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The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
, expressed in units of
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
per temperature increment per
amount of substance In chemistry, the amount of substance ''n'' in a given sample of matter is defined as the quantity or number of discrete atomic-scale particles in it divided by the Avogadro constant ''N''A. The particles or entities may be molecules, atoms, io ...
, i.e. the pressure–volume product, rather than energy per temperature increment per ''particle''. The constant is also a combination of the constants from Boyle's law,
Charles's law Charles's law (also known as the law of volumes) is an experimental gas law that describes how gases tend to expand when heated. A modern statement of Charles's law is: When the pressure on a sample of a dry gas is held constant, the Kelvin t ...
,
Avogadro's law Avogadro's law (sometimes referred to as Avogadro's hypothesis or Avogadro's principle) or Avogadro-Ampère's hypothesis is an experimental gas law relating the volume of a gas to the amount of substance of gas present. The law is a specific ca ...
, and Gay-Lussac's law. It is a
physical constant A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant ...
that is featured in many fundamental equations in the physical sciences, such as the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first s ...
, the
Arrhenius equation In physical chemistry, the Arrhenius equation is a formula for the temperature dependence of reaction rates. The equation was proposed by Svante Arrhenius in 1889, based on the work of Dutch chemist Jacobus Henricus van 't Hoff who had noted in ...
, and the
Nernst equation In electrochemistry, the Nernst equation is a chemical thermodynamical relationship that permits the calculation of the reduction potential of a reaction ( half-cell or full cell reaction) from the standard electrode potential, absolute tempe ...
. The gas constant is the
constant of proportionality In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality const ...
that relates the energy scale in physics to the temperature scale and the scale used for
amount of substance In chemistry, the amount of substance ''n'' in a given sample of matter is defined as the quantity or number of discrete atomic-scale particles in it divided by the Avogadro constant ''N''A. The particles or entities may be molecules, atoms, io ...
. 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 The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining ...
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''A multiplied by the Boltzmann constant ''k'' (or ''k''B): R = N_ k. Since the 2019 redefinition of SI base units, both ''N''A and ''k'' are defined with exact numerical values when expressed in SI units. As a consequence, the SI value of the molar gas constant is exactly . Some have suggested that it might be appropriate to name the symbol ''R'' the Regnault constant in honour of the
French French (french: français(e), link=no) may refer to: * Something of, from, or related to France ** French language, which originated in France, and its various dialects and accents ** French people, a nation and ethnic group identified with Franc ...
chemist A chemist (from Greek ''chēm(ía)'' alchemy; replacing ''chymist'' from Medieval Latin ''alchemist'') is a scientist trained in the study of chemistry. Chemists study the composition of matter and its properties. Chemists carefully describe th ...
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 Clausius’ student, A.F. Horstmann (1873) and
Dmitri Mendeleev Dmitri Ivanovich Mendeleev (sometimes transliterated as Mendeleyev or Mendeleef) ( ; russian: links=no, Дмитрий Иванович Менделеев, tr. , ; 8 February Old_Style_and_New_Style_dates">O.S._27_January.html" ;"title="O ...
who reported it first on Sep. 12, 1874. Using his extensive measurements of the properties of gases, Mendeleev also calculated it with high precision, within 0.3% of its modern value. The gas constant occurs in the ideal gas law: pV = nRT = m R_ T where ''P'' is the absolute
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
, ''V'' is the volume of gas, ''n'' is the
amount of substance In chemistry, the amount of substance ''n'' in a given sample of matter is defined as the quantity or number of discrete atomic-scale particles in it divided by the Avogadro constant ''N''A. The particles or entities may be molecules, atoms, io ...
, ''m'' is the
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
, and ''T'' is the
thermodynamic temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic ...
. ''R''specific is the mass-specific gas constant. The gas constant is expressed in the same unit as are molar
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
and
molar heat Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K). Heat capa ...
.


Dimensions

From the ideal gas law ''PV'' = ''nRT'' we get: :R = \frac where ''P'' is pressure, ''V'' is volume, ''n'' is number of moles of a given substance, and ''T'' is
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
. As pressure is defined as force per area of measurement, the gas equation can also be written as: :R = \frac Area and volume are (length)2 and (length)3 respectively. Therefore: :R = \frac = \frac Since force × length = work: :R = \frac The physical significance of ''R'' is work per degree per mole. It may be expressed in any set of units representing work or energy (such as
joule The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force appli ...
s), units representing degrees of temperature on an absolute scale (such as
Kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and ...
or
Rankine Rankine is a surname. Notable people with the surname include: * William Rankine (1820–1872), Scottish engineer and physicist ** Rankine body an elliptical shape of significance in fluid dynamics, named for Rankine ** Rankine scale, an absolute-t ...
), 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 The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining ...
''). Instead of a mole the constant can be expressed by considering the normal cubic meter. Otherwise, we can also say that: :\mathrm = \frac Therefore, we can write ''R'' as: :R = \frac And so, in terms of SI base units: :''R'' = .


Relationship with the Boltzmann constant

The
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
''k''B (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 :R = N_ k_,\, where ''N''A is the
Avogadro constant The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining ...
. For example, the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first s ...
in terms of the Boltzmann constant is :PV = Nk_ T, where ''N'' is the number of particles (molecules in this case), or to generalize to an inhomogeneous system the local form holds: :P = \rho_ k_ T, where ''ρ''N = ''N''/''V'' is the
number density The number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric num ...
.


Measurement and replacement with defined value

As of 2006, the most precise measurement of ''R'' had been obtained by measuring the
speed of sound The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends strongly on temperature as we ...
 ''c''a(''P'', ''T'') in
argon Argon is a chemical element with the symbol Ar and atomic number 18. It is in group 18 of the periodic table and is a noble gas. Argon is the third-most abundant gas in Earth's atmosphere, at 0.934% (9340 ppmv). It is more than twice a ...
at the temperature ''T'' of the triple point of water at different
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
s ''P'', and extrapolating to the zero-pressure limit ''c''a(0, ''T''). The value of ''R'' is then obtained from the relation :c_\mathrm(0, T) = \sqrt, where: *''γ''0 is the heat capacity ratio ( for monatomic gases such as argon); *''T'' is the temperature, ''T''TPW = 273.16 K by the definition of the kelvin at that time; *''A''r(Ar) is the relative atomic mass of argon and ''M''u =  as defined at the time. However, following the 2019 redefinition of the SI base units, ''R'' now has an exact value defined in terms of other exactly defined physical constants.


Specific gas constant

The specific gas constant of a gas or a mixture of gases (''R''specific) is given by the molar gas constant divided by the
molar mass In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, ...
(''M'') of the gas or mixture. : R_ = \frac 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. : R_ = \frac 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. : R_ = c_ - c_\ where ''c''p is the
specific heat capacity In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
for a constant pressure and ''c''v is the specific heat capacity for a constant volume. 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 ' 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. In case of air, using the perfect gas law and the
standard sea-level conditions Standard sea-level conditions (SSL), also known as sea-level standard (SLS), defines a set of atmospheric conditions for physical calculations. The term "standard sea level" is used to indicate that values of properties are to be taken to be the sa ...
(SSL) (air density ''ρ''0 = 1.225 kg/m3, temperature ''T''0 = 288.15  K and pressure ''p''0 = ), we have that ''R''air = ''P''0/(''ρ''0''T''0) = . Then the molar mass of air is computed by ''M''0 = ''R''/''R''air = .


U.S. Standard Atmosphere

The
U.S. Standard Atmosphere The U.S. Standard Atmosphere is a static atmospheric model of how the pressure, temperature, density, and viscosity of the Earth's atmosphere change over a wide range of altitudes or elevations. The model, based on an existing international stand ...
, 1976 (USSA1976) defines the gas constant ''R'' as: Part 1, p. 3, (Linked file is 17 Meg) :''R'' = = . Note the use of kilomoles, with the resulting factor of in the constant. The USSA1976 acknowledges that this value is not consistent with the cited values for the Avogadro constant and the Boltzmann constant. This disparity is not a significant departure from accuracy, and USSA1976 uses this value of ''R'' for all the calculations of the standard atmosphere. When using the ISO value of ''R'', the calculated pressure increases by only 0.62 
pascal Pascal, Pascal's or PASCAL may refer to: People and fictional characters * Pascal (given name), including a list of people with the name * Pascal (surname), including a list of people and fictional characters with the name ** Blaise Pascal, Frenc ...
at 11 kilometers (the equivalent of a difference of only 17.4 centimeters 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


External links

*
Ideal gas calculator
'' – Ideal gas calculator provides the correct information for the moles of gas involved.

– Engineering Toolbox {{DEFAULTSORT:Gas Constant Ideal gas Physical constants Amount of substance Statistical mechanics Thermodynamics