Gas constant

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The gas constant (also known as the molar, universal, or ideal gas constant, denoted by the symbol R or R) is a physical constant that is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation.

It is equivalent to the Boltzmann constant, but expressed in units of energy (i.e. the pressure-volume product) per temperature increment per mole (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.

Physically, the gas constant is the constant of proportionality that happens to relate the energy scale in physics to the temperature scale, when a mole of particles at the stated temperature is being considered. Thus, the value of the gas constant ultimately derives from historical decisions and accidents in the setting of the energy and temperature scales, plus similar historical setting of the value of the molar scale used for the counting of particles. The last factor is not a consideration in the value of the Boltzmann constant, which does a similar job of equating linear energy and temperature scales.

The gas constant value is

8.3144598(48) J?mol^-1?K^-1

The two digits in parentheses are the uncertainty (standard deviation) in the last two digits of the value. The relative uncertainty is 5.7×10^-7. 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 exact reason for the original representation of the constant by the letter R is elusive.

The gas constant occurs in the ideal gas law, as follows:

${\displaystyle PV=nRT=mR_{\rm {specific}}T\,\!}$

where P is the absolute pressure (SI unit pascals), V is the volume of gas (SI unit cubic metres), n is the amount of gas (SI unit moles), m is the mass (SI unit kilograms) contained in V, and T is the thermodynamic temperature (SI unit kelvins). The gas constant is expressed in the same physical units as molar entropy and molar heat capacity.

Video Gas constant

Dimensions of R

From the general equation PV = nRT we get:

${\displaystyle R={\frac {PV}{nT}}}$

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 unit area, the gas equation can also be written as:

${\displaystyle R={\frac {{\dfrac {\mathrm {force} }{\mathrm {area} }}\times \mathrm {volume} }{\mathrm {amount} \times \mathrm {temperature} }}}$

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

${\displaystyle R={\frac {{\dfrac {\mathrm {force} }{(\mathrm {length} )^{2}}}\times (\mathrm {length} )^{3}}{\mathrm {amount} \times \mathrm {temperature} }}={\frac {\mathrm {force} \times \mathrm {length} }{\mathrm {amount} \times \mathrm {temperature} }}}$

Since force × length = work:

${\displaystyle R={\frac {\mathrm {work} }{\mathrm {amount} \times \mathrm {temperature} }}}$

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 joules), other units representing degrees of temperature (such as degrees Celsius or Fahrenheit), 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's number).

Instead of a mole the constant can be expressed by considering the normal cubic meter.

Otherwise, we can also say that:

${\displaystyle \mathrm {force} ={\frac {\mathrm {mass} \times \mathrm {length} }{(\mathrm {time} )^{2}}}}$

Therefore, we can write "R" as:

${\displaystyle R={\frac {\mathrm {mass} \times \mathrm {length} ^{2}}{\mathrm {amount} \times \mathrm {temperature} \times (\mathrm {time} )^{2}}}}$

And so, in SI base units:

R = 8.3144598(48) kg m² mol^-1 K^-1 s^-2.

Maps Gas constant

Relationship with the Boltzmann constant

The Boltzmann constant k_B (often abbreviated k) may be used in place of the gas constant by working in pure particle count, N, rather than amount of substance, n, since

${\displaystyle R=N_{\rm {A}}k_{\rm {B}},\,}$

where N_A is the Avogadro constant. For example, the ideal gas law in terms of Boltzmann's constant is

${\displaystyle PV=k_{\rm {B}}NT.\,\!}$

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

${\displaystyle P=k_{\rm {B}}nT.\,\!}$

where n is the number density.

${\displaystyle {\textrm {Boltzmann's}}\ {\textrm {constant}}={\frac {{\textrm {Planck}}\ {\textrm {mass}}\times ({\textrm {Planck}}\ {\textrm {length}})^{2}}{{\textrm {Planck}}\ {\textrm {temperature}}\times ({\textrm {Planck}}\ {\textrm {time}})^{2}}}}$

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Measurement

As of 2006, the most precise measurement of R is obtained by measuring the speed of sound c_a(p, T) in argon at the temperature T of the triple point of water (used to define the kelvin) at different pressures p, and extrapolating to the zero-pressure limit c_a(0, T). The value of R is then obtained from the relation

${\displaystyle c_{\mathrm {a} }(0,T)={\sqrt {\frac {\gamma _{0}RT}{A_{\mathrm {r} }(\mathrm {Ar} )M_{\mathrm {u} }}}},}$

where:

• ?₀ is the heat capacity ratio ( ⁵/₃ for monatomic gases such as argon);
• T is the temperature, T_TPW = 273.16 K by definition of the kelvin;
• A_r(Ar) is the relative atomic mass of argon and M_u = 10^-3 kg mol^-1.

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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 (M) of the gas or mixture.

${\displaystyle R_{\rm {specific}}={\frac {R}{M}}}$

Just as the ideal 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.

${\displaystyle R_{\rm {specific}}={\frac {k_{\rm {B}}}{m}}}$

Another important relationship comes from thermodynamics. Mayer's relation relates the specific gas constant to the specific heats for a calorically perfect gas and a thermally perfect gas.

${\displaystyle R_{\rm {specific}}=c_{\rm {p}}-c_{\rm {v}}\ }$

where c_p is the specific heat for a constant pressure and c_v is the specific heat 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 R to distinguish it. In any case, the context and/or units of the gas constant should make it clear as to whether the universal or specific gas constant is being referred to.

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U.S. Standard Atmosphere

The U.S. Standard Atmosphere, 1976 (USSA1976) defines the gas constant R* as:

R* = 8.31432×10³ N m kmol^-1 K^-1.

Note the use of kilomole units resulting in the factor of 1,000 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 at 11 kilometers (the equivalent of a difference of only 17.4 centimeters or 6.8 inches) and an increase of 0.292 Pa at 20 km (the equivalent of a difference of only 0.338 m or 13.2 in).

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References

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External links

• Ideal gas calculator - Ideal gas calculator provides the correct information for the moles of gas involved.
• Individual Gas Constants and the Universal Gas Constant -- Engineering Toolbox

Source of article : Wikipedia