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Encyclopedia > Ideal gas

An ideal gas or perfect gas is a hypothetical gas consisting of identical particles of zero volume, with no intermolecular forces, where the constituent atoms or molecules undergo perfectly elastic collisions with the walls of the container and each other and are in constant random motion. Real gases do not behave according to these exact properties, although the approximation is often good enough to describe real gases. Image File history File links Emblem-important. ... For other uses, see Gas (disambiguation). ... In physics, chemistry, and biology, intermolecular forces are forces that act between stable molecules or between functional groups of macromolecules. ... For other uses, see Atom (disambiguation). ... 3D (left and center) and 2D (right) representations of the terpenoid molecule atisane. ... As long as black-body radiation (not shown) doesnâ€™t escape a system, atoms in thermal agitation undergo essentially elastic collisions. ...

These four properties that constitute an ideal gas can be easily remembered by the acronym PRIE, which stands for;

- Point masses (molecules occupy no volume)

- Random Motion (molecules are in constant random motion)

- Intermolecular forces (there are NO intermolecular forces between the particles) In physics, chemistry, and biology, intermolecular forces are forces that act between stable molecules or between functional groups of macromolecules. ...

- Elastic collisions (the collisions involving the gas molecules are totally elastic)

The concept of ideal gas is useful in technology because one mole (6.02214 ×1023 particles) of an ideal gas has a volume of 22.4 liters at the standard conditions for temperature and pressure and many common real gases approach this behaviour in these conditions. The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... In chemistry and other sciences, STP or standard temperature and pressure is a standard set of conditions for experimental measurements, to enable comparisons to be made between sets of data. ...

The conditions in which a real gas will behave more and more like an ideal gas is either at very high temperatures (as the molecules of the gas have so much energy that the intermolecular forces and energy lost in collisions is negligable) and at very low pressures (as the molecules of the gas rarely collide or come into close enough proximity for intermolecular forces to be significant).

## Types of ideal gases GA_googleFillSlot("encyclopedia_square");

There exist three basic types of ideal gas:

The classical ideal gas can be separated into two types: The classical thermodynamic ideal gas and the ideal quantum Boltzmann gas. Both are essentially the same, except that the classical thermodynamic ideal gas is based on classical thermodynamics alone, and certain thermodynamic parameters such as the entropy are only specified to within an undetermined additive constant. The ideal quantum Boltzmann gas overcomes this limitation by taking the limit of the quantum Bose gas and quantum Fermi gas in the limit of high temperature to specify these additive constants. The behavior of a quantum Boltzmann gas is the same as that of a classical ideal gas except for the specification of these constants. The results of the quantum Boltzmann gas are used in a number of cases including the Sackur-Tetrode equation for the entropy of an ideal gas and the Saha ionization equation for a weakly ionized plasma. It has been suggested that the section Physical applications of the Maxwell-Boltzmann distribution from the article Maxwell-Boltzmann distribution be merged into this article or section. ... An ideal Bose gas is a quantum-mechanical version of a classical ideal gas. ... Boson (game) Bosons, named after Satyendra Nath Bose, are particles which form totally-symmetric composite quantum states. ... A Fermi gas is a collection of non-interacting fermions. ... Fermions, named after Enrico Fermi, are particles which form totally-antisymmetric composite quantum states. ... Thermodynamics (from the Greek Î¸ÎµÏÎ¼Î·, therme, meaning heat and Î´Ï…Î½Î±Î¼Î¹Ï‚, dynamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... The Sackur-Tetrode equation is an expression for the entropy of a classical ideal gas which uses quantum considerations to arrive at an exact formula. ... Developed by the Indian astrophysicist Meghnad Saha in 1920, this formula describes the degree of ionization of a gas as a function of the temperature T, density, and ionization energy. ... For other uses, see Plasma. ...

## Classical thermodynamic ideal gas

The thermodynamic properties of an ideal gas can be described by two equations : The equation of state of a classical ideal gas is given by the ideal gas law. In physics and thermodynamics, an equation of state is a relation between state variables. ... Isotherms of an ideal gas The ideal gas law is the equation of state of a hypothetical ideal gas, first stated by BenoÃ®t Paul Ã‰mile Clapeyron in 1834. ...

$pV = N k_B T = nRT.,$

The internal energy of an ideal gas is given by: In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of...

$U = hat{c}_V nRT = hat{c}_V NkT$

where:

• $hat{c}_V$ is a constant dependent on temperature (e.g. equal to 3/2 for a monatomic gas for moderate temperatures)
• U is the internal energy
• p is the pressure
• V is the volume
• n is the amount of gas (moles)
• R is the gas constant, 8.314 J·K−1mol-1
• T is the absolute temperature
• N is the number of particles
• k is the Boltzmann constant, 1.381×10−23J·K−1

The probability distribution of particles by velocity or energy is given by the Boltzmann distribution. In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... The gas constant (also known as the molar, universal, or ideal gas constant, usually denoted by symbol R) is a physical constant which is featured in a large number of fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. ... The joule (IPA: or ) (symbol: J) is the SI unit of energy. ... For other uses, see Kelvin (disambiguation). ... The mole (symbol: mol) is the SI base unit that measures an amount of substance. ... Absolute zero is the lowest temperature that can be obtained in any macroscopic system. ... The Boltzmann constant (k or kB) is the physical constant relating temperature to energy. ... In physics, the Boltzmann distribution predicts the distribution function for the fractional number of particles Ni / N occupying a set of states i which each has energy Ei: where is the Boltzmann constant, T is temperature (assumed to be a sharply well-defined quantity), is the degeneracy, or number of...

The ideal gas law is an extension of experimentally discovered gas laws. Real fluids at low density and high temperature approximate the behavior of a classical ideal gas. However, at lower temperatures or a higher density, a real fluid deviates strongly from the behavior of an ideal gas, particularly as it condenses from a gas into a liquid or solid. The gas laws are a set of laws that describe the relationship between thermodynamic temperature (T), pressure (P) and volume (V) of gases. ... For other uses, see Density (disambiguation). ... For other uses, see Temperature (disambiguation). ... Temperature is the physical property of a system which underlies the common notions of hot and cold; the material with the higher temperature is said to be hotter. ... For other uses, see Density (disambiguation). ...

## Heat capacity

The heat capacity at constant volume of an ideal gas is: To meet Wikipedias quality standards, this article or section may require cleanup. ...

$C_V = left(frac{partial U}{partial T}right)_V = hat{c}_V Nk.$

It is seen that the constant $hat{c}_V$ is just the dimensionless heat capacity at constant volume. It is equal to half the number of degrees of freedom per particle. For moderate temperatures, the constant for a monatomic gas is $hat{c}_V=3/2$ while for a diatomic gas it is $hat{c}_V=5/2$. It is seen that macroscopic measurements on heat capacity provide information on the microscopic structure of the molecules. Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ...

The heat capacity at constant pressure of an ideal gas is:

$C_p = left(frac{partial H}{partial T}right)_p = (hat{c}_V+1) Nk$

where H = U + pV is the enthalpy of the gas. It is seen that $hat{c}_p$ is also a constant and that the dimensionless heat capacities are related by: t In thermodynamics and molecular chemistry, the enthalpy or heat content (denoted as H or Î”H, or rarely as Ï‡) is a quotient or description of thermodynamic potential of a system, which can be used to calculate the useful work obtainable from a closed thermodynamic system under constant pressure. ...

$hat{c}_p-hat{c}_V=1$

## Entropy

Using the results of thermodynamics only, we can go a long way in determining the expression for the entropy of an ideal gas. This is an important step since, according to the theory of thermodynamic potentials, of which the internal energy U is one, if we can express the entropy as a function of U and the volume V, then we will have a complete statement of the thermodynamic behavior of the ideal gas. We will be able to derive both the ideal gas law and the expression for internal energy from it. Thermodynamics (from the Greek Î¸ÎµÏÎ¼Î·, therme, meaning heat and Î´Ï…Î½Î±Î¼Î¹Ï‚, dynamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... For other uses, see: information entropy (in information theory) and entropy (disambiguation). ... In thermodynamics, thermodynamic potentials are parameters associated with a thermodynamic system and have the dimensions of energy. ...

Since the entropy is an exact differential, using the chain rule, the change in entropy when going from a reference state 0 to some other state with entropy S may be written as ΔS where: In mathematics, a differential dQ is said to be exact, as contrasted with an inexact differential, if the function Q exists. ... In calculus, the chain rule is a formula for the derivative of the composite of two functions. ...

$Delta S = int_{S_0}^{S}dS =int_{T_0}^{T} left(frac{partial S}{partial T}right)_V!dT +int_{V_0}^{V} left(frac{partial S}{partial V}right)_T!dV$

where the reference variables may be functions of the number of particles N. Using the definition of the heat capacity at constant volume for the first differential and the appropriate Maxwell relation for the second we have: To meet Wikipedias quality standards, this article or section may require cleanup. ... Maxwells relations are a set of equations in Thermodynamics which are derivable from the definitions of the four thermodynamic potentials. ...

$Delta S =int_{T_0}^{T} frac{C_v}{T},dT+int_{V_0}^{V}left(frac{partial P}{partial T}right)_VdV$

Expressing CV in terms of $hat{c}_V$ as developed in the above section, differentiating the ideal gas equation of state, and integrating yields:

$Delta S = hat{c}_VNklnleft(frac{T}{T_0}right)+Nklnleft(frac{V}{V_0}right) = Nklnleft(frac{VT^{hat{c}_v}}{f(N)}right)$

where all constants have been incorporated into the logarithm as f(N) which is some function of the particle number N having the same dimensions as $VT^{hat{c}_v}$ in order that the argument of the logarithm be dimensionless. We now impose the constraint that the entropy be extensive. This will mean that when the extensive parameters (V and N) are multiplied by a constant, the entropy will be multiplied by the same constant. Mathematically:

$Delta S(T,aV,aN)=aDelta S(T,V,N),$

From this we find an equation for the function f(N)

$af(N)=f(aN),$

Differentiating this with respect to a, setting a equal to unity, and then solving the differential equation yields f(N):

$f(N)=phi N,$

where φ is some constant with the dimensions of $VT^{hat{c}_v}/N$. Substituting into the equation for the change in entropy:

$frac{Delta S}{Nk} = lnleft(frac{VT^{hat{c}_v}}{Nphi}right),$

This is about as far as we can go using thermodynamics alone. Note that the above equation is flawed — as the temperature approaches zero, the entropy approaches negative infinity, in contradiction to the third law of thermodynamics. In the above "ideal" development, there is a critical point, not at absolute zero, at which the argument of the logarithm becomes unity, and the entropy becomes zero. This is unphysical. The above equation is a good approximation only when the argument of the logarithm is much larger than unity — the concept of an ideal gas breaks down at low values of V/N. Nevertheless, there will be a "best" value of the constant in the sense that the predicted entropy is as close as possible to the actual entropy, given the flawed assumption of ideality. It remained for quantum mechanics to introduce a reasonable value for the value of φ which yields the Sackur-Tetrode equation for the entropy of an ideal gas. It too suffers from a divergent entropy at absolute zero, but is a good approximation to an ideal gas over a large range of densities. The third law of thermodynamics (hereinafter Third Law) states that as a system approaches the zero absolute temperature (hereinafter ZAT), all processes cease and the entropy of the system approaches a minimum value. ... For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... The Sackur-Tetrode equation is an expression for the entropy of a classical ideal gas which uses quantum considerations to arrive at an exact formula. ...

## Thermodynamic potentials

Since the dimensionless heat capacity at constant pressure $hat{c}_p$ is a constant we can express the entropy in what will prove to be a more convenient form: This article needs to be cleaned up to conform to a higher standard of quality. ...

$frac{S}{kN}=lnleft( frac{VT^{hat{c}_V}}{NPhi}right)+hat{c}_p$

where Φ is now the undetermined constant. The chemical potential of the ideal gas is calculated from the corresponding equation of state (see thermodynamic potential): In thermodynamics and chemistry, chemical potential, symbolized by Î¼, is a term introduced in 1876 by the American mathematical physicist Willard Gibbs, which he defined as follows: Gibbs noted also that for the purposes of this definition, any chemical element or combination of elements in given proportions may be considered a... In thermodynamics, thermodynamic potentials are parameters associated with a thermodynamic system and have the dimensions of energy. ...

$mu=left(frac{partial G}{partial N}right)_{T,p}$

where G is the Gibbs free energy and is equal to U + pVTS so that: In thermodynamics, the Gibbs free energy is a thermodynamic potential which measures the useful work obtainable from a closed thermodynamic system at a constant temperature and pressure. ...

$mu(T,V,N)=-kTlnleft(frac{VT^{hat{c}_V}}{NPhi}right).$

The thermodynamic potentials for an ideal gas can now be written as functions of T, V, and N as:

 $U,$ $=hat{c}_V NkT,$ $A=,$ $U-TS,$ $=mu N-NkT,$ $H=,$ $U+pV,$ $=hat{c}_p NkT,$ $G=,$ $U+pV-TS,$ $=mu N,$

The most informative way of writing the potentials is in terms of their natural variables, since each of these equations can be used to derive all of the other thermodynamic variables of the system. In terms of their natural variables, the thermodynamic potentials of a single-specie ideal gas are:

$U(S,V,N)=hat{c}_V Nkleft(frac{NPhi,e^{S/Nk-hat{c}_p}}{V}right)^{1/hat{c}_V}$
$A(T,V,N)=-NkTleft(1+lnleft(frac{VT^{hat{c}_V}}{NPhi}right)right)$
$H(S,P,N)=hat{c}_p Nkleft(frac{pPhi,e^{S/Nk-hat{c}_p}}{k}right)^{1/hat{c}_p}$
$G(T,P,N)=-NkTlnleft(frac{kT^{hat{c}_p}}{pPhi}right)$

In statistical mechanics, the relationship between the Helmholtz free energy and the partition function is fundamental, and is used to calculate the thermodynamic properties of matters; see configuration integral for more details. Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... In statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. ... Here is a partial list of thermodynamic properties of fluids: temperature [K] density [kg/m3] specific heat at constant pressure [J/kg·K] specific heat at constant volume [J/kg·K] dynamic viscosity [N/m²s] kinematic viscosity [m²/s] thermal conductivity [W/m·K] thermal diffusivity [m²/s] volumetric...

### Multicomponent systems

By Gibbs theorem, the entropy of a multicomponent system is equal to the sum of the entropies of each chemical species (assuming no surface effects). The entropy of a multicomponent system will be: The entropy of mixing (also known as configurational entropy) is the change in the entropy, an extensive thermodynamic quantity, when two different chemical substances or components are mixed. ...

$S=sum_j S_j = sum_j N_j,k left[lnleft( frac{VT^{hat{c}_V}}{N_jPhi}right)+hat{c}_pright]$

where the sum is over all species. Likewise, the free energies are equal to the sums of the free energies of each species so that if Φ is a thermodynamic potential then

$Phi=sum_j Phi_j,$

where Φj is expressed in terms of its natural variables. For example, the internal energy will be:

$U=sum_j hat{c}_V N_jkleft(frac{N_jPhi,e^{S_j/N_jk-hat{c}_p}}{V}right)^{1/hat{c}_V}=hat{c}_V NkT$

where N is defined as

$N=sum_j N_j,$.

## Speed of sound

Main article: Speed of sound

The speed of sound in an ideal gas is given by For other uses, see Speed of sound (disambiguation). ...

$v_{sound} = sqrt{frac{gamma R T}{M}}$

where

$gamma ,$ is the adiabatic index
$R ,$ is the universal gas constant
$T ,$ is the temperature
$M ,$ is the molar mass for the gas (in kg/mol)

The adiabatic index of a gas, is the ratio of its specific heat capacity at constant pressure (CP) to its specific heat capacity at constant volume (CV). ... Molar gas constant (also known as universal gas constant, usually denoted by symbol R) is the constant occurring in the universal gas equation, i. ... For other uses, see Temperature (disambiguation). ... Molar mass is the mass of one mole of a chemical element or chemical compound. ...

### Equation Table for an Ideal Gas

The following table gives the values for the change in the value of some thermodynamic variables under the specified transformation.

 Variable Constant Pressure (Isobaric) $Delta p=0;$ Constant Volume (Isochoric) $Delta V=0;$ Isothermal $Delta T=0;$ Adiabatic $q=0;$ Work, $begin{matrix}w=-int_{V_1}^{V_2} pdV end{matrix}$ $-pleft ( V_2-V_1 right ),$ $0;$ $-nRTlnfrac{V_2}{V_1};$ $C_Vleft ( T_2!-!T_1 right ),$ Heat Capacity, $C,$ $C_p = frac{5}{2}nR,$ $C_V = frac{3}{2}nR ,$ $C_p;$ or $C_V,$ $0;$ Internal Energy, $Delta U = frac{3}{2}nRT,$ $q+w;$ $q_p+pDelta V,$ $q;$ $C_Vleft ( T_2!-!T_1 right ),$ $0;$ $q=-w;$ $w;$ $C_Vleft ( T_2!-!T_1 right ),$ Enthalpy, $Delta H;$ $H=U+PV,$ $C_pleft ( T_2-T_1 right ),$ $q_V+VDelta P;$ $0;$ $0;$ Entropy, $Delta S=-int_{T_1}^{T_2} frac {C}{T}dT$ $C_plnfrac{T_2}{T_1},$ $C_Vlnfrac{T_2}{T_1},$ $nRlnfrac{V_2}{V_1},$ $C_plnfrac{V_2}{V_1}+C_Vlnfrac{P_2}{P_1};$

## Ideal quantum gases

In the above mentioned Sackur-Tetrode equation, the best choice of the entropy constant was found to be proportional to the quantum thermal wavelength of a particle, and the point at which the argument of the logarithm becomes zero is roughly equal to the point at which the average distance between particles becomes equal to the thermal wavelength. In fact, quantum theory itself predicts the same thing. Any gas behaves as an ideal gas at high enough temperature and low enough density, but at the point where the Sackur-Tetrode equation begins to break down, the gas will begin to behave as a quantum gas, composed of either bosons or fermions. (See the gas in a box article for a derivation of the ideal quantum gases, including the ideal Boltzmann gas.) In physics, the Thermal de Broglie wavelength is defined for a free ideal gas of massive particles in equilibrium as: where h is Plancks constant m is the mass of a gas particle k is Boltzmanns constant T is the Temperature of the gas The thermal de Broglie... For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... The Sackur-Tetrode equation is an expression for the entropy of a classical ideal gas which uses quantum considerations to arrive at an exact formula. ... Boson (game) Bosons, named after Satyendra Nath Bose, are particles which form totally-symmetric composite quantum states. ... Fermions, named after Enrico Fermi, are particles which form totally-antisymmetric composite quantum states. ... The results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number particles which do not interact with each other except for instantaneous thermalizing collisions. ...

### Ideal Boltzmann gas

The ideal Boltzmann gas yields the same results as the classical thermodynamic gas, but makes the following identification for the undetermined constant Φ:

$Phi = frac{T^{3/2}Lambda^3}{g}$

where Λ is the thermal de Broglie wavelength of the gas and g is the degeneracy of states. In physics, the Thermal de Broglie wavelength is defined for a free ideal gas of massive particles in equilibrium as: where h is Plancks constant m is the mass of a gas particle k is Boltzmanns constant T is the Temperature of the gas The thermal de Broglie... The word degeneracy has more than one meaning: In general, degeneracy means reverting to an earlier, simpler, state In mathematics, a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class. ...

### Ideal Bose and Fermi gases

An ideal gas of bosons (e.g. a photon gas) will be governed by Bose-Einstein statistics and the distribution of energy will be in the form of a Bose-Einstein distribution. An ideal gas of fermions will be governed by Fermi-Dirac statistics and the distribution of energy will be in the form of a Fermi-Dirac distribution. An ideal Bose gas is a quantum-mechanical version of a classical ideal gas. ... In physics, a photon gas is a gas-like collection of photons, which come together to form something that has the properties of a conventional gas like hydrogen or neon - including pressure, and temperature. ... In statistical mechanics, Bose-Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. ... For other topics related to Einstein see Einstein (disambig) In statistical mechanics, Bose-Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. ... A Fermi gas is a collection of non-interacting fermions. ... Fermi-Dirac distribution as a function of Îµ/Î¼ plotted for 4 different temperatures. ... In statistical mechanics, Fermi-Dirac statistics determines the statistical distribution of fermions over the energy states for a system in thermal equilibrium. ...

Isotherms of an ideal gas The ideal gas law is the equation of state of a hypothetical ideal gas, first stated by BenoÃ®t Paul Ã‰mile Clapeyron in 1834. ... Kinetic theory or kinetic theory of gases attempts to explain macroscopic properties of gases, such as pressure, temperature, or volume, by considering their molecular composition and motion. ... Fluid Dynamics Compressibility (physics) is a measure of the relative volume change of fluid or solid as a response to a pressure (or mean stress) change: . For a gas the magnitude of the compressibility depends strongly on whether the process is adiabatic or isothermal, while this difference is small in... The Bunimovich stadium is a chaotic dynamical billiard A billiard is a dynamical system where a particle alternates between motion in a straight line and specular reflections with a boundary. ...

Results from FactBites:

 Ideal Gas Law (1049 words) In such a gas, all the internal energy is in the form of kinetic energy and any change in internal energy is accompanied by a change in temperature. The ideal gas law can be viewed as arising from the kinetic pressure of gas molecules colliding with the walls of a container in accordance with Newton's laws. One mole of an ideal gas at STP occupies 22.4 liters.
 Ideal Gas Law (541 words) In perfect or ideal gas the change in density is directly related to the change of temperature and pressure as expressed by the Ideal Gas Law. The Individual Gas Constant - R - depends on the particular gas and is related to the molecular weight of the gas. Gas Mixture Properties Special care must be taken for gas mixtures when using the ideal gas law, calculating the mass, the individual gas constant or the density
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