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The ideal gas law is a useful approximation for calculating temperatures, volumes, pressures or amount of substance for many gases over a wide range of values, as long as the temperatures and pressures are far from the values where [[condensation]] or [[sublimation]] occur.


The '''ideal gas law''' is the [[equation of state]] of an '''ideal gas''' (also known as '''perfect gas'''). As  an equation of state it relates the [[Pressure#Absolute pressure versus gauge pressure|absolute pressure]] ''p'' of an ideal gas to its [[temperature|absolute temperature]] ''T''. Further parameters that enter the equation are the volume ''V'' of the container holding the gas and the number of moles ''n'' in the container. The equation is,
The '''ideal gas law''' is the equation of state of an '''ideal gas''' (also known as a '''perfect gas''') that relates its [[Pressure#Absolute pressure versus gauge pressure|absolute pressure]] ''p'' to its [[temperature|absolute temperature]] ''T''. Further parameters that enter the equation are the [[volume]] ''V'' of the container holding the gas and the amount ''n'' (in [[mole (unit)|moles]]) of gas contained in there. The law reads
:<math> pV = nRT, \,</math>
:<math> pV = nRT \,</math>
where ''R'' is the [[molar gas constant]] defined as the product of the [[Boltzmann constant]] ''k''<sub>B</sub> and  [[Avogadro's constant]] ''N''<sub>A</sub>,
where ''R'' is the [[molar gas constant]], defined as the product of the [[Boltzmann constant]] ''k''<sub>B</sub> and  [[Avogadro's constant]] ''N''<sub>A</sub>
:<math>
:<math>
R \equiv N_\mathrm{A} k_\mathrm{B}.
R \equiv N_\mathrm{A} k_\mathrm{B}
</math>
</math>
Currently, the most accurate value of R is:<ref>[http://physics.nist.gov/cgi-bin/cuu/Value?r Molar gas constant] Obtained from the [[NIST]] website. [http://www.webcitation.org/query?url=http%3A%2F%2Fphysics.nist.gov%2Fcgi-bin%2Fcuu%2FValue%3Fr&date=2009-01-03 (Archived by WebCite® at http://www.webcitation.org/5dZ3JDcYN on Jan 3, 2009)]</ref>  8.314472 ± 0.000015 J·K<sup>-1</sup>·mol<sup>-1</sup>.
Currently, the most accurate value of R is:<ref>[http://physics.nist.gov/cgi-bin/cuu/Value?r Molar gas constant] Obtained from the [[NIST]] website. [http://www.webcitation.org/query?url=http%3A%2F%2Fphysics.nist.gov%2Fcgi-bin%2Fcuu%2FValue%3Fr&date=2009-01-03 (Archived by WebCite® at http://www.webcitation.org/5dZ3JDcYN on Jan 3, 2009)]</ref>  8.314472 ± 0.000015 J·K<sup>-1</sup>·mol<sup>-1</sup>.


The law applies to hypothetical gases that consist of atoms or molecules that do not interact, i.e., that move through the container independently of one anotherThe law  is a useful approximation for calculating temperatures, volumes, pressures or number of [[mole (unit)|moles]] for many gases over a wide range of temperatures and pressures, as long as the temperatures and pressures are far from the values where condensation or sublimation occurs.
The law applies to ''ideal gases'' which are hypothetical gases that consist of [[molecules]]<ref>Atoms may be seen as monatomic molecules.</ref> that do not interact, i.e., that move through the container independently of each otherIn contrast to what is sometimes stated, an ideal gas does not necessarily consist of [[point particle]]s without internal structure, but may be formed by polyatomic molecules with internal rotational, vibrational, and electronic degrees of freedom. The ideal gas law describes the motion of the [[center of mass|centers of mass]] of the molecules and, indeed, mass centers may be seen as structureless point masses. However, for other properties of ideal gases, such as [[entropy (thermodynamics)|entropy]], the internal structure may play a role.


The ideal gas law is the combination of [[Boyle's law]] (given in 1662 and stating that pressure is inversely proportional to volume) and [[Gay-Lussac's law]] (given in 1808 and stating that pressure is proportional to temperature)Gay-Lussac's law was found a few decennia before Gay-Lussac's publication by [[Jacques Charles]]. In some countries the ideal gas law is known as the ''Boyle-Gay-Lussac law''.
Real gases deviate from ideal gas behavior because the intermolecular attractive and repulsive forces cause the motions of the molecules to be correlated. The deviation is especially significant at low temperatures or high pressures, i.e., close to condensationA conventional measure for this deviation is the [[Compressibility factor (gases)|compressibility factor]].  


Real gases deviate from ideal gas behavior because of the intermolecular attractive and repulsive forcesThe deviation is especially significant at low temperatures or high pressures, i.e., close to condensation. There are many equations of state available for use with real gases, the simplest of which is the [[van der Waals equation]].
There are many equations of state available for use with real gases, the simplest of which is the [[van der Waals equation]].
 
== Historic background ==
 
The early work on the behavior of gases began in pre-industrialized [[Europe]] in the latter half of the 17th century by [[Robert Boyle]] who formulated [[Boyle's law]] in 1662 (independently confirmed by Edme Mariotte at about the same time).<ref name=Savidge>[http://www.ceesi.com/docs_techlib/events/ishm2003/Docs/1040.pdf Compressibility of Natural Gas] Jeffrey L. Savidge, 78th International School for Hydrocarbon Measurement (Class 1040), 2003. From the website of the Colorado Engineering Experiment Station, Inc. (CEESI).</ref> Their work on air at low pressures established the inverse relationship between pressure and volume, ''V'' = constant / ''p'' at constant temperature and a fixed amount of air. Boyle's Law is often referred to as the Boyles-Mariotte Law.
 
In 1699, Guillaume Amontons formulated what is now known as Amontons' law, ''p'' = constant / ''T''.
 
Almost a century later, [[Jacques Alexandre César Charles]] experimented with hot-air balloons (around 1780), and additional contributions by [[John Dalton]] (1801) and [[Joseph Louis Gay-Lussac]] (1808) showed that a sample of gas, at a fixed pressure, increases in volume linearly with the temperature, i.e. ''V'' / ''T'' is constant.
 
In 1811, [[Amedeo Avogadro]] reinterpreted ''[[Gay-Lussac's law]]'' of combining volumes to state what is now commonly called ''[[Avogadro's law]]'': equal volumes of any two gases at the same temperature and pressure contain the same number of molecules.
 
In 1834, [[Benoît Paul Émile Clapeyron]] combined the work of Boyle, Mariotte, Charles and Gay-Lussac into an equation of state of a perfect (i.e., ideal) gas: ''PV'' = ''R''<sub>o</sub>''T'' where ''R''<sub>o</sub> was a gas-dependent constant.<ref name=Savidge/><ref>{{cite journal
| author=Emile Clapeyron
|title=Mémoire sur la puissance motrice de la chaleur (The motive power of heat)
|journal=Journal de l'École Polytechnique
|volume=14
|issue=23
|pages=153-190
|year=1834}}</ref> In 1845, [[Victor Regnault]] cast Clapeyron's perfect gas equation into the familiar ideal gas equation form by applying Avagadro's hypothesis on the volume of one mole of an ideal gas, i.e. ''PV'' = ''nRT''.<ref name=Savidge/>
 
Sometimes, the ideal gas law is referred to as the ''Boyle-Gay-Lussac law''. However, with hindsight, the ''Boyle-Gay-Lussac law'', ''Amontons' law'' and ''Avogadro's law'' are all special cases of the ideal gas law.
 
Extrapolation of the volume/temperature relationship of ideal and many real gases to zero volume crosses the temperature axis at about &minus;273 [[Celsius|°C]]. This temperature is defined as the absolute zero temperature. Since any real gas would [[liquefaction|liquefy]] before reaching it, this temperature region remains a theoretical minimum.
 
== Statistical mechanics derivation ==
The [[statistical mechanics]]<ref>{{cite book|author=T.L. Hill|title=An Introduction to Statistical Thermodynamics|edition=|publisher= Dover Publications|date=1987|id=ISBN 0-486-65242-4}}</ref><ref>{{cite book|author=D.A. McQuarrie|title=Statistical Mechanics|edition=| publisher=University Science Books|year=2000|id=ISBN 1-891389-15-7}}</ref> derivation of the ideal gas law provides the most precise insight into the microscopic conditions that a gas must satisfy in order to be called an ideal gas. In the derivation below, we will make the following two assumptions<ref>{{cite book|author=R. H. Fowler|title=Statistical Mechanics|edition=2nd Edition (Reprinted)|publisher= Cambridge University Press|pages=page 31|year=1966|id=ISBN 0-521-09377-5}}</ref>:
#The molecules constituting the gas are practically independent systems, each pursuing its own motion;
#Exchange of energy between molecules occasionally takes place, so that the system can achieve a [[thermal equilibrium]].<ref>Such occasional exchange of energy can proceed via collisions with the walls, through interaction with a radiation field, or sporadic molecule-molecule collisions. This energy exchange is not explicitly included in the following formalism. </ref>
 
Starting with the second assumption, we recall from equilibrium statistical mechanics that the canonical [[Partition function (statistical physics)|partition function]] is a function of ''N'' &equiv; ''nN''<sub>A</sub>, ''V'', and ''T'', defined as
:<math>
Q(N,V,T) = \sum_I e^{-\mathcal{E}_I/(k_\mathrm{B}T)}
</math>
where <math>\mathcal{E}_I</math> is the ''I''-th energy of the ''total'' gas (i.e. the energy of all ''N'' molecules). From [[quantum mechanics]] follows that a gas in a finite-size container has [[discreteness|discrete]] energies; ''I'' is the discrete index labeling the different energies.
The sum is over all eigenstates of the energy operator including degeneracies.  Further, we recall that the [[Helmholtz free energy]] is given
by <math> A = -k_\mathrm{B}T\,\ln Q </math>. The following expression for the absolute pressure ''p'' will be the starting point in the derivation
:<math>
p = -\left(\frac{\partial A}{\partial V}\right)_{N,T} = k_\mathrm{B}T \left( \frac{\partial \ln Q}{\partial V} \right)_{N,T}
</math>
 
The only approximation (but a very drastic one) that must be made is assumption 1 from above, i.e. that the energies <math>\mathcal{E}_I</math> are sums of ''one-molecule energies'' <math>\varepsilon_i</math>. These one-molecule energies are those of a single molecule moving by itself in the vessel, an approximation that is common in many branches of physics and known as the [[independent particle approximation]]. Thus,
:<math>
\mathcal{E}_I = \varepsilon_{i_1} +  \varepsilon_{i_2} + \cdots
</math>
The total partition function ''Q'' will factorize into one-molecule partition functions ''q'' given by,
:<math>
q(N,V,T) = \sum_i e^{-\varepsilon_i/(k_\mathrm{B}T)}
</math>


== An ideal gas ==
In absence of interactions ''Q'' becomes (assuming that the gas consists of one type of molecules only),
To be an ideal gas, several conditions must be met.   First, the size of the gas molecules must be negligible compared to the average distance between them. This condition is not true at extremely high pressures or extremely cold temperatures.  Second, the intermolecular forces of attraction or repulsion between molecules must be very weak or negligible except during collisions. And third, when the gas molecules do collide, thus must do so in an elastic manner.  That is, they bounce right off of each other rather than sticking together.
:<math>
Q = \sum_{i_1,i_2, \ldots} e^{-(\varepsilon_{i_1} +  \varepsilon_{i_2} + \cdots)/(k_\mathrm{B}T)}
= \sum_{i_1} e^{-\varepsilon_{i_1}/(k_\mathrm{B}T)}\sum_{i_2} e^{-\varepsilon_{i_2}/(k_\mathrm{B}T)}\cdots = q^{N}.
</math>
This form of ''Q''  would be correct if the non-interacting gas molecules were non-identical. However, in the early days of quantum mechanics it was discovered that gas molecules of the same type are identical particles, just like electrons, and that a [[factorial]] 1/''N''! must be inserted to avoid overcounting. Later it was shown that this factorial arises from  [[Bose-Einstein statistics]] (obeyed by the majority of stable molecules). Complete Bose-Einstein statistics itself needs only to be applied at temperatures close to the absolute zero. For higher temperatures Bose-Einstein statistics goes over into [[Boltzmann statistics]], which requires the simple factor 1/''N''! that we will insert now ad hoc<ref>This ad hoc insertion cuts short the application of the Bose-Einstein statistics and the proof that it leads to a multiplication by 1/''N''! for higher temperatures</ref> into the expression for ''Q''In summary, from the additivity of the molecular energies and the application of Boltzmann statistics follows
:<math>
Q = \frac{q^{N}}{N!}
</math>
The application of Boltzmann statistics is of no consequence to the equation of state, but modifies expressions for other properties of the gas, such as the [[entropy]]. The factorization of ''Q'' would be exact if (i) the molecules would not interact and if (ii) every molecule had the whole volume ''V'' of the container to its disposal, or in other words, if the molecules themselves had zero volume.


== When the ideal gas law fails ==
Now,
When the ideal gas law fails, an equation of state such as the van der Waals equation may be used.  However, this equation contains constants, <math>a</math> and <math>b</math>, that are unique for each gas. The van der Waals equation also fails at extreme high pressures. When the coefficients <math>a</math> and <math>b</math> are set to zero, the van der Waals equation reduces to the ideal gas law.
:<math>
p = k_\mathrm{B}T \left(\frac{\partial \ln Q}{\partial V}\right)  = k_\mathrm{B}T \left(\frac{\partial (N\ln q - \ln N!)}{\partial V}\right) = N k_\mathrm{B}T \left(  \frac{\partial \ln q}{\partial V}\right)
</math>
where we used the rules [[logarithm|ln]](''a''/''b'') = ln''a'' - ln''b'' and ln''a''<sup>n</sup> = n ln''a''.


'''van der Waals equation''' : &nbsp; <math>\left(p + \frac{n^2 a}{V^2}\right)\left(V-nb\right) = nRT</math>
It follows from both  classical mechanics and quantum mechanics  that the molecular energy <math>\varepsilon_i</math> can be ''exactly separated'' as
:<math>
\varepsilon_i = \varepsilon_i^\mathrm{transl} + \varepsilon_i^\mathrm{internal}
\quad\Longrightarrow\quad q = q^\mathrm{transl}\; q^\mathrm{internal}
</math>
where <math>\varepsilon_i^\mathrm{transl}</math> is the translational energy of the [[center of mass]] of the molecule and <math>\varepsilon_i^\mathrm{internal}</math> is the internal (rotational, vibrational, electronic)  energy  of the molecule. This factorization of the one-molecule partition function into a translational and an internal factor proceeds in the same way as the factorization of the ''N''-molecule partition function ''Q'' into one-molecule partition functions.


== Background ==
The internal energy of the molecule does ''not'' depend on the volume ''V'' (this is an exact result), but the translational energy ''does'', hence
The gas laws were developed in the 1660's, starting with '''Boyle's law''', derived by [[Robert Boyle]].  Boyle's law states that "the volume of a sample of gas at a given temperature varies inversely with the applied pressure, or V = constant / p (at a fixed temperature and amount of gas)".  [[Jacques Alexandre Charles]]' experiments with hot-air balloons, and additional contributions by [[John Dalton]] (1801) and [[Joseph Louis Gay-Lussac]] (1802) showed that a sample of gas, at a fixed pressure, increases in volume linearly with the temperature, or V/T = a constant.  This is known as  '''Charles's law'''. Extrapolations of volume/temperature data for many gases, to a volume of zero, all cross at about -273 degrees [[Celsius]], which is defined as [[absolute zero]]. Since real gases would liquefy before reaching this temperature, this temperature region remains a theoretical minimum.
:<math>
p = N k_\mathrm{B}T\left( \frac{\partial \ln q^\mathrm{transl}}{\partial V} +  \frac{\partial \ln q^\mathrm{internal}}{\partial V}\right)  = N k_\mathrm{B}T\; \frac{\partial \ln q^\mathrm{transl}}{\partial V}
</math>
The determination of the translational energy of one molecule moving in a box of volume ''V'' is one of the few problems in quantum mechanics that can be solved analytically. That is, the energies <math>\varepsilon_i^\mathrm{transl}</math> are known exactly. To a very good approximation, one may replace the sum appearing in <math>q^\mathrm{transl}</math> by an integral, finding


In 1811 [[Amedeo Avogadro]] re-interpreted '''[[Gay-Lussac's law|Gay-Lussac's law of combining volumes]]''' (1808) to state '''Avogadro's law''': equal volumes of any two gases at the same temperature and pressure contain the same number of molecules.
:<math>
q^\mathrm{transl} \equiv \sum_i e^{-\varepsilon_i^\mathrm{transl}/(k_\mathrm{B}T)} = \frac{V}{\Lambda^3}\quad \hbox{with}\quad \Lambda = \left(\frac{h^2}{2\pi M k_\mathrm{B} T}\right)^{1/2}
</math>
where ''h'' is [[Planck's constant]] and ''M'' is the total mass of the molecule. Note that &Lambda;, the thermal [[de Broglie wavelength]], does not depend on the volume ''V'', so that
:<math>
p = N k_\mathrm{B}T\left( \frac{\partial (\ln V - 3\ln \Lambda) }{\partial V}\right) = \frac{N k_\mathrm{B}T}{V}
</math>
Here we applied that
:<math>
\frac{\partial \ln V }{\partial V} = \frac{1}{V}\quad\hbox{and}\quad \frac{\partial \ln \Lambda }{\partial V} = 0
</math>
Using that ''N'' = ''nN''<sub>A</sub> and ''N''<sub>A</sub>''k''<sub>B</sub> = ''R'' (see introduction), we have


== Special cases of the ideal gas law ==
:<math>p\,V = nN_\mathrm{A}\,k_\mathrm{B}\,T = nR\,T</math>
Because the ideal gas law reflects the combined contributions of several scientists, a number of gas laws are special cases of the ideal gas law. [[Amonton's law]] states that at a fixed volume and moles of gas, that the absolute pressure and temperature are inversely related (<math>p \propto 1/T</math>), while [[Boyle's law]] states that at fixed temperature and moles of gas, the pressure and volume are inversely related (<math>p \propto 1/V</math>). [[Charles' law]] states that the volume and temperature are directly related (<math>V \propto T</math>) at fixed absolute pressure and moles of gas.  [[Avogradro's law]] simply states that at fixed temperature and pressure, the volume of gas is related to a molar gas volume.


and that completes the proof of the ideal gas law. 


==References==
==References==
{{reflist}}
{{reflist}}[[Category:Suggestion Bot Tag]]

Latest revision as of 07:00, 31 August 2024

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Values of R Units
8.314472 J·K-1·mol-1
0.082057 L·atm·K-1·mol-1
8.205745 × 10-5 m3·atm·K-1·mol-1
8.314472 L·kPa·K-1·mol-1
8.314472 m3·Pa·K-1·mol-1
62.36367 mmHg·K-1·mol-1
62.36367 torr·K-1·mol-1
83.14472 L·mbar·K-1·mol-1
10.7316 ft3·psi· °R-1·lb-mol-1
0.73024 ft3·atm·°R-1·lb-mol-1

The ideal gas law is a useful approximation for calculating temperatures, volumes, pressures or amount of substance for many gases over a wide range of values, as long as the temperatures and pressures are far from the values where condensation or sublimation occur.

The ideal gas law is the equation of state of an ideal gas (also known as a perfect gas) that relates its absolute pressure p to its absolute temperature T. Further parameters that enter the equation are the volume V of the container holding the gas and the amount n (in moles) of gas contained in there. The law reads

where R is the molar gas constant, defined as the product of the Boltzmann constant kB and Avogadro's constant NA

Currently, the most accurate value of R is:[1] 8.314472 ± 0.000015 J·K-1·mol-1.

The law applies to ideal gases which are hypothetical gases that consist of molecules[2] that do not interact, i.e., that move through the container independently of each other. In contrast to what is sometimes stated, an ideal gas does not necessarily consist of point particles without internal structure, but may be formed by polyatomic molecules with internal rotational, vibrational, and electronic degrees of freedom. The ideal gas law describes the motion of the centers of mass of the molecules and, indeed, mass centers may be seen as structureless point masses. However, for other properties of ideal gases, such as entropy, the internal structure may play a role.

Real gases deviate from ideal gas behavior because the intermolecular attractive and repulsive forces cause the motions of the molecules to be correlated. The deviation is especially significant at low temperatures or high pressures, i.e., close to condensation. A conventional measure for this deviation is the compressibility factor.

There are many equations of state available for use with real gases, the simplest of which is the van der Waals equation.

Historic background

The early work on the behavior of gases began in pre-industrialized Europe in the latter half of the 17th century by Robert Boyle who formulated Boyle's law in 1662 (independently confirmed by Edme Mariotte at about the same time).[3] Their work on air at low pressures established the inverse relationship between pressure and volume, V = constant / p at constant temperature and a fixed amount of air. Boyle's Law is often referred to as the Boyles-Mariotte Law.

In 1699, Guillaume Amontons formulated what is now known as Amontons' law, p = constant / T.

Almost a century later, Jacques Alexandre César Charles experimented with hot-air balloons (around 1780), and additional contributions by John Dalton (1801) and Joseph Louis Gay-Lussac (1808) showed that a sample of gas, at a fixed pressure, increases in volume linearly with the temperature, i.e. V / T is constant.

In 1811, Amedeo Avogadro reinterpreted Gay-Lussac's law of combining volumes to state what is now commonly called Avogadro's law: equal volumes of any two gases at the same temperature and pressure contain the same number of molecules.

In 1834, Benoît Paul Émile Clapeyron combined the work of Boyle, Mariotte, Charles and Gay-Lussac into an equation of state of a perfect (i.e., ideal) gas: PV = RoT where Ro was a gas-dependent constant.[3][4] In 1845, Victor Regnault cast Clapeyron's perfect gas equation into the familiar ideal gas equation form by applying Avagadro's hypothesis on the volume of one mole of an ideal gas, i.e. PV = nRT.[3]

Sometimes, the ideal gas law is referred to as the Boyle-Gay-Lussac law. However, with hindsight, the Boyle-Gay-Lussac law, Amontons' law and Avogadro's law are all special cases of the ideal gas law.

Extrapolation of the volume/temperature relationship of ideal and many real gases to zero volume crosses the temperature axis at about −273 °C. This temperature is defined as the absolute zero temperature. Since any real gas would liquefy before reaching it, this temperature region remains a theoretical minimum.

Statistical mechanics derivation

The statistical mechanics[5][6] derivation of the ideal gas law provides the most precise insight into the microscopic conditions that a gas must satisfy in order to be called an ideal gas. In the derivation below, we will make the following two assumptions[7]:

  1. The molecules constituting the gas are practically independent systems, each pursuing its own motion;
  2. Exchange of energy between molecules occasionally takes place, so that the system can achieve a thermal equilibrium.[8]

Starting with the second assumption, we recall from equilibrium statistical mechanics that the canonical partition function is a function of NnNA, V, and T, defined as

where is the I-th energy of the total gas (i.e. the energy of all N molecules). From quantum mechanics follows that a gas in a finite-size container has discrete energies; I is the discrete index labeling the different energies. The sum is over all eigenstates of the energy operator including degeneracies. Further, we recall that the Helmholtz free energy is given by . The following expression for the absolute pressure p will be the starting point in the derivation

The only approximation (but a very drastic one) that must be made is assumption 1 from above, i.e. that the energies are sums of one-molecule energies . These one-molecule energies are those of a single molecule moving by itself in the vessel, an approximation that is common in many branches of physics and known as the independent particle approximation. Thus,

The total partition function Q will factorize into one-molecule partition functions q given by,

In absence of interactions Q becomes (assuming that the gas consists of one type of molecules only),

This form of Q would be correct if the non-interacting gas molecules were non-identical. However, in the early days of quantum mechanics it was discovered that gas molecules of the same type are identical particles, just like electrons, and that a factorial 1/N! must be inserted to avoid overcounting. Later it was shown that this factorial arises from Bose-Einstein statistics (obeyed by the majority of stable molecules). Complete Bose-Einstein statistics itself needs only to be applied at temperatures close to the absolute zero. For higher temperatures Bose-Einstein statistics goes over into Boltzmann statistics, which requires the simple factor 1/N! that we will insert now ad hoc[9] into the expression for Q. In summary, from the additivity of the molecular energies and the application of Boltzmann statistics follows

The application of Boltzmann statistics is of no consequence to the equation of state, but modifies expressions for other properties of the gas, such as the entropy. The factorization of Q would be exact if (i) the molecules would not interact and if (ii) every molecule had the whole volume V of the container to its disposal, or in other words, if the molecules themselves had zero volume.

Now,

where we used the rules ln(a/b) = lna - lnb and lnan = n lna.

It follows from both classical mechanics and quantum mechanics that the molecular energy can be exactly separated as

where is the translational energy of the center of mass of the molecule and is the internal (rotational, vibrational, electronic) energy of the molecule. This factorization of the one-molecule partition function into a translational and an internal factor proceeds in the same way as the factorization of the N-molecule partition function Q into one-molecule partition functions.

The internal energy of the molecule does not depend on the volume V (this is an exact result), but the translational energy does, hence

The determination of the translational energy of one molecule moving in a box of volume V is one of the few problems in quantum mechanics that can be solved analytically. That is, the energies are known exactly. To a very good approximation, one may replace the sum appearing in by an integral, finding

where h is Planck's constant and M is the total mass of the molecule. Note that Λ, the thermal de Broglie wavelength, does not depend on the volume V, so that

Here we applied that

Using that N = nNA and NAkB = R (see introduction), we have

and that completes the proof of the ideal gas law.

References

  1. Molar gas constant Obtained from the NIST website. (Archived by WebCite® at http://www.webcitation.org/5dZ3JDcYN on Jan 3, 2009)
  2. Atoms may be seen as monatomic molecules.
  3. 3.0 3.1 3.2 Compressibility of Natural Gas Jeffrey L. Savidge, 78th International School for Hydrocarbon Measurement (Class 1040), 2003. From the website of the Colorado Engineering Experiment Station, Inc. (CEESI).
  4. Emile Clapeyron (1834). "Mémoire sur la puissance motrice de la chaleur (The motive power of heat)". Journal de l'École Polytechnique 14 (23): 153-190.
  5. T.L. Hill (1987). An Introduction to Statistical Thermodynamics. Dover Publications. ISBN 0-486-65242-4. 
  6. D.A. McQuarrie (2000). Statistical Mechanics. University Science Books. ISBN 1-891389-15-7. 
  7. R. H. Fowler (1966). Statistical Mechanics, 2nd Edition (Reprinted). Cambridge University Press, page 31. ISBN 0-521-09377-5. 
  8. Such occasional exchange of energy can proceed via collisions with the walls, through interaction with a radiation field, or sporadic molecule-molecule collisions. This energy exchange is not explicitly included in the following formalism.
  9. This ad hoc insertion cuts short the application of the Bose-Einstein statistics and the proof that it leads to a multiplication by 1/N! for higher temperatures