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Gas constant

From Wikipedia, the free encyclopedia
Physical constant equivalent to the Boltzmann constant, but in different units
Value ofR[1]Unit
SI units
8.31446261815324J⋅K−1⋅mol−1
8.31446261815324m3⋅Pa⋅K−1⋅mol−1
8.31446261815324kg⋅m2⋅s−2⋅K−1⋅mol−1
Other common units
8314.46261815324L⋅Pa⋅K−1⋅mol−1
8.31446261815324L⋅kPa⋅K−1⋅mol−1
0.0831446261815324L⋅bar⋅K−1⋅mol−1
8.31446261815324×107erg⋅K−1⋅mol−1
0.730240507295273atm⋅ft3⋅lbmol−1⋅°R−1
10.731577089016psi⋅ft3⋅lbmol−1⋅°R−1
1.985875279009BTU⋅lbmol−1⋅°R−1
297.031214inH2O⋅ft3⋅lbmol−1⋅°R−1
554.984319180torr⋅ft3⋅lbmol−1⋅°R−1
0.082057366080960L⋅atm⋅K−1⋅mol−1
62.363598221529L⋅torr⋅K−1⋅mol−1
1.98720425864083...cal⋅K−1⋅mol−1
8.20573660809596...×10−5m3⋅atm⋅K−1⋅mol−1
Heating gas at constant pressure and constant-volume

Themolar gas constant (also known as thegas constant,universal gas constant, orideal gas constant) is denoted by the symbolR orR. It is the molar equivalent to theBoltzmann constant, expressed in units ofenergy pertemperature increment peramount of substance, rather than energy per temperature increment perparticle. The constant is also a combination of the constants fromBoyle's law,Charles's law,Avogadro's law, andGay-Lussac's law. It is aphysical constant that is featured in many fundamental equations in the physical sciences, such as theideal gas law, theArrhenius equation, and theNernst equation.

The gas constant is theconstant of proportionality that relates the energy scale in physics to the temperature scale and the scale used foramount of substance. Thus, the value of the gas constant ultimately derives from historical decisions and accidents in the setting of units of energy, temperature and amount of substance. The Boltzmann constant and theAvogadro constant were similarly determined, which separately relate energy to temperature and particle count to amount of substance.

The gas constantR is defined as the Avogadro constantNA multiplied by the Boltzmann constantk (orkB):R=NAk=6.02214076⋅1023mol−1⋅1.380649⋅10−23J⋅K−1=8.31446261815324 J⋅K−1⋅mol−1{\displaystyle {\begin{aligned}R&=N_{\text{A}}k\\&=6.02214076\cdot 10^{23}{\text{mol}}^{-1}\cdot 1.380649\cdot 10^{-23}{\text{J}}\cdot {\text{K}}^{-1}\\&=8.31446261815324\ {\text{J}}\cdot {\text{K}}^{-1}\cdot {\text{mol}}^{-1}\end{aligned}}}

Since the2019 revision of the SI, bothNA andk are defined with exact numerical values when expressed in SI units.[2] As a consequence, the SI value of the molar gas constant is exact.

Some have suggested that it might be appropriate to name the symbolR theRegnault constant in honour of the French chemistHenri Victor Regnault, whose accurate experimental data were used to calculate the early value of the constant. However, the origin of the letterR to represent the constant is elusive. The universal gas constant was apparently introduced independently byAugust Friedrich Horstmann (1873)[3][4] andDmitri Mendeleev who reported it first on 12 September 1874.[5] Using his extensive measurements of the properties of gases,[6][7] Mendeleev also calculated it with high precision, within 0.3% of its modern value.[8]

The gas constant occurs in the ideal gas law:PV=nRT=mRspecificT,{\displaystyle PV=nRT=mR_{\text{specific}}T,}whereP is the absolutepressure,V is the volume of gas,n is theamount of substance,m is themass, andT is thethermodynamic temperature.Rspecific is the mass-specific gas constant. The gas constant is expressed in the same unit asmolar heat.

Dimensions

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From the ideal gas lawPV=nRT{\displaystyle PV=nRT} we getR=PVnT,{\displaystyle R={\frac {PV}{nT}},}whereP is pressure,V is volume,n is the amount of a given substance, andT istemperature.

As pressure is defined as force per area, the gas equation can also be written asR=forcearea×volumeamount×temperature.{\displaystyle R={\frac {{\dfrac {\text{force}}{\text{area}}}\times {\text{volume}}}{{\text{amount}}\times {\text{temperature}}}}.}

Area and volume are (length)2 and (length)3 respectively. Therefore:R=force(length)2×(length)3amount×temperature=force×lengthamount×temperature.{\displaystyle {\begin{aligned}R&={\frac {{\dfrac {\text{force}}{({\text{length}})^{2}}}\times ({\text{length}})^{3}}{{\text{amount}}\times {\text{temperature}}}}\\\\&={\frac {{\text{force}}\times {\text{length}}}{{\text{amount}}\times {\text{temperature}}}}.\end{aligned}}}

Since force × length = work,R=workamount×temperature.{\displaystyle R={\frac {\text{work}}{{\text{amount}}\times {\text{temperature}}}}.}

The physical significance ofR is work per mole per kelvin. It may be expressed in any set of units representing work or energy (such asjoules), units representing temperature on an absolute scale (such askelvin orrankine), and any system of units designating a mole or a similar pure number that allows an equation ofmacroscopic mass and fundamental particle numbers in a system, such as an ideal gas (seeAvogadro constant).

Instead of a mole, the constant can be expressed by considering thenormal cubic metre.

Otherwise, we can also say thatforce=mass×length(time)2.{\displaystyle {\text{force}}={\frac {{\text{mass}}\times {\text{length}}}{({\text{time}})^{2}}}.}

Therefore, we can writeR asR=mass×length2amount×temperature×(time)2.{\displaystyle R={\frac {{\text{mass}}\times {\text{length}}^{2}}{{\text{amount}}\times {\text{temperature}}\times ({\text{time}})^{2}}}.}

And so, in terms ofSI base units,R =8.31446261815324 kg⋅m2⋅s−2⋅K−1⋅mol−1.

Relationship with the Boltzmann constant

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TheBoltzmann constantkB (alternativelyk) may be used in place of the molar gas constant by working in pure particle count,N, rather than amount of substance,n, sinceR=NAkB,{\displaystyle R=N_{\text{A}}k_{\text{B}},}whereNA is theAvogadro constant. For example, theideal gas law in terms of the Boltzmann constant isPV=NkBT,{\displaystyle PV=Nk_{\text{B}}T,}whereN is the number of particles (molecules in this case), or to generalize to an inhomogeneous system the local form holds:P=nkBT,{\displaystyle P=nk_{\text{B}}T,}wheren =N/V is thenumber density. Finally, by defining thekinetic energy associated to the temperature,T:=kBT,{\displaystyle T:=k_{\text{B}}T,}the equation becomes simplyP=nT,{\displaystyle P=nT,}which is the form usually encountered instatistical mechanics and other branches oftheoretical physics.

Measurement and replacement with defined value

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As of 2006, the most precise measurement ofR had been obtained by measuring thespeed of soundca(P, T) inargon at the temperatureT of thetriple point of water at differentpressuresP, andextrapolating to the zero-pressure limitca(0, T). The value ofR is then obtained from the relationca(0,T)=γ0RTAr(Ar)Mu,{\displaystyle c_{\text{a}}(0,T)={\sqrt {\frac {\gamma _{0}RT}{A_{\text{r}}({\text{Ar}})M_{\text{u}}}}},}where

  • γ0 is theheat capacity ratio (5/3 formonatomic gases such as argon);
  • T is the temperature,TTPW = 273.16 K by the definition of the kelvin at that time;
  • Ar(Ar) is the relative atomic mass of argon; and
  • Mu = 10−3 kg⋅mol−1 as defined at the time.

However, following the2019 revision of the SI,R now has an exact value defined in terms of other exactly defined physical constants.

Specific gas constant

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Rspecific
for dry air[9]		Unit
287.052874J⋅kg−1⋅K−1
53.3523ft⋅lbf⋅lb−1⋅°R−1
1,716.46ft⋅lbf⋅slug−1⋅°R−1

Thespecific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by themolar mass (M) of the gas or mixture:Rspecific=RM.{\displaystyle R_{\text{specific}}={\frac {R}{M}}.}

Just as the molar gas constant can be related to theBoltzmann constant, so can the specific gas constant by dividing the Boltzmann constant by themolecular mass of the gas:Rspecific=kBm.{\displaystyle R_{\text{specific}}={\frac {k_{\text{B}}}{m}}.}

Another important relationship comes from thermodynamics.Mayer's relation relates the specific gas constant to the specific heat capacities for a calorically or thermallyperfect gas:Rspecific=cP−cV,{\displaystyle R_{\text{specific}}=c_{P}-c_{V},}wherecP is thespecific heat capacity for a constant pressure andcV is the specific heat capacity for a constant volume.[10]

It is common, especially in engineering applications, to represent the specific gas constant by the symbolR. In such cases, the universal gas constant is usually given a different symbol such asR to distinguish it. In any case, the context and/or unit of the gas constant should make it clear as to whether the universal or specific gas constant is being referred to.[11]

In case of air, using the perfect gas law and thestandard sea-level conditions (SSL) (air densityρ0 = 1.225 kg/m3, temperatureT0 = 288.15 K and pressurep0 =101325 Pa), we have thatRair =P0/(ρ0T0) =287.052874247 J·kg−1·K−1. Then the molar mass of air is computed byM0 =R/Rair =28.964917 g/mol.[12]

U.S. Standard Atmosphere

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TheU.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constantR∗ as[13][14]R∗ =8.31432×103 N⋅m⋅kmol−1⋅K−1, equivalent to8.31432 J⋅K−1⋅mol−1.

The use of the kilomole in the unit results in an extra factor of1000 in the constant. The USSA1976 acknowledges that their defined value ofR* is not consistent with the cited values for the Avogadro constant and the Boltzmann constant.[14] This disparity is not a significant departure from accuracy –R∗ is slightly greater than 99.998% of the actual value of the constant,R = 8.31446261815324 J⋅mol−1⋅K−1‍[1] – and USSA1976 uses this value ofR∗ for all the calculations of the standard atmosphere. When using theISO value ofR, the calculated pressure increases by only 0.62 pascal at 11 kilometres (the equivalent of a difference of only 17.4 centimetres or 6.8 inches) and 0.292 Pa at 20 km (the equivalent of a difference of only 33.8 cm or 13.2 in).

This definition was published well before the 2019 SI revision, through which the constant was given an exact value.

References

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  1. ^ab"2022 CODATA Value: molar gas constant".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved2024-05-18.
  2. ^Newell, David B.; Tiesinga, Eite (2019).The International System of Units (SI). NIST Special Publication 330. Gaithersburg, Maryland: National Institute of Standards and Technology.doi:10.6028/nist.sp.330-2019.S2CID 242934226.
  3. ^Jensen, William B. (July 2003). "The Universal Gas ConstantR".J. Chem. Educ.80 (7): 731.Bibcode:2003JChEd..80..731J.doi:10.1021/ed080p731.
  4. ^"Ask the Historian: The Universal Gas Constant — Why is it represented by the letterR?"(PDF).
  5. ^Mendeleev, Dmitri I. (September 12, 1874). "An exert from the Proceedings of the Chemical Society's Meeting on Sept. 12, 1874".Journal of Russian Chemical-Physical Society, Chemical Part.VI (7):208–209.
  6. ^Mendeleev, Dmitri I. (1875).On the elasticity of gases [Объ упругости газовъ]. A. M. Kotomin, St.-Petersburg.
  7. ^D. Mendeleev. On the elasticity of gases. 1875 (in Russian)Free access icon
  8. ^Mendeleev, Dmitri I. (March 22, 1877)."Mendeleef's researches on Mariotte's law 1".Nature.15 (388):498–500.Bibcode:1877Natur..15..498D.doi:10.1038/015498a0.Free access icon
  9. ^Based on a mean molar mass fordry air of 28.964917 g/mol.
  10. ^Anderson,Hypersonic and High-Temperature Gas Dynamics, AIAA Education Series, 2nd ed., 2006.
  11. ^Moran, Michael J.; Shapiro, Howard N.; Boettner, Daisie D.; Bailey, Margaret B. (2018).Fundamentals of Engineering Thermodynamics (9th ed.). Hoboken, New Jersey: Wiley.
  12. ^Manual of the US Standard Atmosphere(PDF) (3 ed.). National Aeronautics and Space Administration. 1962. pp. 7–11.
  13. ^"Standard Atmospheres". Retrieved2007-01-07.
  14. ^abNational Oceanic and Atmospheric Administration; National Aeronautics and Space Administration; United States Air Force (October 1976).U.S. Standard Atmosphere, 1976(PDF). Washington, D.C.: U.S. Government Printing Office. p. 3. NOAA-S/T 76-1562. Archived fromthe original(PDF) on 2007-07-05. Retrieved2007-01-16.

External links

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