ideal gas law

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The ideal gas law, also called the general gas equation, is the
equation of state In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or internal ...
of a hypothetical
ideal gas 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 a ...
. It is a good approximation of the behavior of many
gas Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma). A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
es under many conditions, although it has several limitations. It was first stated by
Benoît Paul Émile Clapeyron Benoît Paul Émile Clapeyron (; 26 January 1799 – 28 January 1864) was a French engineer and physicist, one of the founders of thermodynamics. Life Born in Paris, Clapeyron studied at the École polytechnique, graduating in 1818. Milton Kerker ...
in 1834 as a combination of the empirical
Boyle's law Boyle's law, also referred to as the Boyle–Mariotte law, or Mariotte's law (especially in France), is an experimental gas law that describes the relationship between pressure and volume of a confined gas. Boyle's law has been stated as: The ...
,
Charles's law Charles's law (also known as the law of volumes) is an experimental gas law that describes how gases tend to expand when heated. A modern statement of Charles's law is: When the pressure on a sample of a dry gas is held constant, the Kelvin t ...
,
Avogadro's law Avogadro's law (sometimes referred to as Avogadro's hypothesis or Avogadro's principle) 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 c ...
, and
Gay-Lussac's law Gay-Lussac's law usually refers to Joseph-Louis Gay-Lussac's law of combining volumes of gases, discovered in 1808 and published in 1809. It sometimes refers to the proportionality of the volume of a gas to its absolute temperature at constant pr ...
. The ideal gas law is often written in an empirical form: $pV = nRT$ where $p$, $V$ and $T$ are the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
,
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
and
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
; $n$ is the
amount of substance In chemistry, the amount of substance ''n'' in a given sample of matter is defined as the quantity or number of discrete atomic-scale particles in it divided by the Avogadro constant ''N''A. The particles or entities may be molecules, atoms, ions, ...
; and $R$ is the ideal gas constant. It can also be derived from the microscopic
kinetic theory Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory, describing a gas as particles in random motion * Kinetic energy, the energy of an object that it possesses due to its motion Art and ente ...
, as was achieved (apparently independently) by
August Krönig August Karl Krönig (; 20 September 1822 – 5 June 1879) was a German chemist and physicist who published an account of the kinetic theory of gases in 1856, probably after reading a paper by John James Waterston. Biography Krönig was born i ...
in 1856 and
Rudolf Clausius Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German physicist and mathematician and is considered one of the central founding fathers of the science of thermodynamics. By his restatement of Sadi Carnot's principle ...
in 1857.

# Equation

The
state State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States * ''Our S ...
of an amount of
gas Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma). A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
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: the appropriate
SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
is the
kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...
.

## Common forms

The most frequently introduced forms are:$pV = nRT = n k_\text N_\text T = N k_\text T$where: * $p$ is the absolute
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
of the gas, * $V$ is the
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
of the gas, * $n$ is the
amount of substance In chemistry, the amount of substance ''n'' in a given sample of matter is defined as the quantity or number of discrete atomic-scale particles in it divided by the Avogadro constant ''N''A. The particles or entities may be molecules, atoms, ions, ...
of gas (also known as number of moles), * $R$ is the ideal, or universal,
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
, equal to the product of the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic 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 the
Avogadro constant The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining con ...
, * $k_\text$ is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic 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, ...
, * ''$N_$'' is the
Avogadro constant The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining con ...
, * $T$ is the
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...
of the gas, * $N$ is the number of particles (usually atoms or molecules) of the gas. In
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
, ''p'' is measured in
pascals The pascal (symbol: Pa) is the unit of pressure in the International System of Units (SI), and is also used to quantify internal pressure, stress, Young's modulus, and ultimate tensile strength. The unit, named after Blaise Pascal, is defined ...
, ''V'' is measured in cubic metres, ''n'' is measured in
moles Moles can refer to: * Moles de Xert, a mountain range in the Baix Maestrat comarca, Valencian Community, Spain * The Moles (Australian band) *The Moles, alter ego of Scottish band Simon Dupree and the Big Sound People *Abraham Moles, French engin ...
, and ''T'' in
kelvins The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phy ...
(the
Kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...
scale is a shifted
Celsius scale The degree Celsius is the unit of temperature on the Celsius scale (originally known as the centigrade scale outside Sweden), one of two temperature scales used in the International System of Units (SI), the other being the Kelvin scale. The d ...
, where 0.00 K = −273.15 °C, the lowest possible temperature). ''R'' has for value 8.314 J/( mol· K) = 1.989 ≈ 2 cal/(mol·K), or 0.0821 L⋅ atm/(mol⋅K).

## 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 total mass of the gas (''m'') (in kilograms) divided by the
molar mass In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, p ...
, ''M'' (in kilograms per mole): : $n = \frac.$ By replacing ''n'' with ''m''/''M'' and subsequently introducing
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
''ρ'' = ''m''/''V'', we get: : $pV = \frac RT$ : $p = \frac \frac$ : $p = \rho \frac T$ Defining the
specific gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
''R''specific(''r'') as the ratio ''R''/''M'', : $p = \rho R_\textT$ 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 In thermodynamics, the specific volume of a substance (symbol: , nu) is an intrinsic property of the substance, defined as the ratio of the substance's volume () to its mass (). It is the reciprocal of density (rho) and it is related to the m ...
''v'', the reciprocal of density, as : $pv = R_\textT.$ It is common, especially in engineering and meteorological 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 $\bar R$ or $R^*$ 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 used.

## Statistical mechanics

In
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
the following molecular equation is derived from first principles : $P = nk_\textT,$ where is the absolute
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
of the gas, is the
number density The number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric number ...
of the molecules (given by the ratio , in contrast to the previous formulation in which is the ''number of moles''), is the
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...
, and is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic 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, ...
relating temperature and energy, given by: : $k_\text = \frac$ where is the
Avogadro constant The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining con ...
. From this we notice that for a gas of mass , with an average particle mass of times the atomic mass constant, , (i.e., the mass is u) the number of molecules will be given by : $N = \frac,$ and since , we find that the ideal gas law can be rewritten as : $P = \frac\frac k_\text T = \frac \rho T.$ In SI units, is measured in
pascals The pascal (symbol: Pa) is the unit of pressure in the International System of Units (SI), and is also used to quantify internal pressure, stress, Young's modulus, and ultimate tensile strength. The unit, named after Blaise Pascal, is defined ...
, in cubic metres, in kelvins, and in
SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
s.

## Combined gas law

Combining the laws of Charles, Boyle and Gay-Lussac gives the combined gas law, which takes the same functional form as the ideal gas law says that the number of moles is unspecified, and the ratio of $PV$ to $T$ is simply taken as a constant: :$\frac=k,$ where $P$ is the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
of the gas, $V$ is the
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
of the gas, $T$ is the
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...
of the gas, and $k$ is a constant. When comparing the same substance under two different sets of conditions, the law can be written as : $\frac= \frac.$

# Energy associated with a gas

According to the assumptions of the kinetic theory of ideal gases, one can consider that there are no intermolecular attractions between the molecules, or atoms, of an ideal gas. In other words, its
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
is zero. Hence, all the energy possessed by the gas is the kinetic energy of the molecules, or atoms, of the gas. : $E=\frac n RT$ This corresponds to the kinetic energy of ''n'' moles of a monoatomic gas having 3 degrees of freedom; ''x'', ''y'', ''z''. The table here below gives this relationship for different amounts of a monoatomic gas.

# 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 Classical thermodynamics considers three main kinds of thermodynamic process: (1) changes in a system, (2) cycles in a system, and (3) flow processes. (1)A Thermodynamic process is a process in which the thermodynamic state of a system is change ...
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'', ''S'', or ''H'') 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 (which are 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 Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
(''S'') is constant. Under these conditions, ''p''1''V''1''γ'' = ''p''2''V''2''γ'', where ''γ'' is defined as the
heat capacity ratio 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 volu ...
, which is constant for a calorifically
perfect gas In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. In all perfect gas models, intermolecular forces are neglected. This means that one ...
. The value used for ''γ'' is typically 1.4 for diatomic gases like
nitrogen Nitrogen is the chemical element with the symbol N and atomic number 7. Nitrogen is a nonmetal and the lightest member of group 15 of the periodic table, often called the pnictogens. It is a common element in the universe, estimated at se ...
(N2) and
oxygen Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group in the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements as wel ...
(O2), (and air, which is 99% diatomic). Also ''γ'' is typically 1.6 for mono atomic gases like the
noble gas The noble gases (historically also the inert gases; sometimes referred to as aerogens) make up a class of chemical elements with similar properties; under standard conditions, they are all odorless, colorless, monatomic gases with very low chemi ...
es
helium Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas and the first in the noble gas group in the periodic table. ...
(He), and
argon Argon is a chemical element with the symbol Ar and atomic number 18. It is in group 18 of the periodic table and is a noble gas. Argon is the third-most abundant gas in Earth's atmosphere, at 0.934% (9340 ppmv). It is more than twice as abu ...
(Ar). In internal combustion engines ''γ'' varies between 1.35 and 1.15, depending on constitution gases and temperature. b. In an isenthalpic process, system
enthalpy Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant ...
(''H'') is constant. In the case of
free expansion Free may refer to: Concept * Freedom, having the ability to do something, without having to obey anyone/anything * Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism * Emancipate, to procur ...
for an ideal gas, there are no molecular interactions, and the temperature remains constant. For real gasses, the molecules do interact via attraction or repulsion depending on temperature and pressure, and heating or cooling does occur. This is known as the
Joule–Thomson effect In thermodynamics, the Joule–Thomson effect (also known as the Joule–Kelvin effect or Kelvin–Joule effect) describes the temperature change of a ''real'' gas or liquid (as differentiated from an ideal gas) when it is forced through a valv ...
. For reference, the Joule–Thomson coefficient μJT for air at room temperature and sea level is 0.22 °C/
bar Bar or BAR may refer to: Food and drink * Bar (establishment), selling alcoholic beverages * Candy bar * Chocolate bar Science and technology * Bar (river morphology), a deposit of sediment * Bar (tropical cyclone), a layer of cloud * Bar (u ...
.

# Deviations from ideal behavior of real gases

The equation of state given here (''PV'' = ''nRT'') 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 A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
and intermolecular attractions, it is most accurate for
monatomic In physics and chemistry, "monatomic" is a combination of the words "mono" and "atomic", and means "single atom". It is usually applied to gases: a monatomic gas is a gas in which atoms are not bound to each other. Examples at standard conditions ...
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 In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for ...
, account for deviations from ideality caused by molecular size and intermolecular forces.

# Derivations

## Empirical

The empirical laws that led to the derivation of the ideal gas law were discovered with experiments that changed only 2 state variables of the gas and kept every other one constant. All the possible gas laws that could have been discovered with this kind of setup are: *
Boyle's law Boyle's law, also referred to as the Boyle–Mariotte law, or Mariotte's law (especially in France), is an experimental gas law that describes the relationship between pressure and volume of a confined gas. Boyle's law has been stated as: The ...
() $PV = C_1 \quad \text \quad P_1 V_1 = P_2 V_2$ *
Charles's law Charles's law (also known as the law of volumes) is an experimental gas law that describes how gases tend to expand when heated. A modern statement of Charles's law is: When the pressure on a sample of a dry gas is held constant, the Kelvin t ...
() $\frac = C_2 \quad \text \quad \frac = \frac$ *
Avogadro's law Avogadro's law (sometimes referred to as Avogadro's hypothesis or Avogadro's principle) 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 c ...
() $\frac=C_3 \quad \text \quad \frac=\frac$ *
Gay-Lussac's law Gay-Lussac's law usually refers to Joseph-Louis Gay-Lussac's law of combining volumes of gases, discovered in 1808 and published in 1809. It sometimes refers to the proportionality of the volume of a gas to its absolute temperature at constant pr ...
() $\frac=C_4 \quad \text \quad \frac=\frac$ * $NT = C_5 \quad \text \quad N_1 T_1 = N_2 T_2$ * $\frac = C_6 \quad \text \quad \frac=\frac$ where ''P'' stands for
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
, ''V'' for
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
, ''N'' for number of particles in the gas and ''T'' for
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
; where $C_1, C_2, C_3, C_4, C_5, C_6$ are constants in this context because of each equation requiring only the parameters explicitly noted in them changing. To derive the ideal gas law one does not need to know all 6 formulas, one can just know 3 and with those derive the rest or just one more to be able to get the ideal gas law, which needs 4. Since each formula only holds when only the state variables involved in said formula change while the others (which are a property of the gas but are not explicitly noted in said formula) remain constant, we cannot simply use algebra and directly combine them all. This is why: Boyle did his experiments while keeping ''N'' and ''T'' constant and this must be taken into account (in this same way, every experiment kept some parameter as constant and this must be taken into account for the derivation). Keeping this in mind, to carry the derivation on correctly, one must imagine the
gas Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma). A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
being altered by one process at a time (as it was done in the experiments). The derivation using 4 formulas can look like this: at first the gas has parameters $P_1, V_1, N_1, T_1$ Say, starting to change only pressure and volume, according to
Boyle's law Boyle's law, also referred to as the Boyle–Mariotte law, or Mariotte's law (especially in France), is an experimental gas law that describes the relationship between pressure and volume of a confined gas. Boyle's law has been stated as: The ...
(), then: After this process, the gas has parameters $P_2,V_2,N_1,T_1$ Using then equation () to change the number of particles in the gas and the temperature, After this process, the gas has parameters $P_2,V_2,N_2,T_2$ Using then equation () to change the pressure and the number of particles, After this process, the gas has parameters $P_3,V_2,N_3,T_2$ Using then
Charles's law Charles's law (also known as the law of volumes) is an experimental gas law that describes how gases tend to expand when heated. A modern statement of Charles's law is: When the pressure on a sample of a dry gas is held constant, the Kelvin t ...
(equation 2) to change the volume and temperature of the gas, After this process, the gas has parameters $P_3,V_3,N_3,T_3$ Using simple algebra on equations (), (), () and () yields the result: $\frac = \frac$ or $\frac = k_\text ,$ where $k_\text$ stands for the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic 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, ...
. Another equivalent result, using the fact that $nR = N k_\text$, where ''n'' is the number of
moles Moles can refer to: * Moles de Xert, a mountain range in the Baix Maestrat comarca, Valencian Community, Spain * The Moles (Australian band) *The Moles, alter ego of Scottish band Simon Dupree and the Big Sound People *Abraham Moles, French engin ...
in the gas and ''R'' is the
universal gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
, is: $PV = nRT,$ which is known as the ideal gas law. If three of the six equations are known, it may be possible to derive the remaining three using the same method. However, because each formula has two variables, this is possible only for certain groups of three. For example, if you were to have equations (), () and () you would not be able to get any more because combining any two of them will only give you the third. However, if you had equations (), () and () you would be able to get all six equations because combining () and () will yield (), then () and () will yield (), then () and () will yield (), as well as would the combination of () and () as is explained in the following visual relation: where the numbers represent the gas laws numbered above. If you were to use the same method used above on 2 of the 3 laws on the vertices of one triangle that has a "O" inside it, you would get the third. For example: Change only pressure and volume first: then only volume and temperature: then as we can choose any value for $V_3$, if we set $V_1 = V_3$, equation () becomes: combining equations () and () yields $\frac = \frac$, which is equation (), of which we had no prior knowledge until this derivation.

## Theoretical

### Kinetic theory

The ideal gas law can also be derived from
first principles In philosophy and science, a first principle is a basic proposition or assumption that cannot be deduced from any other proposition or assumption. First principles in philosophy are from First Cause attitudes and taught by Aristotelians, and nua ...
using the
kinetic theory of gases Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory, describing a gas as particles in random motion * Kinetic energy, the energy of an object that it possesses due to its motion Art and enter ...
, 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. The fundamental assumptions of the kinetic theory of gases imply that :$PV = \fracNmv_^2.$ Using the
Maxwell–Boltzmann distribution In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It was first defined and use ...
, the fraction of molecules that have a speed in the range $v$ to $v + dv$ is $f\left(v\right) \, dv$, where :$f\left(v\right) = 4\pi \left\left(\frac\right\right)^v^2 e^$ and $k$ denotes the Boltzmann constant. The root-mean-square speed can be calculated by :$v_^2 = \int_0^\infty v^2 f\left(v\right) \, dv = 4\pi \left\left(\frac\right\right)^\int_0^\infty v^4 e^ \, dv.$ Using the integration formula :$\int_0^\infty x^e^ \, dx = \sqrt \, \frac\left\left(\frac\right\right)^,$ it follows that :$v_^2 = 4\pi\left\left(\frac\right\right)^\sqrt \, \frac\left\left(\frac\right\right)^ = \frac,$ from which we get the ideal gas law: :$PV = \frac Nm\left\left(\frac\right\right) = NkT.$

### 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-averaged kinetic energy of the particle is: : $\begin \langle \mathbf \cdot \mathbf \rangle &= \left\langle q_ \frac \right\rangle + \left\langle q_ \frac \right\rangle + \left\langle q_ \frac \right\rangle\\ &=-\left\langle q_ \frac \right\rangle - \left\langle q_ \frac \right\rangle - \left\langle q_ \frac \right\rangle = -3k_\text T, \end$ where the first equality is
Newton's second law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...
, and the second line uses
Hamilton's equations Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
and the
equipartition theorem In classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. T ...
. Summing over a system of ''N'' particles yields :$3Nk_ T = - \left\langle \sum_^ \mathbf_ \cdot \mathbf_ \right\rangle.$ By
Newton's third law Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in moti ...
and the ideal gas assumption, the net force of the system is the force applied by the walls of the container, and this force is given by the pressure ''P'' of the gas. Hence :$-\left\langle\sum_^ \mathbf_ \cdot \mathbf_\right\rangle = P \oint_ \mathbf \cdot d\mathbf,$ where dS is the infinitesimal area element along the walls of the container. Since the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
of the position vector q is :$\nabla \cdot \mathbf = \frac + \frac + \frac = 3,$ the
divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
implies that :$P \oint_ \mathbf \cdot d\mathbf = P \int_ \left\left( \nabla \cdot \mathbf \right\right) dV = 3PV,$ where ''dV'' is an infinitesimal volume within the container and ''V'' is the total volume of the container. Putting these equalities together yields :$3 N k_\text T = -\left\langle \sum_^ \mathbf_ \cdot \mathbf_ \right\rangle = 3PV,$ which immediately implies the ideal gas law for ''N'' particles: :$PV = Nk_ T = nRT,$ where ''n'' = ''N''/''N''A is the number of
moles Moles can refer to: * Moles de Xert, a mountain range in the Baix Maestrat comarca, Valencian Community, Spain * The Moles (Australian band) *The Moles, alter ego of Scottish band Simon Dupree and the Big Sound People *Abraham Moles, French engin ...
of gas and ''R'' = ''N''A''k''B is the
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
.

# Other dimensions

For a ''d''-dimensional system, the ideal gas pressure is: :$P^ = \frac,$ where $L^d$ is the volume of the ''d''-dimensional domain in which the gas exists. Note that the dimensions of the pressure changes with dimensionality.

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Gas laws The gas laws were developed at the end of the 18th century, when scientists began to realize that relationships between pressure, volume and temperature of a sample of gas could be obtained which would hold to approximation for all gases. Boyl ...
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