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2019-03
12/03/2019 ForsChem Research Reports 2019-03 (1 / 31)
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Calculation of Molecular Fluxes and Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research, 050030 Medellin, Colombia
hugo.hernandez@forschem.org
doi:
Abstract
Molecules are in permanent motion and interaction. Even though a system might be at rest
from a macroscopic point of view, it is restless at the molecular scale. Such continuous
molecular motion results in the flow of molecules through space. The number of molecules
crossing a boundary per unit time per unit area is denoted as a molecular flux. Three different
types of boundaries are considered: Completely permeable, semi-permeable rigid and
impermeable rigid boundaries. When the system is contained by rigid (impermeable or semi-
permeable) boundaries such molecular flux against the walls produce a net force per unit area
that is denoted as pressure. The purpose of this report is deriving mathematical expressions for
estimating the molecular flux and pressure on rigid boundaries in ideal gases, using only the
classical equations of motion and the distribution of molecular positions and velocities in the
system. The pressure at the interior of an ideal gas system is zero because there are no forces
acting on the molecules. Thus, the concept of equivalent pressure, as the pressure that would
be obtained in the presence of an impermeable rigid boundary, is introduced. The
mathematical results presented are based on the following general assumptions: i) Molecular
collisions in the gas are negligible (ideal gas assumption). ii) There are no external forces acting
on the system. iii) There is no net macroscopic motion of the gas parallel to the surface. iv)
Molecular positions are uniformly distributed in the system volume. v) The orthogonal
component of molecular velocity is normally distributed. vi) The system considered is large
compared to the size of the molecules. The results obtained are consistent with the continuity
equation. Furthermore, when the ideal gas system is at macroscopic rest, the classical ideal gas
thermodynamic equation of state is obtained. In addition, three different regimes of molecular
flux and pressure are observed: The thermal motion regime, the macroscopic regime and the
transition regime between the first two.
Keywords
Collisions, Equation of Motion, Equivalent Pressure, Ideal Gas, Maxwell-Boltzmann Distribution,
Molecular Flux, Probability Distribution.
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
12/03/2019 ForsChem Research Reports 2019-03 (2 / 31)
www.forschem.org
1. Introduction
Molecules are in permanent motion and interaction. As a result, a continuous and variable flow
of molecules across any arbitrary boundary is expected. Such molecular flow plays a key role in
many different macroscopic physicochemical phenomena, such as diffusion and mass transfer,
heat transfer, pressure and flowrates, evaporation rates, surface tension, and many others.
Thus, the first purpose of this report is deriving some mathematical expressions useful for
calculating molecular fluxes of different molecular systems composed of ideal gases, given that
their molecular speeds can be described by the generalized Maxwell-Boltzmann distribution.[1]
The term flux previously used is defined as “flow per unit area”.[2] Thus, the molecular flux at a
boundary corresponds to the number of molecules reaching the boundary per unit time and
per unit area of the boundary. Please notice that it is assumed that the molecules just reach the
boundary but do not necessarily cross it. Thus, three different types of boundaries will be
considered: 1) A completely permeable boundary (i.e. an imaginary or virtual boundary), 2) A
rigid impermeable boundary (i.e. a solid boundary), and 3) A rigid semi-permeable boundary.
The second purpose is calculating the pressure of a system from the molecular flux, and
proposing the concept of an equivalent pressure to describe the corresponding pressure of a
system that it would show if it were suddenly enclosed by rigid walls.
2. Mathematical Description of Molecular Flux through Permeable Boundaries
neglecting Molecular Collisions
Let us consider a molecular system where molecular collisions can be neglected (i.e. highly-
diluted ideal gas systems), and any arbitrary permeable boundary, where all molecules pass
unaffected through such virtual boundary. Even though any shape is possible for the boundary,
it can approximately be described by a large set of infinitesimally small planar surfaces as
depicted in Figure 1. Thus, the results obtained for the molecular flux at a small flat surface can
be generalized to any arbitrary shape.
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
12/03/2019 ForsChem Research Reports 2019-03 (3 / 31)
www.forschem.org
Figure 1. Example of decomposition of any arbitrary surface into many infinitesimally small flat
surfaces.
Now, let us consider one of those infinitesimally small flat surfaces with area , located at any
arbitrary position inside the highly-diluted ideal gas system, as can be seen in Figure 2. At a
certain time , each molecule in the system will have its own relative velocity component
perpendicular to the surface , and it will be located at a certain relative position
perpendicular to the surface . Both velocities and positions are considered relative to the
surface. This means that even though the boundary can be in motion, in this analysis it will
seem to remain perfectly still. A positive value of indicates that the molecule is located to
the right of the boundary (considering Figure 2), whereas a negative value indicates that it is
placed to the left of the boundary. Similarly, a positive value of indicates that the molecule
is moving to the right, whereas a negative value indicates that it is moving to the left.
Figure 2. Small flat permeable boundary with area , located inside a highly-diluted ideal gas
system. Green spheres represent ideal gas molecules. The red arrow corresponds to the
relative velocity of a certain molecule
, whereas the blue arrow represents the relative
molecular velocity component perpendicular to the surface .
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
12/03/2019 ForsChem Research Reports 2019-03 (4 / 31)
www.forschem.org
Given that for the highly-diluted ideal gas considered, collisions with other gas molecules are
neglected, then it is possible to assume that the molecular velocities are not changing (in the
absence of external forces) and that such molecule will reach the surface at a time:
(2.1)
Previously,[3] it was shown that the probability of hitting a flat surface at a time ,
with by a molecule whose initial position and velocity are unknown is given by:
(2.2)
where is the surface ratio between the area of the infinitesimally small flat surface and the
parallel area of the system considered.
On the other hand, the probability density function of the time , for a uniform
distribution of initial positions and normal distribution of molecular velocity in the
perpendicular direction to the wall
§
is given by:
(2.3)
§
A molecule with a speed described by a Maxwell-Boltzmann distribution has a normal distribution of
each velocity component in space.[4,5]
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
12/03/2019 ForsChem Research Reports 2019-03 (5 / 31)
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is the expected initial position of the molecule perpendicular to the surface, is its
expected velocity in the direction perpendicular to the surface. is the standard deviation
of the distribution of initial molecular position. is the standard deviation in the distribution
of molecular velocities perpendicular to the surface.
is a particular realization of the
standard random variable representing the molecular velocity component.
Integrating results in:
(2.4)
Thus, the number of molecules of species , composed by all molecules identical to molecule
that will be reaching the boundary (from any direction) in the time interval will be
given by:
(2.5)
where is the number density of molecules of species in the system, is the total
volume of the system, and is the size of the system in the direction perpendicular to the
infinitesimally small boundary. Eq. (2.5) is obtained considering that: 1) there are molecules
of species in the system; 2) the probability of any of these molecules of reaching the surface
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
12/03/2019 ForsChem Research Reports 2019-03 (6 / 31)
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plane for any possible velocity is ; and 3) the probability of any molecule reaching
the surface plane of hitting the surface is .
The total molecular flux of species through the permeable boundary will then be:
(2.6)
Now, since the initial position of the molecules is uniformly distributed, its standard deviation
will be:
(2.7)
as is the range of possible positions of the molecules in the direction perpendicular to the
surface.
On the other hand, assuming that the molecular speeds relative to the system can be described
by a generalized Maxwell-Boltzmann distribution,[1] then the standard deviation in the
molecular velocity in one direction can be expressed as:
(2.8)
where is Boltzmann’s constant, is the temperature of the
molecular system, and is the mass of molecules of species .
Thus, the total molecular flux of species becomes:
(2.9)
Now, if only the molecules crossing the boundary with a positive velocity (forward, ) are
considered, that is, only the molecules of species coming from the left in Figure 2, then:
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
12/03/2019 ForsChem Research Reports 2019-03 (7 / 31)
www.forschem.org
(2.10)
Similarly, the backward flux of molecules of species reaching the boundary with a negative
(backward, ) velocity is:
(2.11)
Finally, the net molecular flux of species will be given by:
(2.12)
where a positive value of the molecular flux indicates a net flux of molecules to the right side of
the surface, and a negative value indicates a net flux of molecules to the left side of the
surface. This result is consistent with the continuity equation.[2]
2.1. Molecular Fluxes in Pure Systems
Now, assuming that all molecules in the system are identical,then the different molecular
fluxes across a certain permeable boundary simply become:
(2.13)
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
12/03/2019 ForsChem Research Reports 2019-03 (8 / 31)
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where
(2.14)
is a dimensionless velocity, given by the ratio between the average relative velocity of the
system perpendicular to the surface and the standard deviation in thermal speed of the
system, also perpendicular to the surface.
Similarly,
(2.15)
(2.16)
(2.17)
Figure 3 shows the behavior of the net molecular flux and the total molecular flux as a function
of the average to standard deviation velocity ratio . The total molecular flux when ,
, is used as a reference molecular flux. Three different regimes can be identified:
i) Macroscopic motion regime : In this case, the magnitudes of both
the net and total fluxes are identical. The macroscopic flow is so high that all
molecules reaching the boundary cross it in the same direction, which is the
direction of flow. Thus, thermal motion is unable to cause a significant
backflow.
ii) Thermal motion regime : Even though the net molecular flux
changes with the average velocity of the system relative to the boundary, the
total molecular flux remains almost constant as a result of thermal motion. The
minimum value of the total molecular flux corresponds to the condition when
the system is at rest with respect to the boundary.
iii) Transition regime : This regime shows a soft transition between the
thermal and the macroscopic motion regimes. The total molecular flux begins
to increase as the average velocity of the system increases.
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
12/03/2019 ForsChem Research Reports 2019-03 (9 / 31)
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Figure 3. Relative net and total molecular flux as a function of the ratio . The reference flux
corresponds to the total molecular flux when . Total molecular flux
calculated from Eq. (2.13). Net molecular flux calculated from Eq. (2.17).
Figure 4. Relative forward, backward and total molecular flux as a function of the ratio . The
reference flux
corresponds to the total molecular flux when . Forward
molecular flux calculated from Eq. (2.15). Backward molecular flux calculated from Eq. (2.16).
Total molecular flux calculated from Eq. (2.13).
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
12/03/2019 ForsChem Research Reports 2019-03 (10 / 31)
www.forschem.org
Figure 4 presents a comparison between the forward () and the backward () molecular
fluxes, as a function of the velocity ratio. It can be seen that during the macroscopic motion
regimes, the molecular flux exists only in one direction (the direction of flow). Also, for the
thermal motion regime, both forward and backward flux ratios change linearly with the
velocity ratio, at slopes of and , respectively. For that reason, the total flux
remains almost constant during the thermal motion regime.
2.2. Molecular Fluxes in Multicomponent Systems
The results previously obtained in Section 2.1 for pure systems, can be extended to
multicomponent systems by summing the individual molecular fluxes of each species. For a
system composed of different species, the overall forward molecular flux can be calculated
as follows:
(2.18)
where is the molecular mass of species , is the number density of species , and
is the ratio between the average velocity of species and the standard
deviation in the thermal velocity of species perpendicular to the surface.
On the other hand, the overall backward molecular flux will be:
(2.19)
Resulting in the corresponding overall net and total molecular fluxes for the multicomponent
system:
(2.20)
(2.21)
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
12/03/2019 ForsChem Research Reports 2019-03 (11 / 31)
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2.3. Molecular Fluxes in a Permeable Boundary with a Density Difference
So far, it was considered that the density of each molecular species was the same at both sides
of the boundary. While this is true when the systems are allowed to reach equilibrium in the
absence of external forces, it is possible to consider the more general case of a density
difference across the boundary. Again, the mathematical expressions will be obtained for a
single component and then extended to multicomponent systems.
In this case, the forward molecular flux for a single component will be given by:
(2.22)
whereas the backward molecular flux will be:
(2.23)
Now, assuming that the temperature and average velocities are similar at both sides of the
boundary, then the net flux across the boundary becomes:
(2.24)
and the total molecular flux:
(2.25)
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
12/03/2019 ForsChem Research Reports 2019-03 (12 / 31)
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where is the number density of the system at the left side of the boundary, is the
number density at the right side of the boundary, is the difference in number density
between the right side and the left side of the boundary, and is the average number density
between both sides of the boundary.
Generalizing for a multicomponent system:
(2.26)
(2.27)
Please notice that the molecular fluxes are relative to the boundary. Now, given that the net
molecular flux of species relative to the average system velocity is defined as:[2]
(2.28)
then, it can be concluded that:
(2.29)
This last equation indicates that there is a net molecular flux (relative to the system) for each
molecular species caused by a difference in molecular densities at both sides of the boundary.
For a stationary system (), then Eq. (2.29) becomes:
(2.30)
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
12/03/2019 ForsChem Research Reports 2019-03 (13 / 31)
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2.4. Dynamic Effects in Finite Systems with a Permeable Boundary
Let us now consider that the permeable boundary of area virtually separates two sub-
systems at the same temperature with constant finite volumes and , to the left and to
the right of the boundary, respectively. The whole system with volume is
assumed closed. For this system, a non-zero net molecular flux across the boundary will cause a
change in the molecular density of the sub-systems, given by:
(2.31)
Clearly, a steady-state condition will only be possible when . Given that the whole
system is closed, a non-zero velocity of the system cannot sustain a steady state, therefore
. Thus, Eq. (2.31) becomes:
(2.32)
Integrating the first two terms of Eq. (2.32), and rearranging results in:
(2.33)
Replacing in Eq. (2.32):
(2.34)
which can be solved giving:
(2.35)
and combining this result with Eq. (2.33) yields:
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
12/03/2019 ForsChem Research Reports 2019-03 (14 / 31)
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(2.36)
This means that from any arbitrary initial non-equilibrium condition, the equilibrium condition
for each molecular species is reached by an exponential decay function with a characteristic
relaxation time given by:
(2.37)
and reaching the equilibrium values of:
(2.38)
corresponding to the overall number density of molecular species in the total system.
Lighter molecules will reach equilibrium faster, whereas heavier molecules will take longer to
equilibrate. Equilibration times are also reduced by increasing the temperature of the system,
by increasing the ratio of boundary area to total system volume, and by reducing the ratio
. That is, as the relative volume of any of the sub-systems is reduced, the relaxation
times are also reduced. Please notice that so far, molecular interactions have been neglected.
Figure 5 shows an example of two cubic sub-systems of identical volumes (1 liter each) at
, containing initially pure Helium (left sub-system) and pure Radon (right sub-system)
respectively, and with the same initial number density of atoms/m3 (corresponding
to a system pressure of ). Each subsystem is assumed as continuously perfectly mixed.
Both sub-systems are connected at the initial time by a permeable boundary of 0.01 m2. At
these conditions, the relaxation time for Helium is , and for Radon is
. The dynamic behavior of this system is summarized in Figure 6.
It can be observed that although the equilibration occurs very fast (in less than 5 ms), Helium
equilibrates much faster than Radon. It is also interesting noticing that even though the total
initial and equilibrium number densities for both subsystems were the same, the overall
number density of each subsystem changed during the equilibration process. Since Helium
moves faster, the subsystem initially containing Helium loses atoms faster, causing an initial
decrease in its number density followed by an increase in density as Radon atoms begin to
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
12/03/2019 ForsChem Research Reports 2019-03 (15 / 31)
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arrive. Consequently, the subsystem initially containing Radon, increased its number density at
the beginning and then reduced it again until the equilibrium value is reached.
Figure 5. Example system composed of two cubic sub-systems, each containing initially a
different gas at the same number density (Helium and Radon). The system is at medium
vacuum (~ 1 Pa), and at a temperature of 298 K. No collisions between the gas atoms are
considered (highly-diluted ideal behavior).
Figure 6. Dynamic behavior of the Helium-Radon system. Top left: Number density of Helium
atoms. Top right: Overall subsystem number density. Bottom left: Mole fraction of Helium.
Bottom right: Overall subsystem mass density.
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
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Helium mole fraction and the overall mass density of each subsystem presented both a
monotonically convergent behavior, as can be expected. The mole fraction of Helium quickly
increases at the beginning, until Helium concentration equilibrates, and then it continues
increasing slowly as the heavier Radon continued to equilibrate. On the other hand, the mass
density of the left subsystem increases in spite of its decrease in number density, because for
one single Radon atom entering in the subsystem from the right, 56 atoms of Helium should
exit the subsystem in order to keep a constant mass. However, from the characteristic
relaxation times, it can be inferred that for each Radon atom displacing from the right, initially
only between 7 and 8 atoms of Helium would be leaving the left subsystem.
3. Systems contained by Rigid Impermeable Boundaries
When the boundary considered is completely impermeable (no molecule can pass through) and
rigid (all molecular collisions are elastic and the resulting speed of the boundary with respect to
the system is always zero), then all molecules reaching the boundary will bounce with the same
incoming momentum but in the opposite direction for the perpendicular component.
Boltzmann previously suggested this assumption in his analysis of ideal gases.[6]
For this case, the number of molecules hitting the left side of the boundary per unit area per
unit time corresponds to the previously determined forward molecular flux (Eq. 2.18).
Similarly, the number of molecules hitting the right side of the boundary per unit area per unit
time, will be the backward molecular flux (Eq. 2.19).
3.1. Force and Pressure Exerted on the Boundary
The force exerted on the boundary by the collision of a molecule of a species will be:
(3.1)
where is the momentum of the molecule in the direction perpendicular to the
boundary surface, and is the velocity of the molecule before the collision with the rigid
surface. The negative sign indicates that the considered force is exerted on the wall. Now,
since the molecule will have the same momentum but in the opposite direction, then:
(3.2)
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
12/03/2019 ForsChem Research Reports 2019-03 (17 / 31)
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Thus, the total force exerted by collisions of molecules of species with the boundary in the
forward direction is:
(3.3)
where is the number of molecules of species reaching the boundary from the left during
the time interval considered, and is the molecular flux of species reaching the
boundary from the left at positive velocities between and .
The forward flux of molecules of species reaching the boundary with positive velocity
between
and can be determined as:
(3.4)
And therefore, Eq. (3.3) becomes:
(3.5)
where is the maximum distance that must be travelled by a molecule in order to reach the
boundary during the time interval . Now, considering that
and , the
force exerted by molecules of species on the boundary from the left is:
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
12/03/2019 ForsChem Research Reports 2019-03 (18 / 31)
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(3.6)
where
.
Proceeding in a similar way for the molecules reaching the boundary from the right, the force
exerted by species is found to be:
(3.7)
The pressure exerted by the molecules of species on the left side of the rigid boundary is
therefore:
(3.8)
and the pressure exerted by the molecules of species j on the right side of the boundary is:
(3.9)
In closed systems, or in systems where the average velocity of the molecules of is zero, the
pressures at both sides of the rigid boundary become:
(3.10)
(3.11)
corresponding to the ideal gas equation of state.[6]
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
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Figure 7 shows the effect of the dimensionless velocity on the ratio between the pressure
exerted on a rigid impermeable boundary and the corresponding static pressure when .
The forward and backward pressures exerted on the boundary, presented in Eq. (3.8) and
(3.9), can be expressed in terms of the corresponding forward and backward molecular fluxes
as:
(3.12)
(3.13)
Figure 7. Effect of dimensionless velocity ratio on the pressure ratio exerted over a rigid
impermeable boundary. Blue line: Left side of the boundary. Green line: Right side of the
boundary.
For multicomponent systems, the total pressure exerted by the system on both sides of the
boundary will be:
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
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(3.14)
(3.15)
where
and
.
3.2. Equivalent Pressure Definition
In Section 3.1 it was shown that the concept of pressure emerged as the result of the forces
involved during molecular collisions against a physical boundary. On the other hand, for an
ideal gas passing through a virtual or completely permeable boundary, collisions are not taking
place, so there are no forces and thus, there should be zero pressure. However, there is the
apparent idea that a system has a non-zero pressure not only at physical boundaries but also at
any point inside the system. The reason is that pressure can be measured, using a suitable
instrument, anywhere inside the system. However, by placing any instrument for measuring
pressure, a physical boundary is imposed to the system, resulting in molecular collisions against
the boundary and therefore, in pressure.
Thus, as long as no pressure measurements are done in the original system, the pressure of an
ideal gas on any arbitrary virtual permeable boundary is zero. However, it is possible to define
an equivalent pressure of the system at any virtual boundary, as the magnitude of the pressure
that would be exerted by the system on the boundary if it were rigid and impermeable. Or in
other words, the pressure that would be obtained if it were measured at that point.
So basically, the equivalent pressure of a multicomponent system at both sides of a virtual
boundary can be defined as:
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
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(3.16)
(3.17)
Now, assuming that the system conditions at both sides of the boundary are identical, then the
difference in equivalent pressures acting on the boundary will be:
(3.18)
Replacing Eq. (2.9) in (3.18) results in:
(3.19)
Figure 8 illustrates the effect of the dimensionless velocity , on the equivalent pressure
difference for a pure system:
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
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Figure 8. Effect of dimensionless velocity ratio on the dimensionless equivalent pressure
difference for a virtual boundary. Solid blue line: Dimensionless equivalent pressure difference
as the ratio of the equivalent pressure difference to the static ideal gas equivalent pressure.
Purple dashed line: Dimensionless equivalent pressure from the left. Green dashed line:
Dimensionless equivalent pressure from the right.
For large positive system velocities, the pressure difference approximates to:
(3.20)
and for large negative system velocities to:
(3.21)
That is, for fast flowing systems, the magnitude of the equivalent pressure difference is
approximately 4 times the volumetric density of the macroscopic kinetic energy of the system.
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
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3.3. Average Velocity of Molecules reaching the Boundary
The average velocity of the molecules of species colliding with a rigid boundary from the left
can be obtained from the forward pressure, as follows:
(3.22)
Similarly, the average velocity of molecules of species j colliding with the rigid boundary from
the right
will be:
(3.23)
For systems at rest (static), the average velocity of colliding molecules is:
(3.24)
(3.25)
On the other hand, for systems with large macroscopic velocities in the direction of the
boundary, the average velocities of colliding molecules become:
(3.26)
(3.27)
The validity of these expressions can be extended also for virtual boundaries.
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
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4. Molecular Fluxes through Semi-Permeable Boundaries
The results obtained in the previous example of Section 2.4 show an extremely fast mixing of
both systems (less than 10 ms). In practice, the real molecular fluxes are reduced by two main
factors:
a) Influence of molecular collisions, which have been completely neglected for ideal
gases. Such influence can be observed in two different ways: By creating a barrier for
crossing the permeable boundary, and by limiting the motion of molecules in the long
range (molecular diffusion).[7]
b) Influence of additional system boundaries, because even though finite systems were
considered, the effect of system boundaries on the molecular motion was also
neglected.
The barrier effect of molecular collisions at the permeable boundary is depicted in Figure 9. At
any given instant, there will be a certain number of molecules located just at the boundary
(that is, the corresponding Clausius’ molecular sphere of action [8] crosses the boundary at
some point), which cause a physical barrier for all other molecules to cross the boundary.
Figure 9. Molecules at the permeable boundary (red circles) create a barrier for the motion of
all other molecules (green circles) across the boundary.
The fraction of the boundary surface which is unavailable for crossing of molecules of type
because of the physical limitation caused by the presence of molecules of type already at the
boundary can be estimated as the total projected area of the molecules of type at the
boundary divided by the total boundary area:
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
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(4.1)
where is the number density of molecules of type at the boundary, is the volume where
the molecules are considered to be at the boundary, is the area projected on the boundary
by each molecule of type , and is the diameter of Clausius’ molecular sphere of action
between molecules of types and (diameter of collision).
The total unavailable surface fraction for molecules of type j will be:
(4.2)
The unavailable surface fraction can be interpreted as a relative permeability of the
boundary to molecules of type . Thus, such relative permeability might be different for each
molecular species present in the system, a result that makes sense because larger molecules
are more easily obstructed compared to smaller molecules, and therefore larger molecules are
expected to present lower relative permeability values.
Eq. (4.1) and (4.2) are valid approximations only for low-density systems. As the density
increases, superposition of projected areas for the molecules at the boundary will cause
significant deviations in the estimation of the unavailable surface fraction. Also, please notice
that the unavailable surface fraction is limited to the range , so even if the density
increases further, the maximum unavailable surface fraction is 1.
Considering the correction in the effective boundary surface, the corresponding
forward and backward molecular fluxes for molecules of type become (from Eq. ):
(4.3)
(4.4)
Assuming that the average macroscopic motion and temperature of the system are the same
at both sides of the boundary, but the number densities are different, then the net flux across
the boundary is:
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
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(4.5)
This means that if the available surface fraction tends to zero , the flow across
the boundary tends to zero. This effect can be interpreted as “clogging”, and will result in an
impermeable boundary. Thus, flow requires a minimum free space available for molecular
motion.
In the case of a semi-permeable boundary, it is possible to determine the number of molecules
which reach the boundary and do not cross it, but rebound instead. Therefore, we have
forward rebound and backward rebound molecular fluxes:
(4.6)
(4.7)
It is also possible that certain molecules are fixed at the boundary, as can be seen in Figure 10.
Two examples are presented, first there is a grid covering the whole boundary, which can be
considered as a simple representation of a membrane; second, there is a valve (e.g. gate valve),
where the effective area is gradually modified by creating a partial solid barrier. Please notice
that the barrier created in both cases is ultimately a molecular barrier. Thus, it is also possible to
describe their effect on the relative permeability by means of the unavailable surface fraction
concept, as follows:
(4.8)
where
is the total unavailable surface fraction for molecules of type ,
is the
unavailable surface fraction caused by the fixed molecular barrier, and
is the unavailable
surface fraction caused by the self-barrier created by the flowing system.
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
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www.forschem.org
Figure 10. Examples of semi-permeable boundaries: a) Grid, b) Valve
Furthermore,
(4.9)
and
(4.10)
For the case of a perfectly squared grid,
(4.11)
where is the size of the mesh and is the width of the grid. Eq. (4.11) is considered only as an
approximation, because for smaller mesh sizes, additional steric limitations further reduce the
relative permeability of larger molecules.
On the other hand, for the case of valves,
(4.12)
the unavailable surface fraction is a function of the stem opening . Such function
depends on the geometry of the valve. Thus, for linear valves:
(4.13)
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
12/03/2019 ForsChem Research Reports 2019-03 (28 / 31)
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The molecular collisions against the fixed barriers at the boundary, give rise to a pressure
acting on the barrier. The forward pressure will be:
(4.14)
Similarly, the backward pressure can be found to be:
(4.15)
Assuming that the temperature and macroscopic velocity is the same at both sides of the
boundary, then, the pressure difference across the boundary will be:
(4.16)
If the system is at rest with respect to the boundary, the forward and backward pressure acting
on the fixed barrier become:
(4.17)
(4.18)
Finally, there is also a pressure acting on the molecules located at the permeable surface of the
boundary. Such self-exerted pressure can be calculated as:
(4.19)
(4.20)
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
12/03/2019 ForsChem Research Reports 2019-03 (29 / 31)
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Therefore, the total pressures acting on the boundary are:
(4.21)
(4.22)
These pressures correspond to the osmotic pressure of the system at both sides of the semi-
permeable boundary. For static systems they become:
(4.23)
(4.24)
Now, using Eq. (4.8) and (4.10) results in:
(4.25)
(4.26)
For ideal gases, the second term at the right hand of Eq. (4.25) and (4.26) can be neglected. As
density increases, a second order effect of molecular density is expected to arise.
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
12/03/2019 ForsChem Research Reports 2019-03 (30 / 31)
www.forschem.org
5. Conclusion
Molecular fluxes can be calculated for ideal gases using only the equation of motion and the
probability distribution of molecular positions and velocities. Even for systems at rest, there is a
permanent molecular flux reaching the surface of any impermeable boundary inside the
system, in both directions. The net molecular flux is consistent with the continuity equation. At
the limits of the system, where molecules collide with solid boundaries (walls), a pressure on
the walls is developed which is consistent with the ideal gas equation of state for static
systems. For ideal gas systems moving macroscopically perpendicular to the wall, the pressure
will also be a function of the macroscopic velocity. On the other hand, inside the ideal gas
system, as no forces are present, the pressure should be zero. As long as a measurement
device is placed in the system, a rigid boundary is introduced and a non-zero pressure is
measured. Thus, in the absence of a measuring device, it is possible to calculate the equivalent
pressure that would be measured, although the real pressure is zero. For non-ideal gases, a
non-zero internal pressure would exist although much smaller than the corresponding
equivalent pressure. Several different assumptions were considered for obtaining the present
results including: i) Molecular collisions in the gas are negligible (ideal gas assumption). ii)
There are no external forces acting on the system. iii) There is no net macroscopic motion of
the gas parallel to the surface. iv) Molecular positions are uniformly distributed in the system
volume. v) The orthogonal component of molecular velocity is normally distributed. vi) The
system considered is large compared to the size of the molecules. However, future reports will
present the results obtained considering different assumptions. For example: Incorporating the
effect of molecular collisions on the determination of molecular fluxes and pressure;
considering a macroscopic motion of the system parallel to the surface; and assuming smaller
systems.
Acknowledgments
The author gratefully acknowledges fruitful discussions with Prof. Jaime Aguirre (Universidad
Nacional de Colombia) and Prof. Silvia Ochoa (Universidad de Antioquia).
This research did not receive any specific grant from funding agencies in the public,
commercial, or not-for-profit sectors.
Calculation of Molecular Fluxes and
Equivalent Pressure in Ideal Gases
Hugo Hernandez
ForsChem Research
hugo.hernandez@forschem.org
12/03/2019 ForsChem Research Reports 2019-03 (31 / 31)
www.forschem.org
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[5] Hernandez, H. (2017). Multivariate Probability Theory: Determination of Probability Density
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