informational desert: Ideal Gas Law

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Daniel Bernoulli Sadi Carnot Benoît Paul Émile Clapeyron Rudolf Clausius Hermann von Helmholtz Constantin Carathéodory James Prescott Joule Julius Robert von Mayer William Rankine John Smeaton Georg Ernst Stahl Benjamin Thompson William Thomson, 1st Baron Kelvin John James Waterston

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 behavior 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]
The state of an amount of gas is determined by its pressure, volume, and temperature. The modern form of the equation is:

$pV = nRT\,$

where p is the absolute pressure of the gas; V is the volume; n is the amount of substance; R is the Regnault constant, better known as universal gas constant; and T is the absolute temperature.
In SI units, p is measured in pascals; V in cubic metres; n in moles; and T in kelvin. R has the value 8.314472 J·K⁻¹·mol⁻¹ in SI units^[4]).
The temperature given in the equation of state must be an absolute temperature that begins at absolute zero. In the metric system of units, we must specify the temperature in Kelvin (degree increments as in Celsius). In the Imperial system, absolute temperature is in Rankine (degree increments as in Fahrenheit). ^[5]

Deviations from real gases

The equation of state given here applies only to an ideal gas, or a real gas that behaves like an ideal gas. There are in fact many different forms for the equation of state for different gases. Since it neglects both molecular size and intermolecular attractions, the ideal gas law 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, allow deviations from ideality caused by molecular size and intermolecular forces to be taken into account.
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.

Alternative forms

Molar form

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):

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

By replacing $n \,$ , we get:

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

from where

$\ p = \rho \frac{R}{M}T$ .

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

Engineering

Some engineers use a slightly different form of the equation of state that is specialized for a particular gas and written in terms of specific volume^{[citation needed]}.
A variation of the gas constant R is used, which is equal to the universal gas constant divided by the mass per mole of the gas. The value of the new constant depends on the type of gas, as opposed to the universal gas constant which is the same for all gases. In most engineering fields,

R

is written with a bar, as $\bar{R}$ , so that a new symbol

R

is defined by $R={\frac{\bar{R}}{MW}}$ , where

M W

is the molecular weight.
If we divide both sides of the general ideal gas equation by the mass of the gas, the volume becomes the specific volume, which is the reciprocal of the gas density. The equation of state can be written in terms of the specific volume or in terms of the air density as

p v = R T

, or

p = ρ R T

where

ρ

is the density [kg/m³] and

v

is the specific volume [m³/kg].

Statistical mechanics

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

$\ pV = NkT$

Here k is the Boltzmann constant, and N is the actual number of molecules, in contrast 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 here 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)

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

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

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

Applications to thermodynamics 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.

process	Constant	Known ratio	p₂	V₂	T₂
Isobaric process	Pressure	V₂/V₁	p₂ = p₁	V₂ = V₁(V₂/V₁)	T₂ = T₁(V₂/V₁)
Isobaric process	Pressure	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)
Isochoric process	Volume	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
Isothermal process	Temperature	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. In an isentropic process, system entropy (S) is constant. Under these conditions, p₁V₁^γ = p₂V₂^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.

Derivations

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

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

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

$pV = nRT \,$

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.

From 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:
$\begin{align}\langle \mathbf{q} \cdot \mathbf{F} \rangle &= \Bigl\langle q_{x} \frac{dp_{x}}{dt} \Bigr\rangle +\Bigl\langle q_{y} \frac{dp_{y}}{dt} \Bigr\rangle +\Bigl\langle q_{z} \frac{dp_{z}}{dt} \Bigr\rangle\\&=-\Bigl\langle q_{x} \frac{\partial H}{\partial q_x} \Bigr\rangle -\Bigl\langle q_{y} \frac{\partial H}{\partial q_y} \Bigr\rangle -\Bigl\langle q_{z} \frac{\partial H}{\partial q_z} \Bigr\rangle = -3k_{B} T,\end{align}$
where the first equality is Newton's second law, and the second line uses Hamilton's equations and the equipartition theorem. 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. Hence

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

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

$p \oint_{\mathrm{surface}} \mathbf{q} \cdot d\mathbf{S} = p \int_{\mathrm{volume}} \left( \boldsymbol\nabla \cdot \mathbf{q} \right) dV = 3pV,$

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

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Tuesday, February 15, 2011

Ideal Gas Law

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