Multicomponent Molecular Collision Kinetics

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Jun 1, 2017 - Kinetic theory, Maxwell-Boltzmann distribution, Molecular collision time, Molecular ..... Evaluating the third and fourth integrals in Eq. (4.11): .... Figure 2 compares the exact solution of ( ) (Eq. 5.1) as well as the ..... Second Ed. San Diego: Academic ... Thermodynamics, Statistical Thermodynamics, & Kinetics.

2017-7 Multicomponent Molecular Collision Kinetics: Rigorous Collision Time Distribution Hugo Hernandez ForsChem Research, 050030 Medellin, Colombia [email protected] doi: 10.13140/RG.2.2.26218.31689

Abstract Fluctuations in the position and velocities of molecules in a system are responsible for variable distances travelled and times between consecutive collisions. In this report, the expected value, the variance, and the probability density function of the time between consecutive collisions is obtained for multicomponent systems at thermal equilibrium (following the generalized Maxwell-Boltzmann molecular speed distribution). The probability density function obtained can be expressed in terms of a dimensionless collision time, leading to a standard function (scopd) that describes the general behavior of the system. The properties of the scopd function are presented and discussed. The scopd function can also be approximated (with less than 1% relative probability error) by a simpler empirical function, which can also be used to generate random molecular collision time values. A comparison of scopd with the exponential distribution conventionally used for describing the collision time is presented. This conventional exponential distribution shows a relative probability error of 7%, a relative deviation of 27% in the expected value of the collision time and almost 250% in its variance with respect to the scopd distribution.

Keywords Kinetic theory, Maxwell-Boltzmann distribution, Molecular collision time, Molecular free path, Molecular speed distribution, Standard collision time probability density

1. Introduction Previously,[1] it was shown that the molecular free path distribution travelled between consecutive collisions by individual molecules in a multicomponent system is determined by the nature of the molecules (masses and radii of action), the composition and concentration of the different molecular species, and the temperature of the system. 01/06/2017

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The system considered had a volume of any arbitrary shape, containing in total molecules, all assumed to be hard spherical particles. In the system, there are different types of molecules, with molecules of type , such that: ∑ (1.1) Molecules of type have a radius

and mass

, moving in random directions and at speeds

following the Maxwell-Boltzmann distribution. Considering that the system is at thermal equilibrium, the probability density distribution of relative molecular speeds ( ) in the system is given by:[2]

∑√

( )

(

) (1.2)

where is Boltzmann’s constant and is the equilibrium temperature of the system. The molecular speed ( ) is considered as the magnitude of the molecular velocity relative to the macroscopic velocity of the system. The overall relative free path travelled between collisions by a certain reference molecule ( ) is described by the following exponential probability density function:[1] (

)





(1.3) where

is the number density of molecules of type , and

at which a collision between the reference molecule

is the distance

and a molecule of type takes place.

For this distribution, the expected value and variance of the overall free path are: (

)

∑ (1.4)

(

)

(



) (1.5)

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Multicomponent Molecular Collision Kinetics: Rigorous Collision Time Distribution Hugo Hernandez ForsChem Research [email protected]

For pure (single-component) systems, or systems where the molecular sizes are similar, the mean overall relative free path and its variance become: (

) (1.6)

(

)

(

) (1.7)

Although it is also possible to determine the actual free path travelled by a molecule between collisions, we are presently interested only in the relative free path because it is completely independent of the molecular speed distribution (whereas the actual free path is not). The purpose of the present work is the derivation of the molecular collision time distribution function for multicomponent systems at thermal equilibrium, whose molecular speeds follow the generalized Maxwell-Boltzmann distribution (Eq. 1.2). In Section 2, only the expected value and variance of the molecular collision time are obtained, whereas in Section 3, the full probability density function is derived. In Section 4, the concept of a standard molecular collision time probability density function ( ) is introduced, and its most important properties are presented. Section 5 contains a simple empirical approximation for the calculation of , as well as a simple method for generating random collision times following the distribution obtained in Section 3. Section 6 presents a generalization for multicomponent systems of the results previously obtained. Finally, Section 7 includes a comparison and brief discussion between the molecular collision time distribution and two different exponential distribution approximations, the first one matching exactly the expected value of scopd, and the second, considering the conventional method used in the literature for determining the exponential distribution of collision times.

2. Time between collisions for Maxwell-Boltzmann molecular speed distributions: Expected value and Variance Assuming that the speeds of all molecules remain constant (or practically constant) during the whole path travelled before a collision, then the time to collision ( ( ) ) between the reference molecule

and molecules of type can be calculated as: ( )

( )

(2.1)

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where

( )

is the relative free path between the reference molecule and a molecule of type ,

and is the speed of the molecule of type relative to the reference molecule , given (for a Maxwell-Boltzmann distribution) by:[1] √ (2.2) where

is the reduced mass of the two molecules involved in the collision, and

is the standard Maxwell-Boltzmann random variable.[3] Given that the relative free path is independent of the molecular speed, then, the expected value of ( ) can be expressed as: (

( ))

( )

(

)

(

( ))

(

) (2.3)

The expected value of the relative free path can be calculated as:[1] (

( ))

(2.4) whereas the expected value of the inverse of the molecular speed will be (from Eq. 2.2):

(

)

√ (



( )

) (2.5)

The expected value of the reciprocal of the standard Maxwell-Boltzmann distribution can be obtained from its probability density function:[3] ( ) (2.6) where is any particular realization of .

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Thus, ( )



| (2.7)

Therefore, Eq. (2.5) becomes: (

)





(2.8) Replacing Eq. (2.4) and (2.8) in Eq. (2.3) yields: (



( ))

(2.9) On the other hand, the variance of the collision time (

( ))

(

( ))

( )

is:

( (

( ) ))

(2.10) where (

( ))

((

( )

) )

(

( ))

(

) (2.11)

(

( ))

is given by:[1] (

( ))

(

(2.12)

)

whereas (

)

(

)

(

) (2.13)

and

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(

)



(

)( ) (2.14)

The last result is obtained given that: ∫

[4]

Therefore, Eq. (2.13) becomes (

) (2.15)

Replacing Eq. (2.12) and (2.15) in Eq. (2.11) results in: (

( ))

(

)

(

) (2.16)

Thus, the variance can be expressed by (using Eq. 2.9 and 2.16 in Eq. 2.10): (

( ))

(

) (

(

)

)( (

(

(

)

(

)

)

(

)

(

)

( ) ))

(2.17) with a coefficient of variation:

(

( ))

(

√ (

( ))

√(

( ))

)( ( (

( ) ))

( ))

√ (2.18)

Furthermore, the average molecular collision time ( ̅

( ))

after

consecutive collisions of

molecule with a molecule of type is a random variable that can be expressed as (according to the Central Limit Theorem):[5] ̅

( )

(

( ))



(

( ))



(



) (2.19)

where

is the standard random variable, with mean

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and variance .

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For pure systems, Eq. (2.9), (2.17) and (2.19) become: ( )

√ (2.20)

( )

(

) ( )

(

(

)

)( ( )) (2.21)

̅

(





) (2.22)

3. Time between collisions for Maxwell-Boltzmann molecular speed distributions: Probability density function In order to obtain the probability density function of the collision time, we need to calculate the probability of each particular collision time. Since the relative free path ( ( ) ) and the molecular speed ( ) are independent variables, then the probability of obtaining a time to collision ( ) can be obtained by summing (integrating) the probability of all possible combinations of relative free paths and molecular speeds that result in a time to collision with the value ( ) , as follows: (

( ))



(

( )

( )

)

(

) (3.1)

Using Eq. (2.2) and considering that

(

)

(

)

( )

,* then Eq. (3.1)

becomes:

*

The probability of obtaining a value of

when the velocity is in

, then (

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)

is the same probability of having a velocity

will its reciprocal be

. Similarly, since there is only one value of

because only that results

( ). ForsChem Research Reports www.forschem.org

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(

( ))



(

)

( )√

( )

(3.2) Now, considering that the probability of the relative free path can be described by the following exponential distribution: (

(

( )

( )√

)

)√

( )

(3.3) and that, the differential of the relative free path for a constant value of the molecular speed is: ( )|



( )



(3.4) then, Eq. (3.2) can be expressed as: (

( ))

( )√

( )∫



(3.5) By completing the square in the exponential, the following expression is obtained:

(

( ))



(

( ))

( )∫

( √



( ))

(3.6) Let us now use the following substitutions: √

( )

(3.7)

√ (3.8)

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where is a dimensionless collision time and represents the combined exponential effect of random dimensionless molecular speed and collision time. Then, Eq. (3.6) becomes: ( )





(



) (

)

[∫



∫ (



)







]

(3.9) Integrating by parts, ( )



[

(



) [

√ √

(

∫ (

|

)

∫ (



)

(

)

(



) |



( )] )

|

( )] [

√ √

∫ (

(

)



( ))

( )] √

[





( )

( )]

(3.10) Thus,

( )

[



(

)

(

)

( )]

( ) (3.11)

where

( )





, is the complementary error function.[4]

Figure 1 shows a plot for the probability density function ( ) of the dimensionless collision time ( ) given by Eq. (3.11). Expressing Eq. (3.11) in terms of the collision time

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( )

(from Eq. 3.7) results in:

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(

( ))

[√

(

(

)

( (

( ))

( ))

( )

)

(

(

(√

( ))

( ) )]

)

( )

(3.12) Clearly, it is more convenient dealing with the dimensionless collision time (Eq. 3.11) instead.

Figure 1. Probability density function ( ) of the dimensionless collision time . The plot is presented for the dimensionless time interval .

4. Standard Collision Probability Density Function: Definition and Properties By analyzing Eq. (3.11), it can be observed that the probability density function obtained does not depend on the particular conditions of the system, as they are all considered in the dimensionless collision time variable. Therefore, it is possible to assume that the probability density function of the dimensionless collision time is a standard function (standard collision probability density function, ), which would be defined as: ( )



(

)

(

)

( ) (4.1)

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Given that this function is a probability density function, the integration from equal to : ( )



∫ [ √



(

)





(



)

( )]



( )

to

should be

( )



(4.2) The third integral can be solved as: ∫

( )

( )( )

∫ [

(

( )

)



[

( )



]

|





]

(4.3) This result is obtained because (

( ))

.



Now, considering the following asymptotic expansion for large values of :[6] ( )



(

(

)

(

)

) (4.4)

Then, (

( )

) (4.5)

Thus, Eq. (4.3) becomes: ∫

( )



∫ (4.6)

Similarly, the fourth integral can be evaluated as follows:

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( )



( )( )

∫ ( )

[ [

| ( )

(

( )

∫ ( )

)

√ ( )





] ∫

] (4.7)

From the asymptotic expansion (Eq. 4.4): ( )

(

)





∫ (4.8)

Using the results of Eq. (4.6) and (4.8) in Eq. (4.7) yields: ( )



[ ∫ √ √

(









)





]

∫ (4.9)

Replacing the evaluation of the third and fourth integrals (Eq. 4.6 and 4.9) in Eq. (4.2) results in: ( )





√ (

√ ∫







(





√ ∫







)

)











∫ (4.10)

Thus, the standard collision probability density function ( ) satisfies the two basic conditions of a probability density function: Being non-negative and having and the complete integral equal to . On the other hand, the expected value of the dimensionless collision time is defined as: ( )



( ) √









( )



( ) (4.11)

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Evaluating the third and fourth integrals in Eq. (4.11): ( )



( )( )

∫ [

( )

( (

|

( )

∫ (

)

( )





)

(



( )

[





(

]

)



] ( )



))

)

( )

∫ (









(4.12) ( )



( )( )

∫ ( )

[ [

| ( )

( (

(





)



( )



)





( )









( )



]

))











)









(

( )





)

]

)



( )

∫ (



( √

)

(



( )

∫ (









∫ ∫



(4.13) Thus, Eq. (4.11) becomes: ( )







√ (





(

∫ ∫

( )



√ ( )





√ √

( )















)



( )

∫ √



)







√ (4.14)

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This results means that the expected value of the collision time

(

( ))

(√



)

( )

is (from Eq. 3.7 and 4.14):

) ( )

(



(

) (4.15)

which is a result consistent with Eq. (2.9). Similarly, the variance of the dimensionless collision time is: ( )

( )

( ( ))

( ) (4.16)

where the expected value of the square of the dimensionless collision time is: ( )

( )











( )



( )



(4.17) The third integral of Eq. (4.17) is given by Eq. (4.9), whereas the fourth integral can be evaluated as: ∫

( )

( )( )

∫ [

( )

(

)

( ( √ √ √



√ √



)

( ) ∫ √

(

(

( ) ∫ √



))



√ √

( )

∫ (



)

√ ∫

|

( )

∫ (





( )

[



)



]

]

√ ∫



√ √

√ ∫



)









(4.18) Replacing Eq. (4.9) and (4.18) in Eq. (4.17) results in:

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( )





√ (

√ √



(

√ ∫







√ √



















) )













(4.19) And therefore, ( ) (4.20) Now, the corresponding variance of the collision time

(

( ))

(√

is (from Eq. 3.7 and 4.20):

( )

)

(

)(

)(

(

)

( )

) (4.21)

which, again, is consistent with Eq. (2.17). These results indicate that the standard probability density function for the molecular collision dimensionless time ( , Eq. 4.1) has been correctly identified. Other interesting properties of the

function include: (



) (4.22)



(

)

√ (4.23)



(

) (4.24)

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(

∫ (

)

(

)) (4.25)

(

∫ (

)

(

)) (4.26)



(

(

)

(

))



√ (4.27)



(

(

)

(

))



√ (4.28)



(

(

)

(

)) (4.29)



(

(

)

(

)) (4.30)

From Eq. (4.26) and (4.28) it is possible to make the following approximation: (

)

(

)

(

) (4.31)

where

(4.32) (

) (4.33)

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This approximation predicts exactly the expected value of the distribution, but introduces an ) for the variance. For error of the order of ( , the approximation works pretty well.

5. Empirical Approximation and Random Number Generation of Molecular Collision Times Random numbers belonging to the standard dimensionless collision time distribution can be obtained from its cumulative probability function, assuming that the cumulative function behaves as a uniform random number between and . The cumulative probability function of the dimensionless collision time is given by: ( )



( )



√ √

(

√ ( )

(



|

(

)

( )



( )

|

(

( )

( )

√ √







√ |

( )



)





) ( )







) ( )





( )

(5.1) In order to reduce the complexity in the calculation of the cumulative probability distribution of and facilitate the generation of random molecular collision times, ( ) can be approximated by the following empirical function: ( ) (

)√ (5.2)

Figure 2 compares the exact solution of ( ) (Eq. 5.1) as well as the empirical approximation presented in Eq. (5.2).

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Figure 2. Comparison between the exact solution (Eq. 5.1) of the cumulative probability function for the dimensionless molecular collision time (solid line) and the empirical approximation given in Eq. (5.2) (dotted line).

From Eq. (5.2) it is now possible to generate random realizations of the dimensionless molecular collision time as: ( )

(



) (5.3)

where is a uniform random number between

and and



.

Additionally, random realizations of the actual collision time (3.7) and (5.3): ( ( )(

)



( )

can be obtained from Eq.

)

√ (5.4)

Figure 3 shows a histogram of 10.000 random realizations of good agreement to the expected behavior according to the

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generated from Eq. (5.3), in function (Eq. 4.1).

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Figure 3. Histogram of 10.000 realizations of the dimensionless collision time using Eq. (5.3) (bars), compared to the probability density function described by Eq. (4.1) (solid line) The derivative of the empirical approximation presented in Eq. (5.2) can be used as an approximation of the scopd function, as follows:

( )

( ) (

(

)√



(

)



√ (

)

)



(5.5) Interestingly, this approximation also satisfies the conditions of a probability density function, as it is non-negative (for ) and

∫ √ (

)





∫ (

)



(

)



| (5.6)

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Figure 4 presents a comparison between the exact scopd function (Eq. 4.1) and the approximation given in Eq. (5.5). The difference between both functions can be quantified by the relative probability error ( ) defined as: ( (

) )

( (

)

( )



( )

) (5.7)

where

and

are the probability density functions of two probability distributions

and

.

Figure 4. Comparison between the exact function (Eq. 4.1) of the dimensionless molecular collision time (solid line) and the empirical approximation given in Eq. (5.5) (dotted line). The relative probability error between the exact

∫ ||

and its approximation is

||

( ) √ (

)



(5.8) The integral is evaluated numerically, yielding a value of largest error is found at a dimensionless collision time of in units of probability density.

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relative probability error. The , with an absolute deviation of

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Furthermore, the expected value of the dimensionless collision time using the approximated is: ( )

∫ √ (





)



(



)



(



)





||

∫ (

)

[

(





)

||



]



(5.9) Given that the exact expected value of

is ( )

, then the relative error in the



determination of the expected value using the empirical approximation is

.

Similarly, the variance of the dimensionless collision time obtained using the approximation in Eq. (5.5), is: ( ) √ ( √

) (



) (5.10)

Given that the exact variance of

( )

is

, then the relative error in the

determination of the variance using the empirical approximation is

.

6. Molecular Collision Time Distribution in Multicomponent Systems So far, all the results obtained have been for the time to collision between a reference molecule and any molecule of type . Considering that different species are present, the probability of a time to collision between the reference molecule and any other molecule in the system will be given by: ( )

( ) (6.1)

where

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∑√ (6.2) ∑√ (6.2) Thus, Eq. (6.1) becomes: ( )

(∑ √

)

(∑ √

) (6.3)

and therefore, the probability density function for the overall collision time of reference molecule is: ( )

(∑ √

)

(∑ √

) (6.4)

For pure systems, Eq. (6.4) becomes: ( )



(√

) (6.5)

The expected value of the overall collision time will be (using Eq. 4.15): ( )



( )





(∑ √

)

(∑ √

)

√ (6.6)

For pure systems, the expected value of the collision time becomes (given that

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⁄ ):

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( )

(



) (6.7)

Similarly, the variance of the collision time for a multicomponent system is (using Eq. 4.15 and 4.21): ( )



( ) ∫

( ( )) (∑ √

)

(∑ √

)

( ( )) (∑ ( (∑

)



(∑



)

) )



(6.8) For a pure system, Eq. (6.8) becomes: ( )

(

)(

)(

) (6.9)

7. Comparison with the Exponential Approximation for the Molecular Collision Time Distribution Traditionally, the distribution of the time between events has been considered, in the scientific literature, as an exponential distribution (or negative exponential distribution).[5] However, in this work it has been found that although the distribution of molecular collision time is a decaying function similar to the probability density function in an exponential distribution, the mathematical expressions are quite different. Let us first consider that the function is approximated by an exponential probability density distribution, such that the expected value of the collision time is exactly the same.

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For an exponential distribution, the probability density function is given by: ( ) (7.1) with an expected value: ( )





(

)

[



]

∫ (

)

(7.2) Considering the dimensionless collision time, the expected value previously obtained is (Eq. 4.14)

( )



. It can be concluded that the parameter for an equivalent exponential

distribution will be: √

(7.3)

Thus, the equivalent exponential probability density is: ( )





(7.4) The comparison of the exact presented in Figure 6.

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function and the equivalent exponential function is

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Figure 6. Comparison between the exact function (Eq. 4.1) of the dimensionless molecular collision time (solid line) and the exponential approximation given in Eq. (7.4) (dotted line). The relative probability error (Eq. 5.7) between the exact approximation is ( )

∫ |





and the exponential

| (7.5)

The integral is evaluated numerically, yielding a value of relative probability error. The largest error is found at a dimensionless collision time of , with an absolute deviation of in units of probability density, times larger than the maximum deviation obtained with the empirical approximation presented in Eq. (5.5). The expected value is exactly the same because that was the condition given for the determination of the exponential function. On the other hand, the variance of the dimensionless collision time obtained using the exponential approximation in Eq. (7.4), is: ( )

( )

( ( ))

∫ √



(7.6) Given that the exact variance of

is

( )

, then the relative error in the

determination of the variance using the exponential approximation is

.

These results indicate that the exponential distribution is a good approximation only if the expected value is of interest, but it may introduce significant errors if the whole distribution needs to be considered. Please notice that the parameter in the exponential distribution for the actual collision time ( ) of the reference molecule with molecules of type (not in dimensionless units) is:



(

)

(

( ))

(

( ))

(7.7) This is an important consideration, because the conventional definition used for the frequency parameter is:[7]

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Multicomponent Molecular Collision Kinetics: Rigorous Collision Time Distribution Hugo Hernandez ForsChem Research [email protected]

√ (7.8) which corresponds to the average molecular speed divided by Maxwell’s actual mean free path. Thus, the probability density function of the molecular collision time

(

(

))





( )

becomes:

( )

(7.9) which in terms of the dimensionless collision time defined in Eq. (3.7) is: ( )





(7.10) Figure 7 compares the distribution obtained by using Eq. (7.10), and the scopd function (Eq. 4.1). It is quite surprising to find such a good agreement between both results, despite the conceptual differences between both approaches.

Figure 6. Comparison between the exact function (Eq. 4.1) of the dimensionless molecular collision time (solid line) and the conventional exponential distribution given in Eq. (7.10) (dotted line). 01/06/2017

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Multicomponent Molecular Collision Kinetics: Rigorous Collision Time Distribution Hugo Hernandez ForsChem Research [email protected]

The relative probability error between the exact distribution is: ( )

∫ |

and the conventional exponential





| (7.11)

The integral is evaluated numerically, yielding a value of relative probability error. The largest error is found at a dimensionless collision time of , with an absolute deviation of in units of probability density, almost times larger than the maximum deviation obtained with the empirical approximation presented in Eq. (5.5). The expected value of the dimensionless collision time using the conventional exponential distribution is: √

( )

(7.12) And given that the exact expected value of

is ( )



, then the relative error in

the determination of the expected value using the conventional exponential approximation is . Similarly, the variance of the dimensionless collision time obtained using the conventional exponential approximation (Eq. 7.10), is: ( ) (7.13) And given that the exact variance of

is

( )

, then the error in the

determination of the variance using the conventional exponential approximation is

.

8. Summary The time between consecutive collisions of molecules in a system is a random variable influenced by the probability distribution of the molecular speeds and the probability distribution of the relative position between molecules. If the distribution of molecular speeds and positions are known, it is possible to identify the probability distribution of collision times. 01/06/2017

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Multicomponent Molecular Collision Kinetics: Rigorous Collision Time Distribution Hugo Hernandez ForsChem Research [email protected]

For the particular case of a multicomponent system at thermal equilibrium, following the generalized Maxwell-Boltzmann distribution, it was found that the collision time probability density of a single type of collision (between a reference molecule and any molecule of type ) can be described by the scopd function (Eq. 4.1): ( )



(

)

(

)

( ) (8.1)

where is a dimensionless collision time given by (Eq. 3.7): √

( )

(8.2) such that (from Eq. 4.14 and 4.20): ( )

√ (8.3)

( ) (8.4) The

function can be empirically approximated by (Eq. 5.5): ( ) √ (

)



(8.5) with a relative probability error of , a relative error in the determination of the expected value of , and a relative error in the determination of the variance of . This empirical approximation can be used to generate random dimensionless collision numbers using the following equation (Eq. 5.3): ( )

(



) (8.6)

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Multicomponent Molecular Collision Kinetics: Rigorous Collision Time Distribution Hugo Hernandez ForsChem Research [email protected]

where is a uniform random number between

and and



.

For multicomponent systems, the following expressions are obtained (from Eq. 6.4, 6.6, 6.8, and 2.19):

( )

(∑ √

)

(∑ √

) (8.7)

( ) ∑





(

( ))

(8.8) (

( ) (∑

) )



(8.9)

̅

( )



(

( ) ∑



(∑



) )

√ (8.10) For binary systems, the following property of the (

)

function results useful (Eq. 4.31): (

)

(

) (8.11)

where

and

(

)

.

Two exponential distributions are possible approximations to scopd. The first approach is (Eq. 7.4): ( )





(8.12) 01/06/2017

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Multicomponent Molecular Collision Kinetics: Rigorous Collision Time Distribution Hugo Hernandez ForsChem Research [email protected]

with a relative probability error of

, ( )



(exact) and

( )

(17.2% error).

The second is the conventional approach (Eq. 7.10): ( )





(8.13) with a relative probability error of

, ( )



(27% error) and

( )

(247% error).

This model is obtained by considering the rate parameter as the average molecular speed divided by Maxwell’s actual mean free path. Despite the fact that certain correction factors are missing, the approximation to the actual distribution is very good. However, significant errors are introduced in the determination of the expected value and variance of the collision times. In conclusion, a more precise description of the random behavior of the molecular collision time will certainly provide more accurate predictions of the properties of molecular systems, although different simpler approximations (including the conventional exponential approximation) can be used for practical purposes.

Acknowledgments The author gratefully acknowledges the helpful discussions with Prof. Jaime Aguirre (Universidad Nacional de Colombia), as well as his contributions by checking the analytical integrations and reviewing the whole manuscript. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

References [1] Hernandez, H. (2017). Molecular Free Path Statistical Distribution of Multicomponent Systems. ForsChem Research Reports 2017-6. doi: 10.13140/RG.2.2.15605.58088. [2] Hernandez, H. (2017). On the generalized validity of the Maxwell-Boltzmann distribution and the zeroth Law of Thermodynamics. ForsChem Research Reports 2017-4. doi: 10.13140/RG.2.2.26937.16480. 01/06/2017

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Multicomponent Molecular Collision Kinetics: Rigorous Collision Time Distribution Hugo Hernandez ForsChem Research [email protected]

[3] Hernandez, H. (2017). Standard Maxwell-Boltzmann distribution: Definition and properties. ForsChem Research Reports 2017-2. doi: 10.13140/RG.2.2.29888.74244. [4] Polyanin, A. D., & Manzhirov, A. V. (2007). Handbook of Mathematics for Engineers and Scientists. Boca Raton: Chapman & Hall/CRC. [5] Roussas, G. G. (1997). A Course in Mathematical Statistics. Second Ed. San Diego: Academic Press. [6] Shepherd, M. M., & Laframboise, J. G. (1981). Chebyshev approximation of (1+ 2𝑥) 𝑥 (𝑥²) 𝑥 in 0≤ 𝑥

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