Gas constant

Last updated
Value of R [1] Unit
SI units
8.31446261815324 JK −1mol −1
8.31446261815324 m3PaK −1mol −1
8.31446261815324 kgm 2s −2K −1mol −1
Other common units
8314.46261815324 LPaK −1mol −1
8.31446261815324 LkPaK −1mol −1
0.0831446261815324 LbarK −1mol −1
8.31446261815324×107 ergK −1mol −1
0.730240507295273 atmft 3lbmol −1°R −1
10.731577089016 psift 3lbmol −1°R −1
1.985875279009 BTUlbmol −1°R −1
297.031214 inH2Oft 3lbmol −1°R −1
554.984319180 torrft 3lbmol −1°R −1
0.082057366080960 LatmK −1mol −1
62.363598221529 LTorrK −1mol −1
1.98720425864083... calK −1mol −1
8.20573660809596...×10−5 m3atmK −1mol −1

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.

Contents

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 NA multiplied by the Boltzmann constant k (or kB):

Since the 2019 redefinition of SI base units, both NA and k are defined with exact numerical values when expressed in SI units. [2] As a consequence, the SI value of the molar gas constant is exactly 8.31446261815324 J⋅K−1⋅mol−1.

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 Clausius' student, A.F. Horstmann (1873) [3] [4] and Dmitri Mendeleev who reported it first on Sep. 12, 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:

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. Rspecific is the mass-specific gas constant. The gas constant is expressed in the same unit as are molar entropy and molar heat.

Dimensions

From the ideal gas law PV = nRT we get:

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:

Area and volume are (length)2 and (length)3 respectively. Therefore:

Since force × length = work:

The physical significance of R is work per mole per degree. It may be expressed in any set of units representing work or energy (such as joules), units representing degrees of 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 meter.

Otherwise, we can also say that:

Therefore, we can write R as:

And so, in terms of SI base units:

R = 8.314462618... kg⋅m2⋅s−2⋅K−1⋅mol−1.

Relationship with the Boltzmann constant

The Boltzmann constant kB (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:

where NA is the Avogadro constant. For example, the ideal gas law in terms of the Boltzmann constant is:

where N is the number of particles (molecules in this case), or to generalize to an inhomogeneous system the local form holds:

where ρN = N/V is the number density.

Measurement and replacement with defined value

As of 2006, the most precise measurement of R had been obtained by measuring the speed of sound  ca(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 ca(0, T). The value of R is then obtained from the relation:

where:

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

Rspecific
for dry air
Unit
287.052874J⋅kg−1⋅K−1
53.3523ft⋅lbflb −1⋅°R−1
1,716.46ft⋅lbfslug −1⋅°R−1
Based on a mean molar mass
for dry air of 28.964917 g/mol.

The specific gas constant of a gas or a mixture of gases (Rspecific) is given by the molar gas constant divided by the molar mass (M) of the gas or mixture:

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:

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:

where cp is the specific heat capacity for a constant pressure and cv 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 ρ0 = 1.225 kg/m3, temperature T0 = 288.15  K and pressure p0 = 101325  Pa ), we have that Rair = P0/(ρ0T0) = 287.052874247 J·kg−1·K−1. Then the molar mass of air is computed by M0 = R/Rair = 28.964917 g/mol. [11]

U.S. Standard Atmosphere

The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R as: [12] [13]

R = 8.31432×103 N⋅m⋅kmol−1⋅K−1 = 8.31432 J⋅K−1⋅mol−1.

Note the use of kilomoles, 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. [13] 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 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.

Related Research Articles

<span class="mw-page-title-main">Specific heat capacity</span> Heat required to increase temperature of a given unit of mass of a substance

In thermodynamics, the specific heat capacity of a substance is the amount of heat that must be added to one unit of mass of the substance in order to cause an increase of one unit in temperature. It is also referred to as massic heat capacity or as the specific heat. More formally it is the heat capacity of a sample of the substance divided by the mass of the sample. The SI unit of specific heat capacity is joule per kelvin per kilogram, J⋅kg−1⋅K−1. For example, the heat required to raise the temperature of 1 kg of water by 1 K is 4184 joules, so the specific heat capacity of water is 4184 J⋅kg−1⋅K−1.

The laws describing the behaviour of gases under fixed pressure, volume and absolute temperature conditions are called Gas Laws. The basic gas laws were discovered by the end of the 18th century when scientists found out that relationships between pressure, volume and temperature of a sample of gas could be obtained which would hold to approximation for all gases. These macroscopic gas laws were found to be consistent with atomic and kinetic theory.

<span class="mw-page-title-main">Stefan–Boltzmann law</span> Physical law on the emissive power of black body

The Stefan–Boltzmann law, also known as Stefan's law, describes the intensity of the thermal radiation emitted by matter in terms of that matter's temperature. It is named for Josef Stefan, who empirically derived the relationship, and Ludwig Boltzmann who derived the law theoretically.

<span class="mw-page-title-main">Boltzmann constant</span> Physical constant relating particle kinetic energy with temperature

The Boltzmann constant is the proportionality factor that relates the average relative thermal energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, and in Planck's law of black-body radiation and Boltzmann's entropy formula, and is used in calculating thermal noise in resistors. The Boltzmann constant has dimensions of energy divided by temperature, the same as entropy. It is named after the Austrian scientist Ludwig Boltzmann.

<span class="mw-page-title-main">Ideal gas law</span> Equation of the state of a hypothetical ideal gas

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 stated by Benoît Paul Émile Clapeyron in 1834 as a combination of the empirical Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. The ideal gas law is often written in an empirical form:

<span class="mw-page-title-main">Ideal gas</span> Mathematical model which approximates the behavior of real gases

An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics. The requirement of zero interaction can often be relaxed if, for example, the interaction is perfectly elastic or regarded as point-like collisions.

In chemistry and related fields, the molar volume, symbol Vm, or of a substance is the ratio of the volume occupied by a substance to the amount of substance, usually given at a given temperature and pressure. It is equal to the molar mass (M) divided by the mass density (ρ):

<span class="mw-page-title-main">Van der Waals equation</span> Gas equation of state which accounts for non-ideal gas behavior

The van der Waals equation, named for its originator, the Dutch physicist Johannes Diderik van der Waals, is an equation of state that extends the ideal gas law to include the non-zero size of gas molecules and the interactions between them. As a result the equation is able to model the phase change from liquid to gas, and vice versa. It also produces simple analytic expressions for the properties of real substances that shed light on their behavior. One common way to write this dimensional equation is:

<span class="mw-page-title-main">Effusion</span> Process of a gas escaping through a small hole

In physics and chemistry, effusion is the process in which a gas escapes from a container through a hole of diameter considerably smaller than the mean free path of the molecules. Such a hole is often described as a pinhole and the escape of the gas is due to the pressure difference between the container and the exterior. Under these conditions, essentially all molecules which arrive at the hole continue and pass through the hole, since collisions between molecules in the region of the hole are negligible. Conversely, when the diameter is larger than the mean free path of the gas, flow obeys the Sampson flow law.

In physical chemistry, Henry's law is a gas law that states that the amount of dissolved gas in a liquid is directly proportional to its partial pressure above the liquid. The proportionality factor is called Henry's law constant. It was formulated by the English chemist William Henry, who studied the topic in the early 19th century.

Avogadro's law 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 case of the ideal gas law. A modern statement is:

Avogadro's law states that "equal volumes of all gases, at the same temperature and pressure, have the same number of molecules."

For a given mass of an ideal gas, the volume and amount (moles) of the gas are directly proportional if the temperature and pressure are constant.

The Knudsen number (Kn) is a dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length scale. This length scale could be, for example, the radius of a body in a fluid. The number is named after Danish physicist Martin Knudsen (1871–1949).

<span class="mw-page-title-main">Isobaric process</span> Thermodynamic process in which pressure remains constant

In thermodynamics, an isobaric process is a type of thermodynamic process in which the pressure of the system stays constant: ΔP = 0. The heat transferred to the system does work, but also changes the internal energy (U) of the system. This article uses the physics sign convention for work, where positive work is work done by the system. Using this convention, by the first law of thermodynamics,

The density of air or atmospheric density, denoted ρ, is the mass per unit volume of Earth's atmosphere. Air density, like air pressure, decreases with increasing altitude. It also changes with variations in atmospheric pressure, temperature and humidity. At 101.325 kPa (abs) and 20 °C, air has a density of approximately 1.204 kg/m3 (0.0752 lb/cu ft), according to the International Standard Atmosphere (ISA). At 101.325 kPa (abs) and 15 °C (59 °F), air has a density of approximately 1.225 kg/m3 (0.0765 lb/cu ft), which is about 1800 that of water, according to the International Standard Atmosphere (ISA). Pure liquid water is 1,000 kg/m3 (62 lb/cu ft).

The Sackur–Tetrode equation is an expression for the entropy of a monatomic ideal gas.

<span class="mw-page-title-main">Heat capacity ratio</span> Thermodynamic quantity

In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor and is denoted by γ (gamma) for an ideal gas or κ (kappa), the isentropic exponent for a real gas. The symbol γ is used by aerospace and chemical engineers.

The Loschmidt constant or Loschmidt's number (symbol: n0) is the number of particles (atoms or molecules) of an ideal gas per volume (the number density), and usually quoted at standard temperature and pressure. The 2014 CODATA recommended value is 2.6867811(15)×1025 per cubic metre at 0 °C and 1 atm and the 2006 CODATA recommended value was 2.686 7774(47)×1025 per cubic metre at 0 °C and 1 atm. It is named after the Austrian physicist Johann Josef Loschmidt, who was the first to estimate the physical size of molecules in 1865. The term "Loschmidt constant" is also sometimes used to refer to the Avogadro constant, particularly in German texts.

<span class="mw-page-title-main">Volume (thermodynamics)</span> Extensive parameter used to describe a thermodynamic systems state

In thermodynamics, the volume of a system is an important extensive parameter for describing its thermodynamic state. The specific volume, an intensive property, is the system's volume per unit mass. Volume is a function of state and is interdependent with other thermodynamic properties such as pressure and temperature. For example, volume is related to the pressure and temperature of an ideal gas by the ideal gas law. The physical region covered by a system may or may not coincide with a control volume used to analyze the system.

<span class="mw-page-title-main">Dulong–Petit law</span> Empirical thermodynamic law

The Dulong–Petit law, a thermodynamic law proposed by French physicists Pierre Louis Dulong and Alexis Thérèse Petit, states that the classical expression for the molar specific heat capacity of certain chemical elements is constant for temperatures far from the absolute zero.

<span class="mw-page-title-main">Entropy production</span> Development of entropy in a thermodynamic system

Entropy production is the amount of entropy which is produced during heat process to evaluate the efficiency of the process.

References

  1. "2018 CODATA Value: molar gas constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
  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 Constant R". 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 letter R?" (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) Lock-green.svg
  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 . Lock-green.svg
  9. Anderson, Hypersonic and High-Temperature Gas Dynamics, AIAA Education Series, 2nd Ed, 2006
  10. Moran, Michael J.; Shapiro, Howard N.; Boettner, Daisie D.; Bailey, Margaret B. (2018). Fundamentals of Engineering Thermodynamics (9th ed.). Hoboken, New Jersey: Wiley.
  11. Manual of the US Standard Atmosphere (PDF) (3 ed.). National Aeronautics and Space Administration. 1962. pp. 7–11.
  12. "Standard Atmospheres" . Retrieved 2007-01-07.
  13. 1 2 NOAA, NASA, USAF (1976). U.S. Standard Atmosphere, 1976 (PDF). U.S. Government Printing Office, Washington, D.C. NOAA-S/T 76-1562.{{cite book}}: CS1 maint: multiple names: authors list (link) Part 1, p. 3, (Linked file is 17 Meg)