Kinetic pressure

The kinetic pressure , or pressure of Daniel Bernoulli (1700 - 1782), is the Pression resulting from the inelastic Choc S of the particles on the thermalized wall. This calculation of 1738 in kinetic Théorie of the gases is remarkable, because it precedes one century this theory, primarily initiated by Clausius in 1857.

The result for a simple monoatomic perfect gas is:

p = \ frac {2} {3} \ cdot \ frac {U} {V}

where

p is the pressure;
U is the energy interns system;
V is the Volume system.

Concretely, the pressure represents two thirds of voluminal energy (Recall: the Pascal is homogeneous with Joule S by Cubic meter).

Demonstration

there are two contributions: that of the particles which adsorb by impacting the wall:

p_1 = \ frac {1} {3} \ cdot \ frac {U} {V}

and that identical of the particles which desorb by taking support on the wall:

p_2 = \ frac {1} {3} \ cdot \ frac {U} {V}

maybe, if < is called; U > the quadratic Speed Average, a kinetic energy

E_ {c1} = \ frac {1} {6} \ cdot \ frac {NR} {V} \ cdot m \ cdot u^2

in impact, and idem in support.

For the demonstration, one will use the Théorème of the virial:

2 < E_c > + < virial > = 0

Here, the average force D ƒ on the whole of the particles of gas on an element dS of surface east $df = - p \ cdot dS$ . Summoned on the whole of the wall S which encloses gas:

= \ int \ int_S (- p \ cdot OM) \ cdot dS

The pressure p is homogeneous, therefore leaves the integral; the remainder is traditional (1/3 V ). On the whole, the virial is negative of course, and is worth -1/3· statement , is

pV = \ frac {2} {3} < E_c >

One can also refer to the demonstration made in the kinetic article Théorie of gases > Pression and kinetic energy .

Length of Loschmidt

The length of Loschmidt is the average distance separating two molecules in the Normal conditions from pressure and temperature; it is defined as being the cubic Racine average free volume around a molecule. This length is independent of perfect gas considered, and is worth

D = 333 pm.

See the article Perfect gas.

Corrections of covolume of Bernoulli

As of 1738, Bernoulli had included/understood the role of the size of the Atome S: reasoning on a Gaz with a dimension - made “straws” length has - it had understood that real “volume” available for the whole of the straws was not the size L segment where they were included, but L - Na .

With two dimensions, it included/understood just as if the surface of each disc were S , surface really available for the whole of the discs was not surface S plate where they were enclose, but S - 2Ns , for S rather large (factor 2 will be explained further).

Lastly, with 3 dimensions, it had as included/understood as volume available would be lower than V :

pV - Nb = NR K T

;

with

b = \ frac43 \ pi \ frac {(2 r _0) ^3} {2}

The parameter NR B is called “Co-volume” or “residual volume”.

One calls “compactness”, and one notes X , the relationship between the clean volume of the particles of ray R ₀ and total volume. The preceding equation can be written in series compactness:

$\ frac {statement} {NR K T} = 1 + 4x + \ mathrm {O} \ left (X \ right)$ .

Thus the study of the isotherms with low pressure makes it possible to evaluate this covolume, therefore the ray R ₀ of the particles.

A more thorough study made it possible Boltzmann analytically to calculate the 2 following terms of the series:

$\ frac {statement} {NkT} = 1 + 4 X + 10 x^2 + 18.6 x^3 + \ ldots$ .

One knows, with the current computers, to find the first 8 terms of this development known as of the “virial” of a gas modelled by hard balls. One can also calculate the ray of convergence, and approach it of it by one approximating of Padé, which gives a better comprehension of the asymptote, which is obviously not V = NR B ! Indeed, compactness tends towards X = π/(18) ^1/2 ~ 74%. Moreover, simulations reveal sometimes a phase shift towards X = 60%.

The Pression interns Van der Waals

Van der Waals, being based on work of Laplace, included/understood the part which attraction between 2 particles of the gas played (attraction which bears its name today). On average, a particle located at long distance from the wall feels only one NULL average force. On the other hand, a particle going towards the wall is decelerated by this attraction, multiple of the density: thus the kinetic energy of impact will be reduced of a factor F.L.N/V. And for the pressure which had with the whole of the particles, there will be thus a reduction of a factor (N/V) ^2:

$p + N^2 \ cdot \ frac {has} {V^2} = P_c$

The whole of these considerations brought Van der Waals to its famous equation:

$(p + N^2 \ cdot \ frac {has} {V^2}) (V-Nb) = NkT$

One should not believe the internal pressure like negligible, because it varies considerably: let us give an evaluation:

Maybe of liquid water in balance with 473K with its vapor: for the liquid NkT/V is very large: 55000/22.4 (473/373) ~2000 bars, therefore internal pressure, 2000-1, too. but for the vapor, it will be approximately 1/10^6 weaker time, that is to say 1/1000 bar and thus negligible.

Equation of state of the virial

$b = 4v$ , then $10x^2$

Other equations of state

Perceived-Yevick, Redlich-Kwong, factor acentric…

See too

Internal bonds

Theorem of the kinetic virial
Theory of the gases

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