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Ideal gas law
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.
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. Obviously 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.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.
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 ():
Then, replacing gives:
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:
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 μ times the atomic mass constant mu (i.e., the mass is μ u)
and since ρ = m / V, we find that the ideal gas law can be re-written as:
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
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.
Derivation from the statistical mechanics
Let q = (qx, qy, qz) and p = (px, py, pz) 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. Summing over a system of N particles yields
where dS is the infinitesimal area element along the walls of the container. Since the divergence of the position vector q is
the divergence theorem 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 for N particles:
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Ideal_gas_law". A list of authors is available in Wikipedia.|