Ideal gas law

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Ideal gas law

The ideal gas law is the equation of state

Equation of state

In physics and thermodynamics, an equation of state is a relation between thermodynamic variables. More specifically, an equation of state is a thermodynamic equations describing the state of matter under a given set of physical conditions....
of a hypothetical ideal gas

Ideal gas

The ideal gas model is a model of matter in which the molecules are treated as non-interacting point particles which are engaged in a random motion that obeys conservation of energy....
, first stated by Benoît Paul Émile Clapeyron

Benoit Paul Émile Clapeyron

Beno?t Paul ?mile Clapeyron was a France engineer and physicist, one of the founders of thermodynamics....
in 1834. The law is derived from the fact that in the ideal state of any gas a given number of its "particles" occupy the same volume, and that volume changes are inverse to pressure changes and linear to temperature changes.

The state

State function

In thermodynamics, a state function, state quantity, or a function of state, is a physical quantity of a system that depends only on the current Thermodynamic state, not on the way in which the system got to that state....
of an amount of gas

Gas

In physics, a gas is a state of matter, consisting of a collection of particles without a definite shape or volume that are in more or less random motion....
is determined by its pressure, volume, and temperature according to the equation:

where is the absolute pressure

Pressure

Pressure is the force per unit area applied to an object in a direction surface normal to the surface. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure....
of the gas, is the volume

Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
of the gas, is the number of moles

Mole (unit)

The mole is a Units of measurement of amount of substance: it is an SI base unit, and one of the few units used to measure this physical quantity....
of gas, is the universal gas constant

Gas constant

The ideal gas law mathematically follows from a statistical mechanical

Statistical mechanics

Statistical mechanics is the application of probability theory, which includes Mathematics tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force....
treatment of primitive identical particles (point particles without internal structure) which do not interact, but exchange momentum

Momentum

In classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section Momentum#Modern definitions of momentum on this page....
(and hence kinetic energy

Kinetic energy

The kinetic energy of an object is the extra energy which it possesses due to its motion. It is defined as the mechanical work needed to accelerate a body of a given mass from rest to its current velocity....
) in elastic collision

Elastic collision

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Encyclopedia

The ideal gas law is the equation of state

Equation of state

Ideal gas

Benoit Paul Émile Clapeyron

State function

Gas

Pressure

Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
of the gas, is the number of moles

Mole (unit)

The mole is a Units of measurement of amount of substance: it is an SI base unit, and one of the few units used to measure this physical quantity....
of gas, is the universal gas constant

Gas constant

R	=	8.314472	J Joule The joule is the SI derived unit of energy in the International System of Units. It is defined as:One joule is the amount of energy required to perform the following actions:... ·mol Mole (unit) The mole is a Units of measurement of amount of substance: it is an SI base unit, and one of the few units used to measure this physical quantity.... ^-1·K Kelvin The kelvin is a Units of measurement of temperature and is one of the seven SI base units. The Kelvin scale is a Thermodynamic temperature scale where absolute zero, the theoretical absence of all thermal energy, is zero .... ^-1
=	8.314472	m³·Pa·K^-1·mol^-1
=	8.314472	kPa·L Litre The litre or liter is a unit of volume. There are two official symbols: the Latin letter L in lower and upper case . The lower case L is often written as a cursive l to avoid confusion with the number 1 in antiqua fonts.... ·mol Mole (unit) The mole is a Units of measurement of amount of substance: it is an SI base unit, and one of the few units used to measure this physical quantity.... ^-1·K^-1
=	0.08205746	L Litre The litre or liter is a unit of volume. There are two official symbols: the Latin letter L in lower and upper case . The lower case L is often written as a cursive l to avoid confusion with the number 1 in antiqua fonts.... ·atm Atmosphere (unit) The standard atmosphere is an international reference pressure defined as 101,325 Pascal and formerly used as unit of pressure . For practical purposes it has been replaced by the Bar which is 100,000 Pa.... ·K^-1·mol^-1
=	62.36367	L·mmHg·K^-1·mol^-1
=	10.73159	ft³·psi Pounds per square inch The pound per square inch or, more accurately, pound-force per square inch is a unit of pressure or of stress based on avoirdupois units.... ·°R^-1·lb-mol^-1
=	1545.3490	ft·lbf·°R^-1·lb-mol^-1 (for air)

The ideal gas law mathematically follows from a statistical mechanical

Statistical mechanics

Momentum

Kinetic energy

Elastic collision

An elastic collision is a collision in which the total kinetic energy of the colliding bodies after collision is equal to their total kinetic energy before collision....
s.

Since it neglects both molecular size and intermolecular attractions, the ideal gas law is most accurate for monoatomic gases at high temperatures and low pressures. The neglect of molecular size becomes less important for larger volumes, i.e., for lower pressures. The relative importance of intermolecular attractions diminishes with increasing thermal kinetic energy

Thermal energy

Thermal energy is a form of energy that manifests itself as an increase of temperature. It is also the sum of sensible heat and latent heat....
i.e., with increasing temperatures. More sophisticated equations of state
Equation of state

In physics and thermodynamics, an equation of state is a relation between thermodynamic variables. More specifically, an equation of state is a thermodynamic equations describing the state of matter under a given set of physical conditions....
, such as the van der Waals equation

Van der Waals equation

The 'van der Waals equation' is an equation of state for a fluid composed of particles that have a non-zero size and a pairwise attractive inter-particle force It was derived by Johannes Diderik van der Waals in 1873, based on a modification of the ideal gas law, who received the Nobel prize in 1910 for "his work on the equation of state for...
, allow deviations from ideality caused by molecular size and intermolecular forces to be taken into account.

Alternative Forms

As the amount of substance could be given in mass instead of moles, sometimes an alternative form of the ideal gas law is useful. The number of moles is equal to the mass divided by the molar mass

Molar mass

Molar mass, symbol M, is the mass of one mole of a substance . It is a physical property which is characteristic of each pure substance. The base SI unit for mass is the kilogram but, for both practical and historical reasons, molar masses are almost always quoted in grams per mole , especially in chemistry....
:

By replacing , we get:

from where

This form of the ideal gas law is very useful because it links pressure, density , and temperature in a unique formula independent from the quantity of the considered gas.

In statistical mechanics

Statistical mechanics

Statistical mechanics is the application of probability theory, which includes Mathematics tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force....
the following molecular equation is derived from first principles:

Here is Boltzmann's constant, and is the actual number of molecules, in contrast to the other formulation, which uses , the number of moles. This relation implies that , and the consistency of this result with experiment is a good check on the principles of statistical mechanics.

From here we can notice that for an average particle mass of times the atomic mass constant

Atomic mass constant

In physics and chemistry, the atomic mass constant, mu, is one twelfth of the mass of an unbound atom of the carbon-12 nuclide at rest and in its ground state....
(i.e., the mass is u

Atomic mass unit

The unified atomic mass unit , or dalton or, sometimes, universal mass unit, is a Units of measurement of mass used to express atomic weight and molecular masses....
) and since , we find that the ideal gas law can be rewritten as:

Calculations

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,T) at state 2 can be calculated from the properties at state 1 using the equations listed.

Process	Constant	Known ratio	P₂	V₂	T₂
Isobaric process Isobaric process An isobaric process is a thermodynamic process in which the pressure stays constant: The term derives from the Greek isos, "equal," and barus, "heavy." The heat transferred to the system does work but also changes the internal energy of the system:...	Pressure	V₂/V₁	P₂ = P₁	V₂ = V₁ (V₂/V₁)	T₂ = T₁ (V₂/V₁)
"	"	T₂/T₁	P₂ = P₁	V₂ = V₁ (T₂/T₁)	T₂ = T₁ (T₂/T₁)
Isochoric process Isochoric process An isochoric process, also called an isovolumetric process, is a process during which volume remains constant. The name is derived from the Greek isos, "equal", and khora, "place."...	Volume	P₂/P₁	P₂ = P₁ (P₂/P₁)	V₂ = V₁	T₂ = T₁ (P₂/P₁)
"	"	T₂/T₁	P₂ = P₁ (T₂/T₁)	V₂ = V₁	T₂ = T₁ (T₂/T₁)
Isothermal process Isothermal process An isothermal process is a thermodynamic process in which the temperature of the system stays constant: ΔT = 0. This typically occurs when a system is in contact with an outside thermal reservoir , and the change occurs slowly enough to allow the system to continually adjust to the temperature of the reservoir through heat exchange....	Temperature	P₂/P₁	P₂ = P₁ (P₂/P₁)	V₂ = V₁ / (P₂/P₁)	T₂ = T₁
"	"	V₂/V₁	P₂ = P₁ / (V₂/V₁)	V₂ = V₁ (V₂/V₁)	T₂ = T₁
Isentropic process Isentropic process In thermodynamics, an isentropic process or isoentropic process is one during which the entropy of the system remains constant. It can be proved that any Reversible process adiabatic process is an isentropic process.... (Reversible adiabatic process Adiabatic process In thermodynamics, an adiabatic process or an isocaloric process is a thermodynamic process in which no heat is transferred to or from the working fluid.... )	Entropy Entropy In many branches of science, entropy is a measure of the disorder of a system. The concept of entropy is particularly notable as it is applied across physics, information theory and mathematics....	P₂/P₁	P₂ = P₁ (P₂/P₁)	V₂ = V₁ (P₂/P₁)^-₁/	T₂ = T₁ (P₂/P₁)^(-₁)/
"	"	V₂/V₁	P₂ = P₁ (V₂/V₁)^-	V₂ = V₁ (V₂/V₁)	T₂ = T₁ (V₂/V₁)^₁-
"	"	T₂/T₁	P₂ = P₁ (T₂/T₁)^/(-₁)	V₂ = V₁ (T₂/T₁)^1/(1-)	T₂ = T₁ (T₂/T₁)

a. In an isentropic process, system entropy (Q) is constant. Under these conditions, P₁ V₁ = P₂ V₂, where is defined as the heat capacity ratio

Heat capacity ratio

The heat capacity ratio or adiabatic index or ratio of specific heats, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume ....
, which is constant for an ideal gas. The value used for is typically 1.4 for diatomic gases like nitrogen

Nitrogen

Nitrogen is a chemical element that has the symbol N and atomic number 7 and atomic mass 14.00674?. Elemental nitrogen is a colorless, odorless, tasteless and mostly inert diatomic gas at standard conditions, constituting 78% by volume of Earth's atmosphere....
(N₂) and oxygen

Oxygen

Oxygen no O2 produced; 2) O2 produced, but absorbed in oceans & seabed rock; 3) O2 starts to gas out of the oceans, but is absorbed by land surfaces and formation of ozone layer; 4-5) O2 sinks filled and the gas accumulates]]...
(O₂), (and air, which is 99% diatomic). Also is typically 1.6 for monatomic gases like the noble gas

Noble gas

|}The noble gases are a group of chemical elements with very similar properties: under standard conditions, they are all odorless, colorless, monatomic gases, with a very low chemical reactivity....
es helium

Helium

Helium is a colorless, odorless, tasteless, non-toxic, inert monatomic chemical element that heads the noble gas group in the periodic table and whose atomic number is 2....
(He), and argon

Argon

Argon is a chemical element designated by the symbol Ar. Argon has atomic number 18 and is the third element in group 18 of the periodic table ....
(Ar). In internal combustion engines varies between 1.35 and 1.15, depending on constitution gases and temperature.

Derivations

Empirical

The ideal gas law can be derived from combining two empirical gas laws

Gas laws

The gas laws are a set of empirical laws that describe the relationship between thermodynamic temperature , absolute pressure and volume of gases....
: the combined gas law

Combined gas law

The combined gas law is a gas law which combines Charles's law, Boyle's law, and Gay-Lussac's law. These laws each relate one Conjugate_variables_%28thermodynamics%29 to another mathematically while holding everything else constant....
and Avogadro's law

Avogadro's law

Avogadro's law is a gas law named after Amedeo Avogadro who, in 1811, hypothesized that:Thus, the number of molecules in a specific volume of gas is independent of the size or mass of the gas molecules....
. The combined gas law states that

where C is a constant which is directly proportional to the amount of gas, n (Avogadro's law

Avogadro's law

Avogadro's law is a gas law named after Amedeo Avogadro who, in 1811, hypothesized that:Thus, the number of molecules in a specific volume of gas is independent of the size or mass of the gas molecules....
). The proportionality factor is the universal gas constant, R, i.e. .

Hence the ideal gas law

Theoretical

The ideal gas law can also be derived from first principles

First principles

In philosophy, a first principle is a basic, foundational proposition or assumption that cannot be deduced from any other proposition or assumption....
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.

Derivation from the 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, and let F denote the net force on that particle, then

where the first equality is Newton's second law, and the second line uses Hamilton's equations and the equipartition theorem

Equipartition theorem

In classical physics statistical mechanics, the equipartition theorem is a general formula that relates the temperature of a system with its average energy....
. Summing over a system of N particles yields

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

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

Divergence

In vector calculus, the divergence is an operator that measures the magnitude of a vector field's source or sink at a given point; the divergence of a vector field is a scalar....
of the position vector q is

the divergence theorem

Divergence theorem

In vector calculus, the divergence theorem, also known as Gauss?s theorem , Ostrogradsky?s theorem , or Gauss-Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface....
implies that

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

Putting these equalities together yields

which immediately implies the ideal gas law

Ideal gas law

The ideal gas law is the equation of state of a hypothetical ideal gas, first stated by Beno?t Paul ?mile Clapeyron in 1834. The law is derived from the fact that in the ideal state of any gas a given number of its "particles" occupy the same volume, and that volume changes are inverse to pressure changes and linear to temperature changes....
for N particles:

where n=N/N_A is the number of moles

Mole (unit)

The mole is a Units of measurement of amount of substance: it is an SI base unit, and one of the few units used to measure this physical quantity....
of gas and R=N_Ak_B is the gas constant

Gas constant

The gas constant is a physical constant which is featured in a large number of fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation....
.

The readers are referred to the comprehensive article where an alternative statistical mechanics derivation of the ideal-gas law, using the relationship between the Helmholtz free energy

Helmholtz free energy

In thermodynamics, the Helmholtz free energy is a thermodynamic potential which measures the ?useful? work obtainable from a closed system thermodynamic thermodynamic system at a constant temperature and volume....
and the partition function

Partition function (statistical mechanics)

In statistical mechanics, the partition function Z is an important quantity that encodes the statistics properties of a system in thermodynamic equilibrium....
, but without using the equipartition theorem, is provided.

Ideal gas law

Alternative Forms

Calculations

Derivations

Empirical

Theoretical

Derivation from the statistical mechanics

See also