Derivation of Fundamental Equation Kinetic Theory and Kinetic Molecular Theory

Part 1

Derivation of The Fundamental Equation of The Kinetic Theory

Kinetic-Molecular Theory (model & fundamental laws) N number of molecules of ideal gas m mass of each molecules present in a cube has length l cm Each molecule move in deferent speed u1, u2, u3, uN (cm s-1) Each molecule move in three dimensions (u1x, u1y, u1z) and each molecule make one collision when it cut distance l cm 3

4

Thus, the number of collisions that one molecule make in this direction in one second is?1? ? The impulse before collision is +mu1xand after collision is mu1x, thus, the change in impulse due to collision with the wall of cube is 2mu1x. The change of the impulse in one second equal 2mu1x ?1? ? 2 ? =2??1? 5

Finally, the sum of change of molecules impulse in all dimensions in one second is: 2? ? ?2 ? 2 2+?3 2+ +?? ? 2+?2 Where: ?2=?1 ?2 called mean square speed of one molecule, and it s the exerted force by all molecules in all dimensions : ? =2???2 ? 6

The pressure is the exerted force on the area, so the sum of area in the cube is 6l2 ? ? = 6?2 ? =2???2 l 3is the volume that the molecules move in it, so: 6?3 ???2 ? ? =1 3 ?? =1 3???2 This equation known as The Fundamental Equation of The Kinetic Theory 7

Derivation of kinetic energy based on temperature only 8

Based on the hypotheses of the theory: ?? ? ? ?? = ???????? ? 1 2??2= ???????? ? (a) From the Fundamental Equation of the Kinetic Theory: ?? =1 3???2 ?= ??2 ???= ??2 3?? 3?? 9

3?? 2???=1 ?? ? From equation (a): 2 ??2 2??=1 3 2 ??2 ?? ? 3 2??= ???????? ? ?? ?= ???????? ? The constant is R, Then the equation become: PV=nRT 10

Again, from the Fundamental Equation of The Kinetic Theory: ?? =1 3???2 And from the ideal gas equation: PV = nRT 1 3???2= ??? ?? ??2= 3?? ????2= 3?? (b) 11

1 2????2=3 1 2????2= ????= ?? KE is the kinetic energy of one mole, while ?? for one molecule. ?? =3 2?? ???????=3 2?? 2??? ?? =3 ? ??? 2 ?? =3 2?? ? Where ? = ?? called Boltzmann s constant and has the value 1.37 10-23 J.K-1, R = 8.314 J.mol-1K-1 and NA = 6.022 1023 mol-1 12

Derivation of ideal gas speed from Boltzmann Distribution and Maxwell-Boltzmann Distribution 13

Distribution of Molecular Speeds in an Ideal Gas Root mean square speed is assumed that all molecules move at the same speed. The motions of gas molecules should have distribution of molecular speeds in equilibrium. Statistical mechanics help us to understand gas molecules behaviour. 14

Boltzmann Distribution: ?????????? ?? ????? ?? ?????? ? ?????????? ?? ????? ?? ?????? 0= ? ? ?? ?????????? ?? ????? ?? ?????? ?2 ?????????? ?? ????? ?? ?????? ?1 ?1 ?2 ?? = ? Kinetic energy E = ?? =1 2??2 ?? ?= ? ? ?? Ni : number of molecules between E and E+dE N: total number of molecules 15

16

Maxwell-Boltzmann Distribution The M-B distribution function can be written in terms of energy as: ? ? = ?? ? ?? The distribution function can be written in terms of velocity: ? ? = ?? ??2 2?? But the velocity is in three dimensional: u = ux , uy , uz and du = dux .duy .duz = d3u The constant A is determined by normalization. We will treat this as a probability distribution function normalized so that: ? ? ?3? = ?? ??2 2???3? = 1 17

1 ? = ? ??2 2???3? So: From calculus : ? ?2= ? 3 2 2??? ? Therefore: ? = 3 2 ? ??2 2?? 2??? ? ? ? = In spherical coordinates. The velocity will be: ??= ?sin?cos? ??= ?sin?sin? ??= ?cos? 18

And in spherical coordinates the 3D differential volume element is: ?? = ?2??sin????? ? ? ? ? = ?(?,?,?)?2??sin????? 0 < ? < , 0 < ? < ?, 0 < ? < 2? Where: 19

???2 3 2 ? ? =??? ? 2?? ?2?? = 4? ? 2??? dNi: number of gas molecules that have speed between u and u+du N: total number of molecules k: boltzmann constant which equal to R/NA ? ??2 3 2 1 ? ?? ?? ? 2?? ?2?? = 4? 2??? 20

To find the average speed: ? or ? ? ??2 3 2 ???? ? 2?? ?3?? ? = = 4? ? 2??? 0 0 ? 2?? ? = Let 3 2 ???? ? ? ? ??2 ?3?? ? = = 4? ? 0 0 3 2 ? ? 1 ? = 4? 2?2 mNA= M 21

1 2 1 2 8?? ?? 8?? ???? 8?? ?? ? = = = To find the most probable speed uP : By differentiating f with respect to u and looking for the value of u at which the derivative is zero ) ??(? ?? i.e.: = 0 2?? ? Therefore: ??= 22

To find the root mean square (rms) speed urms: ?? =1 2??2=3 2?? ??2= 3?? ?2=3?? ? ?2=3?? =3?? ???=3?? ? ? 3?? ? ????= 23

Note!: ?2 ? Because: 2+ ?2 2+ ?3 2+ + ?? ? 2 ?1 ?2= While: ? =?1+ ?2+ ?3+ + ?? ? 24

Derivation of The collision frequency and the mean free path 25

The collision frequency and the mean free path The collision number (frequency) that happened by one A molecule per one second ZA is: 2? 2????? ?? 2??= collision s-1 ??= 2????? ?? average speed of A molecules in unit (m.s-1) ?? the diameter of A molecule, also call the diameter of collision in unit (m) nA number of A molecules V the volume in unit (m3) 26

To know how many collision that happened by molecules let suppose a vessel its volume is V and contains gas molecules A. The diameter of A molecule is d and move in average speed u in a cylindrical path its diameter is twice of A (i.e. 2d). Also supposed all other molecules which collide with A molecule is solid. 27

Note that A molecule collide with only that molecules which center inside the cylindrical path. If we suppose that length of cylindrical path equal to average speed ?, therefore the volume of the path is ??2 ?. And if we suppose that number of molecules is ?? then the number of A molecules per volume unit is ??=? ?. Note: ??= ? ?? therefore ??=??? ? ?? ??= ?? average speed of A molecules in unit (m.s-1) ?? the diameter of A molecule, also call the diameter of collision in unit (m) nA number of A molecules V the volume in unit (m3) 28

The relative mean speed ????, is the mean speed with which one molecule approaches another, can also be calculated: 1 2= ????= ??2+ ??2 2?? For different molecules A and B 8?? ? ??+ ?? ???? 1 2= ????= ??2+ ??2 ???? ??+?? is known as the reduced molar mass of the two molecules A and B The term 29

The collision number (frequency) that happened by one A molecule per one second ZA is: 2????? ? ??=??? 2??????collision s-1 = ??? Also: 2? 2????? ?? 2??= ??= 2????? just for similar molecules A But for different molecules A and B, we have to find the diameter of collision???: ???=??+ ?? = (??+ ??) 2 30

Therefore the number of collision that happed by one A molecules with B molecules: 1 2?? ??2+ ??2 2??????= ???? 2 ??= ???? If we want to know the total number of collision that happened by all A molecules, ZAA, we have to multiply ZA by the total number of molecules nA as follows: ???=1 2?? ??=1 2=1 8?? ??? 2?? 2 2 2??????? 2??? ?? 2 =1 ?8?? ?? 2 2 2?? ?? 2 31

Note that we divided on 2 to avoid calculating each collision twice because we deal with similar molecules. But if the collided molecules different, such as A molecules collide with B molecules. Therefore the total number of collision (collision frequency) (this time will be denoted as ZAB) will be find through the equation: 1 2???? ??2+ ??2 2 ???= ???? Again, dAB is the diameter of collision and given by equation: ???=??+ ?? = ??+ ?? 2 32

If A molecule collide with another molecule, it will take a specific distance before collide with another molecule. This distance called mean free path ? = ?? ?? 2?????? ?? 1 = = = 2 ???? 2?? ?? ??? 2??? 2??? 1 ?? ?? ? = = = 2?? 2???? 2?? 2??? 2??? 2??? This equation prove that reducing the pressure in a system will decrease the number of collision and increase length of the mean free path 33