is the volume of the d-dimensional domain in which the gas exists. The dimensions of the pressure changes with dimensionality.