Thermodynamics
The classical Carnot heat engine
Branches Classical · Statistical · Chemical Equilibrium / Non-equilibrium
Laws Zeroth · First · Second · Third
Systems State: Equation of state Ideal gas · Real gas Phase of matter · Equilibrium Control volume · Instruments Processes: Isobaric · Isochoric · Isothermal Adiabatic · Isentropic · Isenthalpic Quasistatic · Polytropic Free expansion Reversibility · Irreversibility Endoreversibility Cycles: Heat engines · Heat pumps Thermal efficiency
System properties Property diagrams Intensive and extensive properties Functions of state: Temperature / Entropy (intro.) † Pressure / Volume † Chemical potential / Particle no. † († Conjugate variables) Vapor quality Reduced properties Process functions: Work · Heat
Material properties Specific heat capacity  Compressibility  Thermal expansion  Property database
Equations Carnot’s theorem · Clausius theorem · Fundamental relation · Ideal gas law · Maxwell relations · Onsager reciprocal relations Table of thermodynamic equations
Potentials Free energy · Free entropy Internal energy Enthalpy Helmholtz free energy Gibbs free energy
History and culture Philosophy: Entropy and time · Entropy and life Brownian ratchet Maxwell’s demon Heat death paradox Loschmidt’s paradox Synergetics History: General · Heat · Entropy · Gas laws Perpetual motion Theories: Caloric theory · Vis viva Theory of heat Mechanical equivalent of heat Motive power Publications: “An Experimental Enquiry Concerning … Heat“ “On the Equilibrium of Heterogeneous Substances“ <a “Reflections on the Motive Power of Fire” Timelines of: Thermodynamics · Heat engines Art: Maxwell’s thermodynamic surface Education: Entropy as energy dispersal
Scientists Bernoulli · Carnot · Clapeyron · Clausius · von Helmholtz · Carathéodory · Pierre Duhem · Gibbs · Joule · Maxwell · von Mayer · Onsager · Rankine · Smeaton · Stahl · Thompson · Kelvin · Waterson
v t e

It has been suggested that Combined gas law be merged into this article or section. (<a Discuss) Proposed since August 2012.

Isotherms of an ideal gas. The curved lines represent the relationship between pressure (on the vertical, y-axis) and volume (on the horizontal, x-axis) for an ideal gas at different temperatures: lines which are further away from the origin (that is, lines that are nearer to the top right-hand corner of the diagram) represent higher temperatures.

The ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behaviour of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle’s law and Charles’s law.^[1] It can also be derived from kinetic theory, as was achieved (apparently independently) by August Krönig in 1856^[2] and Rudolf Clausius in 1857.^[3]

[edit] Equation

The state of an amount of gas is determined by its pressure, volume, and temperature. The modern form of the equation relates these simply in two main forms. The temperature used in the equation of state is an absolute temperature: in the SI system of units, kelvin; in the Imperial system, degrees Rankine.^[4]

[edit] Common form

The most frequently introduced form is

where P is the pressure of the gas, V is the volume of the gas, n is the amount of substance of gas (also known as number of moles), T is the temperature of the gas and R is the ideal, or universal, gas constant, equal to the product of Boltzmann’s constant and Avogadro’s constant.

In SI units, P is measured in pascals, V is measured in cubic metres, n is measured in moles, and T in kelvin (273.15 Kelvin = 0.01 degrees Celsius). R has the value 8.314 J·K⁻¹·mol⁻¹ or 0.08206 L·atm·mol⁻¹·K⁻¹ if using pressure in standard atmospheres (atm) instead of pascals, and volume in litres instead of cubic metres.

[edit] Molar form

How much gas is present could be specified by giving the mass instead of the chemical amount of gas. Therefore, an alternative form of the ideal gas law may be useful. The chemical amount (n) (in moles) is equal to the mass (m) (in grams) divided by the molar mass (M) (in grams per mole):

By replacing n with m / M, and subsequently introducing density ρ = m/V, we get:

Defining the specific gas constant R_specific as the ratio R/M,

This form of the ideal gas law is very useful because it links pressure, density, and temperature in a unique formula independent of the quantity of the considered gas. Alternatively, the law may be written in terms of the specific volume v, the reciprocal of density, as

It is common, especially in engineering applications, to represent the ideal 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.^[5]

[edit] Statistical mechanics

In statistical mechanics the following molecular equation is derived from first principles:

where P is the absolute pressure of the gas measured in Pascals; V is the volume (in this equation the volume is expressed in meters cubed, as pascal times cubic meter equals one Joule); N is the number of particles in the gas; k is the Boltzmann constant relating temperature and energy; and T is the absolute temperature.

The actual number of molecules contrasts to the other formulation, which uses n, the number of moles. This relation implies that Nk = nR, and the consistency of this result with experiment is a good check on the principles of statistical mechanics.

From this we can notice that for an average particle mass of μ times the atomic mass constant m_u (i.e., the mass is μ u)

and since ρ = m/V, we find that the ideal gas law can be rewritten as:

In SI units, P is measured in pascals; V in cubic metres; N is a dimensionless number; and T in kelvin. k has the value 1.38·10⁻²³ J·K⁻¹ in SI units.

[edit] Applications to thermodynamic processes

The table below essentially simplifies the ideal gas equation for a particular processes, thus making this equation easier to solve using numerical methods.

A thermodynamic process is defined as a system that moves from state 1 to state 2, where the state number is denoted by subscript. As shown in the first column of the table, basic thermodynamic processes are defined such that one of the gas properties (P, V, T, or S) is constant throughout the process.

For a given thermodynamics process, in order to specify the extent of a particular process, one of the properties ratios (listed under the column labeled “known ratio”) must be specified (either directly or indirectly). Also, the property for which the ratio is known must be distinct from the property held constant in the previous column (otherwise the ratio would be unity, and not enough information would be available to simplify the gas law equation).

In the final three columns, the properties (P, V, or T) at state 2 can be calculated from the properties at state 1 using the equations listed.

^{^} a. In an isentropic process, system entropy (S) is constant. Under these conditions, P₁ V₁^γ = P₂ V₂^γ, where γ is defined as the heat capacity ratio, which is constant for an ideal gas. The value used for γ is typically 1.4 for diatomic gases like nitrogen (N₂) and oxygen (O₂), (and air, which is 99% diatomic). Also γ is typically 1.6 for monatomic gases like the noble gases helium (He), and argon (Ar). In internal combustion engines γ varies between 1.35 and 1.15, depending on constitution gases and temperature.

[edit] Deviations from ideal behavior of real gases

The equation of state given here applies only to an ideal gas, or as an approximation to a real gas that behaves sufficiently like an ideal gas. There are in fact many different forms of the equation of state. Since the ideal gas law neglects both molecular size and intermolecular attractions, it is most accurate for monatomic gases at high temperatures and low pressures. The neglect of molecular size becomes less important for lower densities, i.e. for larger volumes at lower pressures, because the average distance between adjacent molecules becomes much larger than the molecular size. The relative importance of intermolecular attractions diminishes with increasing thermal kinetic energy, i.e., with increasing temperatures. More detailed equations of state, such as the van der Waals equation, account for deviations from ideality caused by molecular size and intermolecular forces.

A residual property is defined as the difference between a real gas property and an ideal gas property, both considered at the same pressure, temperature, and composition.

[edit] Derivations

[edit] Empirical

The ideal gas law can be derived from combining two empirical gas laws: the combined gas law and Avogadro’s law. The combined gas law states that

where C is a constant which is directly proportional to the amount of gas, n (Avogadro’s law). The proportionality factor is the universal gas constant, R, i.e. C = nR.

Hence the ideal gas law

[edit] Theoretical

[edit] Kinetic theory

The ideal gas law can also be derived from first principles using the kinetic theory of gases, in which several simplifying assumptions are made, chief among which are that the molecules, or atoms, of the gas are point masses, possessing mass but no significant volume, and undergo only elastic collisions with each other and the sides of the container in which both linear momentum and kinetic energy are conserved.

[edit] Statistical mechanics

Let q = (q_x, q_y, q_z) and p = (p_x, p_y, p_z) denote the position vector and momentum vector of a particle of an ideal gas, respectively. Let F denote the net force on that particle. Then the time average momentum of the particle is:

<img class="tex" alt="
begin{align}
langle mathbf{q} cdot mathbf{F} rangle &= Bigllangle q_{x} frac{dp_{x}}{dt} Bigrrangle +
Bigllangle q_{y} frac{dp_{y}}{dt} Bigrrangle +
Bigllangle q_{z} frac{dp_{z}}{dt} Bigrrangle
&=-Bigllangle q_{x} frac{partial H}{partial q_x} Bigrrangle –
Bigllangle q_{y} frac{partial H}{partial q_y} Bigrrangle –
Bigllangle q_{z} frac{partial H}{partial q_z} Bigrrangle = -3k_{B} T,
end{align}
” src=”//upload.wikimedia.org/math/2/5/e/25efc0af4fd1c897c8bacb9351df5657.png” />

where the first equality is Newton’s second law, and the second line uses Hamilton’s equations and the equipartition theorem. Summing over a system of N particles yields

<img class="tex" alt="
3Nk_{B} T = – biggllangle sum_{k=1}^{N} mathbf{q}_{k} cdot mathbf{F}_{k} biggrrangle.
” src=”//upload.wikimedia.org/math/4/1/6/4169575599ec3ec20b0994adee417376.png” />

By Newton’s third law and the ideal gas assumption, the net force on the system is the force applied by the walls of their container, and this force is given by the pressure P of the gas. Hence

<img class="tex" alt="
-biggllanglesum_{k=1}^{N} mathbf{q}_{k} cdot mathbf{F}_{k}biggrrangle = P oint_{mathrm{surface}} mathbf{q} cdot dmathbf{S},
” src=”//upload.wikimedia.org/math/0/7/f/07fb904fed61b41defecd94e977b1d68.png” />

where dS is the infinitesimal area element along the walls of the container. Since the divergence of the position vector q is

<img class="tex" alt="
nabla cdot mathbf{q} =
frac{partial q_{x}}{partial q_{x}} +
frac{partial q_{y}}{partial q_{y}} +
frac{partial q_{z}}{partial q_{z}} = 3,
” src=”//upload.wikimedia.org/math/e/c/a/ecae37731ebde0b9d901a3199e371da0.png” />

the divergence theorem implies that

<img class="tex" alt="P oint_{mathrm{surface}} mathbf{q} cdot dmathbf{S} = P int_{mathrm{volume}} left( nabla cdot mathbf{q} right) dV = 3PV,
” src=”//upload.wikimedia.org/math/d/f/8/df851ef4b5ba5a528f999d2e21100ef9.png” />

where dV is an infinitesimal volume within the container and V is the total volume of the container.

Putting these equalities together yields

<img class="tex" alt="
3Nk_{B} T = -biggllangle sum_{k=1}^{N} mathbf{q}_{k} cdot mathbf{F}_{k} biggrrangle = 3PV,
” src=”//upload.wikimedia.org/math/d/a/d/dad82d23e90e1f3a8983d63290557d70.png” />

which immediately implies the ideal gas law for N particles:

<img class="tex" alt="
PV = Nk_{B} T = nRT,,
” src=”//upload.wikimedia.org/math/f/a/0/fa0704457f45293e1dc3c0f95cd9f369.png” />

where n = N/N_A is the number of moles of gas and R = N_Ak_B is the gas constant.

[edit] See also

• Combined gas law
• Van der Waals equation
• Boltzmann’s constant
• Configuration integral
• Dynamic pressure

[edit] References

[edit] Further reading

• Davis and Masten Principles of Environmental Engineering and Science, McGraw-Hill Companies, Inc. New York (2002) ISBN 0-07-235053-9
• Website giving credit to Benoît Paul Émile Clapeyron, (1799–1864) in 1834

[edit] External links

• Ideal Gas Law Calculator
• Configuration integral (statistical mechanics) where an alternative statistical mechanics derivation of the ideal-gas law, using the relationship between the Helmholtz free energy and the partition function, but without using the equipartition theorem, is provided.

v t e Diving medicine, physiology, physics and environment Diving medicine: Injuries and Disorders Pressure Oxygen Deep water blackout Hyperoxia Oxygen toxicity Shallow water blackout Inert gases Atrial septal defect Avascular necrosis Decompression sickness Dysbaric osteonecrosis High-pressure nervous syndrome Hydrogen narcosis Isobaric counterdiffusion Nitrogen narcosis Taravana Uncontrolled decompression Carbon dioxide Hypercapnia Aerosinusitis Air embolism Alternobaric vertigo Barodontalgia Barostriction Barotrauma Decompression illness Dental barotrauma Dysbarism Ear clearing Frenzel maneuver Valsalva maneuver Immersion Asphyxia Drowning Hypothermia Hypoxia (medical) Immersion diuresis Instinctive drowning response Salt water aspiration syndrome Swimming-induced pulmonary edema List of signs and symptoms of diving disorders Seasickness Cramps Treatments Diving medicine Hyperbaric medicine In-water recompression Oxygen therapy Diving chamber Diving physiology Blood–air barrier CO₂ retention Cold shock response Blood shift Decompression (diving) Gas exchange Lipid Mammalian diving reflex Metabolism Normocapnia Oxygen window in technical diving Perfusion Pulmonary circulation Respiratory exchange ratio Respiratory quotient Systemic circulation Tissue (biology) Diving physics Ambient pressure Anti fog Amontons’ law Archimedes’ principle Atmospheric pressure Boyle’s law Breathing performance of regulators Buoyancy Charles’ law Combined gas law Dalton’s law Diffusion Diving physics Force Gay-Lussac’s law Henry’s law Hydrophobe Hydrostatic pressure Ideal gas law Molecular diffusion Neutral buoyancy Partial pressure Permeation Pressure Psychrometric constant Solution Solubility Supersaturation Surfactant Surface tension Torricellian chamber Underwater vision Weight Diving environment Algal bloom Breaking wave Current (ocean) Current (stream) Ekman transport Halocline Longshore current Rip current Stratification Surf Surge (wave action) Thermocline Tides Turbidity Undertow (wave action) Upwelling Researchers in diving medicine, physiology and physics Arthur J. Bachrach Albert R. Behnke Paul Bert George F. Bond Sir Robert Boyle Albert A. Bühlmann William Paul Fife John Scott Haldane Leonard Erskine Hill Felix Hoppe-Seyler Christian J. Lambertsen Simon Mitchell Charles Momsen John Rawlins (Royal Navy officer) Charles Wesley Shilling Jules Triger Edward D. Thalmann Related Portal:Underwater diving Artificial gills (human) Aerospace Medical Association Divers Alert Network Diving Diseases Research Centre List of diving hazards and precautions National Board of Diving and Hyperbaric Medical Technology Naval Submarine Medical Research Laboratory Royal Australian Navy School of Underwater Medicine Rubicon Foundation South Pacific Underwater Medicine Society Undersea and Hyperbaric Medical Society United States Navy Experimental Diving Unit Glossary of underwater diving terminology

<!–
NewPP limit report
Preprocessor visited node count: 7326/1000000
Preprocessor generated node count: 53553/1500000
Post-expand include size: 182362/2048000 bytes
Template argument size: 102878/2048000 bytes
Highest expansion depth: 21/40
Expensive parser function count: 3/500
–>

This article uses material from the Wikipedia article Ideal Gas Law, which is released under the Creative Commons Attribution-Share-Alike License 3.0.