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94-698: Phosphorus trifluoride has an F−P−F bond angle of approximately 96.3°. Gaseous PF 3 has a standard enthalpy of formation of −945 kJ/mol (−226  kcal / mol ). The phosphorus atom has a nuclear magnetic resonance chemical shift of 97 ppm (downfield of H 3 PO 4 ). Phosphorus trifluoride hydrolyzes especially at high pH , but it is less hydrolytically sensitive than phosphorus trichloride . It does not attack glass except at high temperatures, and anhydrous potassium hydroxide may be used to dry it with little loss. With hot metals , phosphides and fluorides are formed. With Lewis bases such as ammonia addition products (adducts) are formed, and PF 3

188-453: A combustion chamber of a jet engine . It may also be useful to keep the elementary reactions and chemical dissociations for calculating emissions . 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

282-402: A gas of bosons in a box . The solid black line is the fraction of excited states 1 − N 0 / N for N = 10 000 and the dotted black line is the solution for N = 1000 . The blue lines are the fraction of condensed particles N 0 / N . The red lines plot values of the negative of the chemical potential μ and the green lines plot the corresponding values of z . The horizontal axis

376-403: 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

470-443: A photon gas and extended to massive particles by Albert Einstein , who realized that an ideal gas of bosons would form a condensate at a low enough temperature, unlike a classical ideal gas. This condensate is known as a Bose–Einstein condensate . Bosons are quantum mechanical particles that follow Bose–Einstein statistics , or equivalently, that possess integer spin . These particles can be classified as elementary: these are

564-422: A box ( α = 3 / 2 {\displaystyle \alpha =3/2} and using the above noted value of E c ) we get: For α ≤ 1 , there is no upper limit on the number of particles ( N m diverges as z approaches 1), and thus for example for a gas in a one- or two-dimensional box ( α = 1/2 and α = 1 respectively) there is no critical temperature. The above problem raises

658-417: A classical ideal gas . The thermodynamics of an ideal Bose gas is best calculated using the grand canonical ensemble . The grand potential for a Bose gas is given by: where each term in the sum corresponds to a particular single-particle energy level ε i ; g i is the number of states with energy ε i ; z is the absolute activity (or "fugacity"), which may also be expressed in terms of

752-538: A comprehensive listing of these exotic states of matter, see list of states of matter . The only chemical elements that are stable diatomic homonuclear molecular gases at STP are hydrogen (H 2 ), nitrogen (N 2 ), oxygen (O 2 ), and two halogens : fluorine (F 2 ) and chlorine (Cl 2 ). 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". The word gas

846-425: 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

940-415: 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

1034-429: A gas of photons, developed by Bose . This model leads to a better understanding of Planck's law and the black-body radiation . The photon gas can be easily expanded to any kind of ensemble of massless non-interacting bosons. The phonon gas , also known as Debye model , is an example where the normal modes of vibration of the crystal lattice of a metal, can be treated as effective massless bosons. Peter Debye used

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1128-473: 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

1222-471: 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. The equation of state for an ideal or perfect gas

1316-556: A result, superconductors behave like having no electrical resistivity at low temperatures. The equivalent model for half-integer particles (like electrons or helium-3 atoms), that follow Fermi–Dirac statistics , is called the Fermi gas (an ensemble of non-interacting fermions ). At low enough particle number density and high temperature, both the Fermi gas and the Bose gas behave like

1410-479: 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

1504-424: 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. The symbol used to represent density in equations is ρ (rho) with SI units of kilograms per cubic meter. This term

1598-790: 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

1692-425: 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

1786-459: Is given by: where V ( r ) = mω r /2 is the harmonic potential. It is seen that E c is a function of volume only. This integral expression for the grand potential evaluates to: where Li s ( x ) is the polylogarithm function. The problem with this continuum approximation for a Bose gas is that the ground state has been effectively ignored, giving a degeneracy of zero for zero energy. This inaccuracy becomes serious when dealing with

1880-469: Is however crucially dependent on the value of α (i.e., dependent on whether the gas is 1D, 2D, 3D, whether it is in a flat or harmonic potential well). For α > 1 , the number of particles only increases up to a finite maximum value, i.e., N m is finite at z = 1 : where ζ ( α ) is the Riemann zeta function (using Li α ( 1 ) = ζ ( α ) ). Thus, for a fixed number of particles N m ,

1974-629: Is not. Such complexes are usually prepared directly from the related metal carbonyl compound, with loss of CO . However, nickel metal reacts directly with PF 3 at 100 °C under 35 MPa pressure to form Ni(PF 3 ) 4 , which is analogous to Ni(CO) 4 . Cr(PF 3 ) 6 , the analogue of Cr(CO) 6 , may be prepared from dibenzenechromium : Phosphorus trifluoride is usually prepared from phosphorus trichloride via halogen exchange using various fluorides such as hydrogen fluoride , calcium fluoride , arsenic trifluoride , antimony trifluoride , or zinc fluoride : Phosphorus trifluoride

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2068-405: Is oxidized by oxidizing agents such as bromine or potassium permanganate . As a ligand for transition metals, PF 3 is a strong π-acceptor. It forms a variety of metal complexes with metals in low oxidation states . PF 3 forms several complexes for which the corresponding CO derivatives (see metal carbonyl ) are unstable or nonexistent. Thus, Pd(PF 3 ) 4 is known, but Pd(CO) 4

2162-405: 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

2256-421: Is similar to carbon monoxide in that it is a gas which strongly binds to iron in hemoglobin , preventing the blood from absorbing oxygen. PF 3 is highly toxic , comparable to phosgene . Gas This is an accepted version of this page 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.

2350-414: Is the ideal gas law and reads 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 where R s {\displaystyle R_{s}}

2444-414: 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

2538-404: 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

2632-513: Is the most simple quantitative model that explains this phase transition . Mainly when a gas of bosons is cooled down, it forms a Bose–Einstein condensate , a state where a large number of bosons occupy the lowest energy, the ground state , and quantum effects are macroscopically visible like wave interference . The theory of Bose-Einstein condensates and Bose gases can also explain some features of superconductivity where charge carriers couple in pairs ( Cooper pairs ) and behave like bosons. As

2726-438: Is the normalized temperature τ defined by It can be seen that each of these parameters become linear in τ in the limit of low temperature and, except for the chemical potential, linear in 1/ τ in the limit of high temperature. As the number of particles increases, the condensed and excited fractions tend towards a discontinuity at the critical temperature. The equation for the number of particles can be written in terms of

2820-403: 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. The intermolecular attractions and repulsions between two gas molecules depend on the distance between them. The combined attractions and repulsions are well-modelled by

2914-426: 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

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3008-413: 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

3102-421: 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

3196-407: Is why it can normally be ignored. This high variance is due to the choice of using the grand canonical ensemble for the entire system, including the condensate state. Expanded out, the grand potential is: All thermodynamic properties can be computed from this potential. The following table lists various thermodynamic quantities calculated in the limit of low temperature and high temperature, and in

3290-597: 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

3384-515: The Higgs boson , the photon , the gluon , the W/Z and the hypothetical graviton ; or composite like the atom of hydrogen , the atom of O , the nucleus of deuterium , mesons etc. Additionally, some quasiparticles in more complex systems can also be considered bosons like the plasmons (quanta of charge density waves ). The first model that treated a gas with several bosons, was the photon gas ,

3478-546: 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

3572-601: 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

3666-475: The Maxwell–Boltzmann distribution . Use of this distribution implies ideal gases near thermodynamic equilibrium for the system of particles being considered. The symbol used to represent specific volume in equations is "v" with SI units of cubic meters per kilogram. The symbol used to represent volume in equations is "V" with SI units of cubic meters. When performing a thermodynamic analysis, it

3760-411: The canonical ensemble , which fixes the total particle number, however the calculations are not as easy. Practically however, the aforementioned theoretical flaw is a minor issue, as the most unrealistic assumption is that of non-interaction between bosons. Experimental realizations of boson gases always have significant interactions, i.e., they are non-ideal gases. The interactions significantly change

3854-470: The chemical potential μ by defining: and β defined as: where k B is the Boltzmann constant and T is the temperature . All thermodynamic quantities may be derived from the grand potential and we will consider all thermodynamic quantities to be functions of only the three variables z , β (or T ), and V . All partial derivatives are taken with respect to one of these three variables while

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3948-458: 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

4042-411: 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

4136-473: The Bose–Einstein condensate and will be dealt with in the next sections. As will be seen, even at low temperatures the above result is still useful for accurately describing the thermodynamics of just the uncondensed portion of the gas. The total number of particles is found from the grand potential by This increases monotonically with z (up to the maximum z = +1). The behaviour when approaching z = 1

4230-727: The French-American historian Jacques Barzun speculated that Van Helmont had borrowed the word from the German Gäscht , meaning the froth resulting from fermentation . 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

4324-408: 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 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

4418-418: The above stated effects which cause these attractions and repulsions, real gases , delineate from the ideal gas model by the following generalization: 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,

4512-444: 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

4606-423: The average energy is large compared to the energy difference between levels so that the above sum may be replaced by an integral. This replacement gives the macroscopic grand potential function Ω m {\displaystyle \Omega _{m}} , which is close to Ω {\displaystyle \Omega } : The degeneracy dg may be expressed for many different situations by

4700-429: 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. There are many mathematical tools available for analyzing gas properties. Boyle's lab equipment allowed

4794-491: 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

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4888-410: 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. 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

4982-466: 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 provides insight into the macroscopic properties of gases by considering their molecular composition and motion. Starting with

5076-479: 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

5170-522: The equation to read pV = 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: 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

5264-439: 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. The symbol used to represent pressure in equations is "p" or "P" with SI units of pascals . When describing

5358-409: The gases produced during geological events as in the image of the 1990 eruption of Mount Redoubt . Bose gas An ideal Bose gas is a quantum-mechanical phase of matter , analogous to a classical ideal gas . It is composed of bosons , which have an integer value of spin and abide by Bose–Einstein statistics . The statistical mechanics of bosons were developed by Satyendra Nath Bose for

5452-408: The general formula: where α is a constant, E c is a critical energy, and Γ is the gamma function . For example, for a massive Bose gas in a box, α = 3/2 and the critical energy is given by: where Λ is the thermal wavelength , and f is a degeneracy factor ( f = 1 for simple spinless bosons). For a massive Bose gas in a harmonic trap we will have α = 3 and the critical energy

5546-483: The grand potential, as in the section below), this gives rise to an unrealistic fluctuation catastrophe: the number of particles in any given state follow a geometric distribution , meaning that when condensation happens at T < T c and most particles are in one state, there is a huge uncertainty in the total number of particles. This is related to the fact that the compressibility becomes unbounded for T < T c . Calculations can instead be performed in

5640-414: The grand potential: which gives instead N 0 = ⁠ g 0 z / 1 − z ⁠ . Now, the behaviour is smooth when crossing the critical temperature, and z approaches 1 very closely but does not reach it. This can now be solved down to absolute zero in temperature. Figure 1 shows the results of the solution to this equation for α = 3/2 , with k = ε c = 1 , which corresponds to

5734-441: 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

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5828-475: The largest possible value that β can have is a critical value β c . This corresponds to a critical temperature T c = 1/ k B β c , below which the Thomas–Fermi approximation breaks down (the continuum of states simply can no longer support this many particles, at lower temperatures). The above equation can be solved for the critical temperature: For example, for the three-dimensional Bose gas in

5922-401: The limit as N approaches infinity, which can be easily determined from these expansions. This approach to modelling small systems may in fact be unrealistic, however, since the variance in the number of particles in the ground state is very large, equal to the number of particles. In contrast, the variance of particle number in a normal gas is only the square-root of the particle number, which

6016-468: The limit of infinite particle number. An equal sign (=) indicates an exact result, while an approximation symbol indicates that only the first few terms of a series in τ α {\displaystyle \tau ^{\alpha }} is shown. It is seen that all quantities approach the values for a classical ideal gas in the limit of large temperature. The above values can be used to calculate other thermodynamic quantities. For example,

6110-420: The macroscopic equation gives an accurate estimate of the number of particles in the excited states, and it is not a bad approximation to simply "tack on" a ground state term to accept the particles that fall out of the continuum: where N 0 is the number of particles in the ground state condensate. Thus in the macroscopic limit, when T < T c , the value of z is pinned to 1 and N 0 takes up

6204-593: 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,

6298-508: 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

6392-473: 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

6486-420: 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

6580-510: 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

6674-491: The normalized temperature as: For a given N and τ , this equation can be solved for τ and then a series solution for z can be found by the method of inversion of series , either in powers of τ or as an asymptotic expansion in inverse powers of τ . From these expansions, we can find the behavior of the gas near T = 0 and in the Maxwell–Boltzmann as T approaches infinity. In particular, we are interested in

6768-514: 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. When observing gas, it

6862-481: The other two are held constant. The permissible range of z is from negative infinity to +1, as any value beyond this would give an infinite number of particles to states with an energy level of 0 (it is assumed that the energy levels have been offset so that the lowest energy level is 0). Following the procedure described in the gas in a box article, we can apply the Thomas–Fermi approximation , which assumes that

6956-422: 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

7050-521: 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. 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

7144-477: 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,

7238-409: 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

7332-417: 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 macro scopic, measurable quantity of pressure, is the direct result of these micro scopic particle collisions with the surface, over which, individual molecules exert a small force, each contributing to

7426-414: The phonon gas model to explain the behaviour of heat capacity of metals at low temperature. An interesting example of a Bose gas is an ensemble of helium-4 atoms. When a system of He atoms is cooled down to temperature near absolute zero , many quantum mechanical effects are present. Below 2.17  K , the ensemble starts to behave as a superfluid , a fluid with almost zero viscosity . The Bose gas

7520-480: The physics of how a condensate of bosons behaves: the ground state spreads out, the chemical potential saturates to a positive value even at zero temperature, and the fluctuation problem disappears (the compressibility becomes finite). See the article Bose–Einstein condensate. For smaller, mesoscopic , systems (for example, with only thousands of particles), the ground state term can be more explicitly approximated by adding in an actual discrete level at energy ε =0 in

7614-407: 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

7708-407: The question for α > 1 : if a Bose gas with a fixed number of particles is lowered down below the critical temperature, what happens? The problem here is that the Thomas–Fermi approximation has set the degeneracy of the ground state to zero, which is wrong. There is no ground state to accept the condensate and so particles simply 'disappear' from the continuum of states. It turns out, however, that

7802-400: 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

7896-812: The relationship between internal energy and the product of pressure and volume is the same as that for a classical ideal gas over all temperatures: A similar situation holds for the specific heat at constant volume The entropy is given by: Note that in the limit of high temperature, we have which, for α = 3/2 is simply a restatement of the Sackur–Tetrode equation . In one dimension bosons with delta interaction behave as fermions, they obey Pauli exclusion principle . In one dimension Bose gas with delta interaction can be solved exactly by Bethe ansatz . The bulk free energy and thermodynamic potentials were calculated by Chen-Ning Yang . In one dimensional case correlation functions also were evaluated. In one dimension Bose gas

7990-406: The remainder of particles. For T > T c there is the normal behaviour, with N 0 = 0 . This approach gives the fraction of condensed particles in the macroscopic limit: The above standard treatment of a macroscopic Bose gas is straightforward, but the inclusion of the ground state is somewhat inelegant. Another approach is to include the ground state explicitly (contributing a term in

8084-467: 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

8178-513: 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

8272-404: The surface area of the wall. 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

8366-493: 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

8460-434: 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 . ) In the kinetic theory of gases , kinetic energy is assumed to purely consist of linear translations according to a speed distribution of particles in

8554-407: 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

8648-411: 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

8742-543: 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

8836-500: 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,

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