The basic gas law for a perfect gas is
V = volume of the gas
represents the average kinetic energy of a molecule
of mass m in any direction, i.e., the average for the three
Cartesian co-ordinates x, y, z.
The distribution of molecular speeds was derived by Maxwell based on three probabilistic assumptions, namely (i) uniform distribution in space, (ii) mutual independence of the three velocity components and (iii) isotropy as regards the directions of the velocities (Ruhla and Barton, 1992). These assumptions were also used in deriving the fundamental gas law at Eq.(1) for a perfect gas. Maxwell's distribution of molecular speeds is given by the following equation.
For a given gas at a fixed temperature T , the probability density r(v) may be written as
r (v) µ exp(-v2 ) v2
where z is the size scale ratio equal to R/r . Considering three-dimensional fluctuations the fractional contribution (probability density) of smaller length scale r fluctuations in the environment of the larger length scale R fluctuation is given by f 3 . The eddy circulation speeds follow the logarithmic law with respect to the length scale ratio z , namely
where k is a constant equal to 1/t2 and t is the golden mean equal to (1+Ö 5)/2 (»1.618 ). The eddy circulation speeds are therefore proportional to log z , that is
W » log z
A graph of f 3 versus log z will give the probability density distribution for molecular speeds. The cell dynamical system model predicted molecular speed distribution in a perfect gas is shown as crosses in Fig.1 (Fig.1). The distributions (Maxwell's and model predicted) are normalised with respect to the maximum speed. There is close agreement between the Maxwell's and model-predicted distributions for molecular speeds in a perfect gas.
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