Gas constant
The gas constant, also known as the molar gas constant or universal gas constant and denoted by the symbol R, is a fundamental physical constant in the equation of state for an ideal gas, expressed as PV = nRT, where P is the pressure, V is the volume, n is the amount of substance (in moles), and T is the absolute temperature (in kelvin).[1] This law, first formulated in its modern form by Benoît Paul Émile Clapeyron in 1834, describes the behavior of gases under conditions where intermolecular forces are negligible.[2] In the International System of Units (SI), the value of R is exactly 8.314462618 J⋅mol⁻¹⋅K⁻¹, a precise figure established following the 2019 revision of the SI base units, which fixed the numerical values of the Boltzmann constant (k) and Avogadro's constant (N_A).[3][4] The gas constant is defined as the product R = N_A k, linking macroscopic thermodynamic properties to microscopic statistical mechanics, where k relates the average kinetic energy of particles to temperature.[5] The constant appears in various forms depending on the units employed; for example, in units involving atmospheres and liters, R = 0.082057 L⋅atm⋅mol⁻¹⋅K⁻¹, while in calories, it is approximately 1.987 cal⋅mol⁻¹⋅K⁻¹.[6] For real gases, a specific gas constant (R_s) is often used, defined as R_s = R / M, where M is the molar mass of the gas, enabling applications in engineering contexts like fluid dynamics and heat transfer.[7] Beyond the ideal gas law, R features prominently in equations for entropy changes, chemical equilibria, and the Gibbs free energy, underscoring its role as a bridge between classical and statistical thermodynamics.[1]Definition and Role in Thermodynamics
Ideal Gas Law
The ideal gas law describes the behavior of an ideal gas under various conditions of pressure, volume, temperature, and amount of substance. It is mathematically expressed as PV = nRT, where P represents the pressure of the gas, V is the volume it occupies, n is the number of moles of the gas, T is the absolute temperature, and R is the universal gas constant that relates these macroscopic properties.[8] This equation assumes that the gas particles have negligible volume and do not interact except through elastic collisions, providing a foundational model for thermodynamic calculations.[8] The law originated empirically from combining earlier observations, including Boyle's law (relating pressure and volume at constant temperature) and Charles's law (relating volume and temperature at constant pressure), which Benoît Paul Émile Clapeyron unified in 1834 in his memoir "Mémoire sur la Puissance Motrice de la Chaleur."[9] Clapeyron formulated it as pv = R(267 + t), where t is the temperature in degrees Celsius and 267 approximates the conversion to an absolute scale, marking the first explicit use of the gas constant R.[9] From a theoretical perspective, the ideal gas law emerges from the kinetic theory of gases, which models gas as a collection of particles in random motion. In this framework, pressure arises from the momentum transfer during collisions of molecules with container walls, leading to P = \frac{1}{3} \rho v_{\text{rms}}^2, where \rho is density and v_{\text{rms}} is the root-mean-square speed.[10] Linking this microscopic view to macroscopic observables via the equipartition theorem—assigning \frac{1}{2} kT of kinetic energy per degree of freedom per molecule—yields the ideal gas law, with R appearing as the proportionality constant that scales the total thermal energy across moles to connect molecular kinetics with bulk properties.[10] An equivalent form of the equation, useful in statistical mechanics, is PV = NkT, where N is the total number of molecules and k is the Boltzmann constant, highlighting the per-molecule perspective while R = N_A k (with N_A as Avogadro's number) bridges the molar and molecular scales.[11]Physical Significance
The gas constant R serves as a fundamental proportionality factor in thermodynamics, quantifying the energy scale associated with thermal motion in ideal gases. Specifically, it represents the amount of energy needed to increase the temperature of one mole of an ideal gas by one kelvin under constant volume conditions, where the molar heat capacity at constant volume C_V for a monatomic ideal gas equals \frac{3}{2} R. This relationship arises from the equipartition theorem, wherein each quadratic degree of freedom contributes \frac{1}{2} R per mole to C_V, underscoring R's role in linking macroscopic thermal energy to molecular kinetics.[12][13] In expressions for entropy and free energy, R provides the scaling for entropic contributions tied to volume and temperature changes in ideal gases. For instance, the Sackur-Tetrode equation for the entropy S of a monatomic ideal gas per mole is S = R \left[ \ln \left( \frac{V}{N} \left( \frac{4 \pi m U}{3 N h^2} \right)^{3/2} \right) + \frac{5}{2} \right], where terms involving \ln V and \ln T (since U = \frac{3}{2} n R T) explicitly incorporate R to normalize the logarithmic contributions on a molar basis. This highlights R's function in making entropy an extensive property, additive for multiple moles, and universal across ideal gases regardless of their molecular identity.[14] The universal applicability of R stems from its derivation under ideal conditions, where intermolecular forces and molecular volumes are negligible, allowing the same constant to describe diverse gases like helium or nitrogen. However, real gases exhibit deviations from this ideality, particularly at high pressures or low temperatures, as captured by corrections in the van der Waals equation \left( P + \frac{a n^2}{V^2} \right) (V - n b) = n R T, where a and b account for attractive forces and excluded volume, respectively, without altering R itself. These deviations emphasize R's idealized nature while affirming its foundational role in thermodynamic scaling.[15] Within the thermodynamic identity dU = T dS - P dV, R emerges implicitly in the integrated forms for ideal gases, such as the relation P V = n [R](/page/R) T derived from combining this identity with the first law, thereby connecting internal energy changes solely to temperature via dU = n C_V d[T](/page/Temperature). This integration reveals R as the bridge between mechanical work, heat, and thermal disorder, enabling predictions of state functions like Gibbs free energy G = H - T S, where entropic terms scale with R.[16]Value and Units
SI Value and Definition
The universal gas constant, denoted as R, has an exact defined value in the International System of Units (SI) of R = 8.314462618 \, \mathrm{J \cdot mol^{-1} \cdot K^{-1}}.[3] This value carries no uncertainty, as it is derived directly from the fixed numerical values of fundamental constants established in the SI.[3] The 2019 revision of the SI, effective from May 20, 2019, fixed R by defining the Boltzmann constant k exactly as k = 1.380649 \times 10^{-23} \, \mathrm{J \cdot K^{-1}} and the Avogadro constant N_A exactly as N_A = 6.02214076 \times 10^{23} \, \mathrm{mol^{-1}}, such that R = N_A k. This redefinition, recommended by the Committee on Data for Science and Technology (CODATA) in 2018, ties the kelvin and mole base units to these constants, ensuring R is invariant and precisely known without reliance on experimental measurement. Before the 2019 redefinition, the CODATA 2018 recommended value for R was $8.314462618(21) \, \mathrm{J \cdot mol^{-1} \cdot K^{-1}}, corresponding to a relative standard uncertainty of $2.5 \times 10^{-8}.[17] This uncertainty arose from experimental determinations involving acoustic thermometry and other methods to measure k and N_A. The elimination of uncertainty in R post-redefinition improves the precision of thermodynamic calculations, particularly in calorimetry, where R relates molar heat capacities to temperature differences, and in precise determinations of standard enthalpies of reaction, reducing error propagation in high-accuracy measurements.Values in Other Unit Systems
The exact value of the gas constant in the International System of Units (SI) is R = 8.314462618 \, \mathrm{J \cdot mol^{-1} \cdot K^{-1}}, which serves as the basis for deriving its numerical values in other unit systems through standard unit conversions.[3] In fields like chemistry and biochemistry, the calorie (cal) unit system remains prevalent due to its historical origins in 19th-century calorimetry experiments measuring heat effects in chemical reactions and nutritional energy, where the international steam table calorie was standardized as exactly 4.1868 J, though the thermochemical calorie of exactly 4.184 J is often used for precise thermodynamic calculations. The corresponding value is R = 1.9872036 \, \mathrm{cal \cdot mol^{-1} \cdot K^{-1}} using the thermochemical definition.[7]/Physical_Properties_of_Matter/States_of_Matter/Properties_of_Gases/Gases_(Waterloo)/The_Ideal_Gas_Law) For volumetric measurements in laboratory chemistry, the liter-bar system is common, as the bar provides a convenient pressure scale close to atmospheric conditions (1 bar = 10^5 Pa exactly), and the liter (1 L = 10^{-3} m^3 exactly) aligns with standard glassware volumes; here, R = 0.08314462618 \, \mathrm{L \cdot [bar](/page/Bar) \cdot [mol](/page/Mol)^{-[1](/page/1)} \cdot K^{-[1](/page/1)}}.[7] In American engineering contexts, particularly for process design and HVAC systems, English units incorporate feet cubed for volume, pounds-mass moles for substance amount, and Rankine for temperature (where 1 °R = 5/9 K interval), with atmosphere or pounds per square inch absolute (psia) for pressure; the atmosphere-foot cubed system yields R = 0.730240 \, \mathrm{atm \cdot ft^3 \cdot lb\text{-}[mol](/page/Mol)^{-[1](/page/1)} \cdot ^\circ R^{-[1](/page/1)}}, while the psia variant is R = 10.73159 \, \mathrm{psia \cdot ft^3 \cdot lb\text{-}[mol](/page/Mol)^{-[1](/page/1)} \cdot ^\circ R^{-[1](/page/1)}}, and the British thermal unit (Btu, defined as approximately 1055.06 J) version is R = 1.985877 \, \mathrm{Btu \cdot lb\text{-}[mol](/page/Mol)^{-[1](/page/1)} \cdot ^\circ R^{-[1](/page/1)}}.[7][18] The following table summarizes values in these and one additional common system (liter-atmosphere, used in gas stoichiometry where 1 atm = 101325 Pa exactly, giving R = 0.08205746 \, \mathrm{L \cdot atm \cdot mol^{-1} \cdot K^{-1}}) for quick reference, derived from the SI value:| Unit System | Value of R | Typical Field of Use |
|---|---|---|
| cal · mol⁻¹ · K⁻¹ | 1.9872036 | Chemistry, biochemistry |
| L · bar · mol⁻¹ · K⁻¹ | 0.08314462618 | Laboratory chemistry |
| L · atm · mol⁻¹ · K⁻¹ | 0.08205746 | Gas stoichiometry |
| atm · ft³ · lb-mol⁻¹ · °R⁻¹ | 0.730240 | Chemical engineering |
| psia · ft³ · lb-mol⁻¹ · °R⁻¹ | 10.73159 | Process engineering |
| Btu · lb-mol⁻¹ · °R⁻¹ | 1.985877 | HVAC, energy systems |
Dimensions and Fundamental Relations
Dimensional Formula
The dimensional formula of the gas constant R expresses its fundamental physical dimensions in terms of the base quantities: mass M, length L, time T, amount of substance N, and thermodynamic temperature \Theta. It is given by[R] = M L^{2} T^{-2} N^{-1} \Theta^{-1}.
This formulation equivalently represents R as having dimensions of energy per mole per unit temperature, underscoring its role as a proportionality factor bridging thermal and mechanical properties of gases.[19] To derive this dimensional formula, consider the ideal gas law PV = nRT, where P is pressure with dimensions [P] = M L^{-1} T^{-2}, V is volume with [V] = L^{3}, n is the amount of substance with = N, and T is temperature with [T] = \Theta. Rearranging for R yields [R] = \frac{[P][V]}{[T]} = \frac{(M L^{-1} T^{-2})(L^{3})}{N \Theta} = M L^{2} T^{-2} N^{-1} \Theta^{-1}, confirming the expression through direct substitution of the base dimensions.[20] In dimensional analysis, the gas constant facilitates the construction of dimensionless groups vital for scaling laws in thermodynamics and gas dynamics, such as in predicting prototype behavior from model tests or deriving similarity criteria for compressible flows. For example, when analyzing the speed of sound in an ideal gas, which depends on temperature T and specific gas constant R, dimensional analysis reveals a dimensionless form involving \sqrt{\gamma R T} (where \gamma is the heat capacity ratio), enabling universal scaling across different conditions. Additionally, the product RT possesses dimensions of energy per mole (M L^{2} T^{-2} N^{-1}), aligning R with energy scales and allowing consistent unit conversions in thermodynamic equations without altering physical relationships.
Connection to Boltzmann Constant
The gas constant R is directly related to the Boltzmann constant k through the equation R = N_A k, where N_A is the Avogadro constant.[4] With the 2019 redefinition of the SI units, N_A is exactly $6.02214076 \times 10^{23} \, \mathrm{mol}^{-1} and k is exactly $1.380649 \times 10^{-23} \, \mathrm{J \cdot K^{-1}}, yielding the exact value R = 8.314462618 \, \mathrm{J \cdot mol^{-1} \cdot K^{-1}}.[21][22] This relation bridges macroscopic thermodynamic quantities to microscopic particle behavior in statistical mechanics. Physically, k represents the energy scale per particle per unit temperature, while R scales this to molar quantities by accounting for the number of particles in one mole of substance. In statistical mechanics, the ideal gas law emerges as PV = NkT, where N is the total number of particles and T is temperature; substituting n = N / N_A (with n as the number of moles) yields the thermodynamic form PV = nRT.[23] This derivation underscores how R facilitates the transition from per-particle descriptions to bulk molar properties. In statistical mechanics, the connection manifests in principles like the equipartition theorem, which assigns an average energy of \frac{1}{2} kT per quadratic degree of freedom per molecule. For a mole of monatomic ideal gas with three translational degrees of freedom, this scales to \frac{3}{2} RT per mole, linking thermal energy distribution across ensembles of particles to observable thermodynamic energies./18%3A_Partition_Functions_and_Ideal_Gases/18.11%3A_The_Equipartition_Principle) The 2019 SI redefinition fixed exact values for both k and N_A, making R exactly defined without reliance on experimental measurements of interdependent quantities like the kilogram or kelvin.[22] This eliminates propagation of uncertainties in prior determinations, enhancing precision in applications spanning thermodynamics and statistical mechanics.[21]Specific Gas Constants
Definition and Calculation
The specific gas constant R_s, also known as the individual or mass-specific gas constant, is defined for a particular gas or gas mixture as the ratio of the universal gas constant R to the molar mass M of the substance, expressed as R_s = \frac{R}{M}.[24] This formulation yields R_s in units of energy per unit mass per unit temperature, such as J/(kg·K) when M is in kg/mol and R is in J/(mol·K).[25] With this definition, the ideal gas law can be rewritten in a mass-based form as P = \rho R_s T, where P is pressure, \rho is mass density, and T is absolute temperature; this adaptation is particularly useful when working with density measurements rather than molar quantities.[26] To compute R_s, one divides R by M; for instance, dry air has a molar mass M \approx 0.02896 kg/mol, resulting in R_s \approx 287 J·kg⁻¹·K⁻¹ when using R = 8.314 J/(mol·K).[27][25] Care must be taken with units: if M is given in g/mol, R_s will be in J/(g·K), requiring conversion (e.g., multiplication by 1000) for consistency with SI mass-specific formulations.[25] Compared to the universal R, the specific R_s offers advantages in engineering contexts by directly incorporating mass properties, thereby simplifying equations for fluids where density \rho is a primary variable, as in analyses of compressible flow.[28] For ideal gases, R_s also connects to the mass-specific heat capacities at constant pressure c_p and constant volume c_v via the relation R_s = c_p - c_v, which underpins derivations in isentropic processes.[29] In such processes, the heat capacity ratio \gamma = \frac{c_p}{c_v} further characterizes the gas behavior, with R_s enabling efficient computation of properties like speed of sound or expansion factors.[29]Examples for Common Gases
The specific gas constant R_s varies for different gases depending on their molar mass, with lighter gases exhibiting higher values. This section presents values for several common gases used in scientific and engineering applications, calculated as R_s = R / M, where R is the universal gas constant and M is the molar mass in kg/mol.[7] The following table lists R_s in SI units (J⋅kg⁻¹⋅K⁻¹) for selected gases, based on standard molar masses and the universal gas constant R = 8.314462618 J⋅mol⁻¹⋅K⁻¹.[7][3]| Gas | Formula | Molar Mass (kg/mol) | R_s (J⋅kg⁻¹⋅K⁻¹) |
|---|---|---|---|
| Hydrogen | H₂ | 0.002016 | 4124 |
| Helium | He | 0.004003 | 2077 |
| Air | - | 0.028965 | 287 |
| Oxygen | O₂ | 0.031999 | 259.8 |
| Nitrogen | N₂ | 0.028013 | 296.8 |
| Carbon Dioxide | CO₂ | 0.04401 | 188.9 |
| Water Vapor | H₂O | 0.018015 | 461.5 |