Ideal gas law

Isotherms of an ideal gas

The ideal gas law is the equation of state of a hypothetical ideal gas, first stated by Benoît Paul Émile Clapeyron in 1834. An isothermal process is a Thermodynamic process in which the Temperature of the System stays Constant: &Delta T = 0 In Physics and Thermodynamics, an equation of state is a relation between state variables More specifically an equation of state is a thermodynamic These four properties that constitute an ideal gas can be easily remembered by the acronym RIPE which stands for - R andom Motion (molecules are in constant random motion Benoît Paul Émile Clapeyron (26 February 1799 - 28 January 1864 was a French Engineer and Physicist, one of the founders of Thermodynamics Year 1834 ( MDCCCXXXIV) was a Common year starting on Wednesday (link will display the full calendar of the Gregorian Calendar (or a Common

The state of an amount of gas is determined by its pressure, volume, and temperature according to the equation:

$\ PV = nRT$

where

$\ P$ is the absolute pressure of the gas,

$\ V$ is the volume of the gas,

$\ n$ is the number of moles of gas,

$\ R$ is the universal gas constant,

$\ T$ is the absolute temperature. In Thermodynamics, a state function, state quantity, or a function of state, is a property of a system that depends only on the current This page is about the physical properties of gas as a state of matter Relationship with the Boltzmann constant The Boltzmann constant kB (often abbreviated k) may be used in place of the gas constant by working Thermodynamic temperature is the absolute measure of Temperature and is one of the principal parameters of Thermodynamics.

The value of the ideal gas constant, R, is found to be as follows. Relationship with the Boltzmann constant The Boltzmann constant kB (often abbreviated k) may be used in place of the gas constant by working

R	=	8. 314472	J·mol⁻¹·K⁻¹
	=	8. The joule (written in lower case ˈdʒuːl or /ˈdʒaʊl/ (symbol J) is the SI unit of Energy measuring heat, Electricity The mole (symbol mol) is a unit of Amount of substance: it is an SI base unit, and almost the only unit to be used to measure this The kelvin (symbol K) is a unit increment of Temperature and is one of the seven SI base units The Kelvin scale is a thermodynamic 314472	m³·Pa·K⁻¹·mol⁻¹
	=	8. 314472	kPa·L·mol^-1·K^-1
	=	0. 08205746	L·atm·K⁻¹·mol⁻¹
	=	62. The litre or liter (see spelling differences) is a unit of Volume. The Standard atmosphere is an international reference pressure defined as 101325 Pa and formerly used as unit of Pressure (symbol atm 36367	L·mmHg·K⁻¹·mol⁻¹
	=	10. The torr (symbol Torr) is a non- SI unit of Pressure defined as 1/760 of an atmosphere. 73159	ft³·psi·°R⁻¹·lb-mol⁻¹
	=	53. A foot (plural feet or foot; symbol or abbreviation ft or sometimes &prime – the prime symbol) is a non-SI unit The pound per square inch or more accurately pound-force per square inch (symbol psi or lbf/in² or lbf/in²) is a unit of Rankine is a thermodynamic (absolute temperature scale named after the Scottish Engineer and Physicist William John Macquorn Rankine The mole (symbol mol) is a unit of Amount of substance: it is an SI base unit, and almost the only unit to be used to measure this 34	ft·lbf·°R⁻¹·lbm⁻¹ (for air)

The ideal gas law mathematically follows from a statistical mechanical treatment of primitive identical particles (point particles without internal structure) which do not interact, but exchange momentum (and hence kinetic energy) in elastic collisions. A foot (plural feet or foot; symbol or abbreviation ft or sometimes &prime – the prime symbol) is a non-SI unit This article deals with the unit of force For the unit of mass see Pound (mass. Rankine is a thermodynamic (absolute temperature scale named after the Scottish Engineer and Physicist William John Macquorn Rankine The pound or pound-mass (abbreviation lb, lbm, or sometimes in the United States #) is a unit of Mass Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product The kinetic energy of an object is the extra Energy which it possesses due to its motion 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

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. In Physics and Chemistry, monatomic is a combination of the words "mono" and "atomic" and means "single Atom. 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 i. Thermal energy is the sum of the sensible energy and latent energy. e. , with increasing temperatures. More sophisticated equations of state, such as the van der Waals equation, allow deviations from ideality caused by molecular size and intermolecular forces to be taken into account. In Physics and Thermodynamics, an equation of state is a relation between state variables More specifically an equation of state is a thermodynamic 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

1 Alternative forms
2 Calculations
3 Derivations
4 See also
5 References

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 ( $n\,$ ) is equal to the mass ( $\, m$ ) divided by the molar mass ( $\, M$ ):

$n = {\frac{m}{M}}$

Then, replacing $\, n$ gives:

$\ PV = \frac{m}{M}RT$

from where

$\ P = \rho \frac{R}{M}T$ . Molar mass, symbol M, is the Mass of one mole of a substance ( Chemical element or Chemical compound)

This form of the ideal gas law is particularly useful because it links pressure, density $ρ = m / V$ , and temperature in a unique formula independent from the quantity of the considered gas.

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

$\ PV = NkT .$

Here $\,k$ is Boltzmann's constant, and $\,N$ is the actual number of molecules, in contrast to the other formulation, which uses $\,n$ , the number of moles. Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics Bridge from macroscopic to microscopic physics Boltzmann's constant k is a bridge between Macroscopic and microscopic physics This relation implies that $N\,k = nR$ , 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 $μ$ times the atomic mass constant $m u$ (i. In Physics and Chemistry, the atomic mass constant, m u is one twelfth of the mass of an unbound atom of the carbon-12 nuclide at rest and in e. , the mass is $μ$ u)

$N = \frac{m}{\mu m_\mathrm{u}}$

and since $ρ = m / V$ , we find that the ideal gas law can be re-written as:

$p = \frac{1}{V}\frac{m}{\mu m_\mathrm{u}} kT = \frac{k}{\mu m_\mathrm{u}} \rho T .$

One more equation involves density where:

$\ PM = DRT .$

where M is the mass and D is the density. The unified atomic mass unit ( u) or Dalton ( Da) or sometimes universal mass unit, is an unit of Mass used to express

Calculations

Process	Constant	Known ratio	P₂	V₂	T₂
Isobaric process	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	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	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 (Reversible adiabatic process)	Entropy^[a]	P₂/P₁	P₂ = P₁ (P₂/P₁)	V₂ = V₁ (P₂/P₁) ^{-₁/ $γ$}	T₂ = T₁ (P₂/P₁)^{( $γ$ -₁)/ $γ$}
"	"	V₂/V₁	P₂ = P₁ (V₂/V₁) ^{- $γ$}	V₂ = V₁ (V₂/V₁)	T₂ = T₁ (V₂/V₁) ^{₁- $γ$}
"	"	T₂/T₁	P₂ = P₁ (T₂/T₁) ^{$γ$ /( $γ$ -₁)}	V₂ = V₁ (T₂/T₁) ^{1/(1- $γ$ )}	T₂ = T₁ (T₂/T₁)

^{^} a. An isobaric process is a Thermodynamic process in which the pressure stays constant \Delta p = 0 The term derives from the Greek isos "equal" An isochoric process, also called an isovolumetric process, is a process during which volume remains constant An isothermal process is a Thermodynamic process in which the Temperature of the System stays Constant: &Delta T = 0 In Thermodynamics, an isentropic process ( iso = "equal" (Greek Entropy = "disorder" is one during which the entropy of the system This article covers adiabatic processes in Thermodynamics. For adiabatic processes in Quantum mechanics, see Adiabatic process (quantum mechanics In an isentropic process, system entropy (Q) 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. Ideal gas relations For an ideal gas the heat capacity is constant with temperature

Derivations

Empirical

The ideal gas law can be derived from combining two empirical gas laws: the combined gas law and Avogadro's law. This articles outlines the historical development of the laws describing ideal gases The combined gas law is a Gas law which combines Charles's law, Boyle's law, and Gay-Lussac's law. Avogadro's law ( Avogadro's Hypothesis, or Avogadro's Principle) is a Gas law named after Amedeo Avogadro, who in 1811 hypothesized The combined gas law states that

$\frac {pV}{T}= C$

where C is a constant which is directly proportional to the amount of gas, n (Avogadro's law). Avogadro's law ( Avogadro's Hypothesis, or Avogadro's Principle) is a Gas law named after Amedeo Avogadro, who in 1811 hypothesized The proportionality factor is the universal gas constant, R, i. Relationship with the Boltzmann constant The Boltzmann constant kB (often abbreviated k) may be used in place of the gas constant by working e. $C = n R$ .

Hence the ideal gas law

$pV = nRT \,$

Theoretical

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. First Principles is also the title of a work by Herbert Spencer. Kinetic theory (or kinetic theory of gases) attempts to explain Macroscopic properties of Gases such as pressure temperature or volume by considering

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

$\begin{align} \langle \mathbf{q} \cdot \mathbf{F} \rangle &= \Bigl\langle q_{x} \frac{dp_{x}}{dt} \Bigr\rangle + \Bigl\langle q_{y} \frac{dp_{y}}{dt} \Bigr\rangle + \Bigl\langle q_{z} \frac{dp_{z}}{dt} \Bigr\rangle\\ &=-\Bigl\langle q_{x} \frac{\partial H}{\partial q_x} \Bigr\rangle - \Bigl\langle q_{y} \frac{\partial H}{\partial q_y} \Bigr\rangle - \Bigl\langle q_{z} \frac{\partial H}{\partial q_z} \Bigr\rangle = -3k_{B} T, \end{align}$

where the first equality is Newton's second law, and the second line uses Hamilton's equations and the equipartition theorem. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. In classical Statistical mechanics, the equipartition theorem is a general formula that relates the Temperature of a system with its average energies Summing over a system of N particles yields

$3Nk_{B} T = - \biggl\langle \sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k} \biggr\rangle.$

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. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Hence

$-\biggl\langle\sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k}\biggr\rangle = P \oint_{\mathrm{surface}} \mathbf{q} \cdot \mathbf{dS},$

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

$\boldsymbol\nabla \cdot \mathbf{q} = \frac{\partial q_{x}}{\partial q_{x}} + \frac{\partial q_{y}}{\partial q_{y}} + \frac{\partial q_{z}}{\partial q_{z}} = 3,$

the divergence theorem implies that

$P \oint_{\mathrm{surface}} \mathbf{q} \cdot \mathbf{dS} = P \int_{\mathrm{volume}} \left( \boldsymbol\nabla \cdot \mathbf{q} \right) dV = 3PV,$

where dV is an infinitesimal volume within the container and V is the total volume of the container. In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the In Vector calculus, the divergence theorem, also known as Gauss&rsquos theorem ( Carl Friedrich Gauss) Ostrogradsky&rsquos theorem ( Mikhail

Putting these equalities together yields

$3Nk_{B} T = -\biggl\langle \sum_{k=1}^{N} \mathbf{q}_{k} \cdot \mathbf{F}_{k} \biggr\rangle = 3PV,$

which immediately implies the ideal gas law for N particles:

$PV = Nk_{B} T = nRT,\,$

where n=N/N_A is the number of moles of gas and R=N_Ak_B is the gas constant. The mole (symbol mol) is a unit of Amount of substance: it is an SI base unit, and almost the only unit to be used to measure this Relationship with the Boltzmann constant The Boltzmann constant kB (often abbreviated k) may be used in place of the gas constant by working

The readers are referred to the comprehensive article 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. In Thermodynamics, the Helmholtz free energy is a Thermodynamic potential which measures the “useful” work obtainable from a closed thermodynamic In Statistical mechanics, the partition function Z is an important quantity that encodes the statistical properties of a system in Thermodynamic

References

Davis and Masten Principles of Environmental Engineering and Science, McGraw-Hill Companies, Inc. The combined gas law is a Gas law which combines Charles's law, Boyle's law, and Gay-Lussac's law. These four properties that constitute an ideal gas can be easily remembered by the acronym RIPE which stands for - R andom Motion (molecules are in constant random motion In Physics and Thermodynamics, an equation of state is a relation between state variables More specifically an equation of state is a thermodynamic 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 Bridge from macroscopic to microscopic physics Boltzmann's constant k is a bridge between Macroscopic and microscopic physics New York (2002) ISBN 0-07-235053-9
Website giving credit to Benoît Paul Émile Clapeyron, (1799-1864) in 1834
Website containing Ideal Gas Law Calculator & a host of other scientific calculators, Rex Njoku & Dr. Anthony Steyermark -University of St.Thomas

Benoît Paul Émile Clapeyron (26 February 1799 - 28 January 1864 was a French Engineer and Physicist, one of the founders of Thermodynamics Year 1834 ( MDCCCXXXIV) was a Common year starting on Wednesday (link will display the full calendar of the Gregorian Calendar (or a Common

Dictionary

-noun

(physics) the equation of state of an ideal gas - PV = nRT

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