From en.wikipedia.org:
[Physical constant equivalent to the Boltzmann constant, but in different
units] {| class="wikitable" style="margin: 0 0 0 0.5em; float: right;" ! Value
of [_R_][R] ! Unit |- | colspan="2" | SI UNITS |- | [8.31446261815324] |
J⋅K<sup>−1</sup>⋅mol<sup>−1</sup> |- | [8.31446261815324] | m<sup> 3
</sup>⋅Pa⋅K<sup>−1</sup>⋅mol<sup>−1</sup> |- | [8.31446261815324] |
kg⋅m<sup>2</sup>⋅s<sup>−2</sup>⋅K<sup>−1</sup>⋅mol<sup>−1</sup> |- |
colspan="2" | OTHER COMMON UNITS |- | [8314.46261815324] |
L⋅Pa⋅K<sup>−1</sup>⋅mol<sup>−1</sup> |- | [8.31446261815324] |
L⋅kPa⋅K<sup>−1</sup>⋅mol<sup>−1</sup> |- | [0.0831446261815324] |
L⋅bar⋅K<sup>−1</sup>⋅mol<sup>−1</sup> |- | [8.31446261815324] |
erg⋅K<sup>−1</sup>⋅mol<sup>−1</sup> |- | [0.730240507295273] |
atm⋅ft<sup>3</sup>⋅lbmol<sup>−1</sup>⋅°R<sup>−1</sup> |- | [10.731577089016] |
psi⋅ft<sup>3</sup>⋅lbmol<sup>−1</sup>⋅°R<sup>−1</sup> |- | [1.985875279009] |
BTU⋅lbmol<sup>−1</sup>⋅°R<sup>−1</sup> |- | [297.031214] | inH <sub> 2 </sub>
O⋅ft<sup>3</sup>⋅lbmol<sup>−1</sup>⋅°R<sup>−1</sup> |- | [554.984319180] |
torr⋅ft<sup>3</sup>⋅lbmol<sup>−1</sup>⋅°R<sup>−1</sup> |- |
[0.082057366080960] | L⋅atm⋅K<sup>−1</sup>⋅mol<sup>−1</sup> |- |
[62.363598221529] | L⋅torr⋅K<sup>−1</sup>⋅mol<sup>−1</sup> |- |
[1.98720425864083] | cal⋅K<sup>−1</sup>⋅mol<sup>−1</sup> |- |
[8.20573660809596] | m<sup> 3 </sup>⋅atm⋅K<sup>−1</sup>⋅mol<sup>−1</sup> |- |}
The MOLAR GAS CONSTANT (also known as the GAS CONSTANT, UNIVERSAL GAS
CONSTANT, or IDEAL GAS CONSTANT) is denoted by the symbol [_R_] or [[_R_]].
It is the molar equivalent to the Boltzmann constant, expressed in units of
energy per temperature increment per amount of substance, rather than energy
per temperature increment per _particle_. The constant is also a combination
of the constants from Boyle's law, Charles's law, Avogadro's law, and
Gay-Lussac's law. It is a physical constant that is featured in many
fundamental equations in the physical sciences, such as the ideal gas law, the
Arrhenius equation, and the Nernst equation.
The gas constant is the constant of proportionality that relates the energy
scale in physics to the temperature scale and the scale used for amount 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 the Avogadro
constant were similarly determined, which separately relate energy to
temperature and particle count to amount of substance.
The gas constant _R_ is defined as the Avogadro constant _N_<sub>A</sub>
multiplied by the Boltzmann constant _k_ (or _k_<sub>B</sub>):
<math> R = N_\text{A} k </math>
= [NA] × [k]
= [8.31446261815324] .
Since the 2019 revision of the SI, both _N_<sub>A</sub> and _k_ are defined
with exact numerical values when expressed in SI units.[1] As a consequence,
the SI value of the molar gas constant is exact.
Some have suggested that it might be appropriate to name the symbol _R_ the
REGNAULT CONSTANT in honour of the French chemist Henri Victor Regnault, whose
accurate experimental data were used to calculate the early value of the
constant. However, the origin of the letter _R_ to represent the constant is
elusive. The universal gas constant was apparently introduced independently
by August Friedrich Horstmann (1873)[2][3] and Dmitri Mendeleev who reported
it first on 12 September 1874.[4] Using his extensive measurements of the
properties of gases,[5][6] Mendeleev also calculated it with high precision,
within 0.3% of its modern value.[7]
The gas constant occurs in the ideal gas law: <math display="block">PV = nRT =
m R_\text{specific} T,</math> where _P_ is the absolute pressure, _V_ is the
volume of gas, _n_ is the amount of substance, _m_ is the mass, and _T_ is the
thermodynamic temperature. _R_<sub>specific</sub> is the mass-specific gas
constant. The gas constant is expressed in the same unit as molar heat.
** Dimensions
From the ideal gas law _PV_ = _nRT_ we get
<math> R = \frac{PV}{nT}, </math>
where _P_ is pressure, _V_ is volume, _n_ is number of moles of a given
substance, and _T_ is temperature.
As pressure is defined as force per area of measurement, the gas equation can
also be written as
<math> R = \frac{ \dfrac{\text{force }} {\text{area }} \times \text{volume} }
{ \text{amount} \times \text{temperature} }.
</math>
Area and volume are (length)<sup>2</sup> and (length)<sup>3</sup>
respectively. Therefore:
<math> 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} }.
</math>
Since force × length = work,
<math> R = \frac{ \text{work} }
{ \text{amount} \times \text{temperature} }.
</math>
The physical significance of _R_ is work per mole per kelvin. It may be
expressed in any set of units representing work or energy (such as joules),
units representing temperature on an absolute scale (such as kelvin or
rankine), and any system of units designating a mole or a similar pure number
that allows an equation of macroscopic mass and fundamental particle numbers
in a system, such as an ideal gas (see Avogadro constant).
Instead of a mole the constant can be expressed by considering the normal
cubic metre.
Otherwise, we can also say that
<math> \text{force} = \frac{ \text{mass} \times \text{length} }
{ (\text{time})^2 }.
</math>
Therefore, we can write _R_ as
<math> R = \frac{ \text{mass} \times \text{length}^2 }
{ \text{amount} \times \text{temperature} \times (\text{time})^2 }.
</math>
And so, in terms of SI base units,
_R_ = [R] .
** Relationship with the Boltzmann constant
The Boltzmann constant _k_<sub>B</sub> (alternatively _k_) may be used in
place of the molar gas constant by working in pure particle count, _N_, rather
than amount of substance, _n_, since
<math> R = N_\text{A} k_\text{B}, </math>
where _N_<sub>A</sub> is the Avogadro constant. For example, the ideal gas
law in terms of the Boltzmann constant is
<math> pV = Nk_\text{B} T, </math>
where _N_ is the number of particles (molecules in this case), or to
generalize to an inhomogeneous system the local form holds:
<math> p = n k_\text{B} T, </math>
where _n_ = _N_/_V_ is the number density. Finally, by defining the kinetic
energy associated to the temperature,
<math> T := k_\text{B} T, </math>
the equation becomes simply
<math> p = n T, </math>
which is the form usually encountered in statistical mechanics and other
branches of theoretical physics.
** Measurement and replacement with defined value
As of 2006, the most precise measurement of _R_ had been obtained by measuring
the speed of sound _c_<sub>a</sub>(_P_, _T_) in argon at the temperature _T_
of the triple point of water at different pressures _P_, and extrapolating to
the zero-pressure limit _c_<sub>a</sub>(0, _T_). The value of _R_ is then
obtained from the relation
<math> c_\text{a}(0, T) = \sqrt{\frac{\gamma_0 R T}{A_\text{r}(\text{Ar}) M_\text{u }} }, </math>
where
_γ_ <sub> 0 </sub> is the heat capacity ratio (5/3 for monatomic gases such as argon);
_T_ is the temperature, _T_ <sub> TPW </sub> = 273.16 K by the definition of the kelvin at that time;
_A_ <sub> r </sub> (Ar) is the relative atomic mass of argon, and _M_ <sub> u </sub> = [e=-3] as defined at the time.
However, following the 2019 revision of the SI, _R_ now has an exact value
defined in terms of other exactly defined physical constants.
** Specific gas constant
{| class="wikitable" style="float: right;" ! _R_<sub>specific</sub><br />for
dry air[8] ! Unit |- | 287.052874 | J⋅kg<sup>−1</sup>⋅K<sup>−1</sup> |- |
53.3523 | ft⋅lbf⋅lb<sup>−1</sup>⋅°R<sup>−1</sup> |- | 1,716.46 |
ft⋅lbf⋅slug<sup>−1</sup>⋅°R<sup>−1</sup> |}
The SPECIFIC GAS CONSTANT of a gas or a mixture of gases
(_R_<sub>specific</sub>) is given by the molar gas constant divided by the
molar mass (_M_) of the gas or mixture:
<math> R_\text{specific} = \frac{R}{M}. </math>
Just as the molar gas constant can be related to the Boltzmann constant, so
can the specific gas constant by dividing the Boltzmann constant by the
molecular mass of the gas:
<math> R_\text{specific} = \frac{k_\text{B }} {m}. </math>
Another important relationship comes from thermodynamics. Mayer's relation
relates the specific gas constant to the specific heat capacities for a
calorically perfect gas and a thermally perfect gas:
<math> R_\text{specific} = c_p - c_V, </math>
where _c<sub>p</sub>_ is the specific heat capacity for a constant pressure,
and _c<sub>V</sub>_ is the specific heat capacity for a constant volume.[9]
It is common, especially in engineering applications, to represent the
specific gas constant by the symbol _R_. In such cases, the universal gas
constant is usually given a different symbol such as _[R]_ 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.[10]
In case of air, using the perfect gas law and the standard sea-level
conditions (SSL) (air density _ρ_<sub>0</sub> = 1.225 kg/m<sup>3</sup>,
temperature _T_<sub>0</sub> = 288.15 K and pressure _p_<sub>0</sub> =
[101325]), we have that _R_<sub>air</sub> =
_P_<sub>0</sub>/(_ρ_<sub>0</sub>_T_<sub>0</sub>) = [287.052874247]. Then the
molar mass of air is computed by _M_<sub>0</sub> = _R_/_R_<sub>air</sub> =
[28.964917].[11]
** U.S. Standard Atmosphere
The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant
_R_<sup>∗</sup> as[12][13]
_R_ <sup> ∗ </sup> = [8.31432] = [8.31432] .
Note the use of the kilomole, with the resulting factor of [1000] in the
constant. The USSA1976 acknowledges that this value is not consistent with
the cited values for the Avogadro constant and the Boltzmann constant.<ref
name="USSA1976"/> This disparity is not a significant departure from accuracy,
and USSA1976 uses this value of _R_<sup>∗</sup> for all the calculations of
the standard atmosphere. When using the ISO value of _R_, 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).
Also note that this was well before the 2019 SI redefinition, through which
the constant was given an exact value.
** References
<!--See Wikipedia:Footnotes for an explanation of how to generate footnotes
using the <ref(erences/)> tags--> [reflist]
** External links
- _Ideal gas calculator (see http://calculator.tutorvista.com/chemistry/567/ideal-gas-law-calculator.html) [url=https://web.archive.org/web/20120715222930/http://calculator.tutorvista.com/chemistry/567/ideal-gas-law-calculator.html ]_ – Ideal gas calculator provides the correct information for the moles of gas involved.
- Individual Gas Constants and the Universal Gas Constant (see http://www.engineeringtoolbox.com/individual-universal-gas-constant-d_588.html) – Engineering Toolbox
[Mole concepts]
[DEFAULTSORT:Gas Constant] Category:Ideal gas Category:Physical constants
Category:Amount of substance Category:Statistical mechanics
Category:Thermodynamics Category:Molar quantities