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PHYS 260 4. Kinetic theory of ideal gases Ideal gas law decoded

Kinetic theory of ideal gases • Provides molecular interpretation for macroscopic properties (P, V, T); • Establishes quantitative relationships between kinetic energy and pressure and kinetic energy and temperature; • Allows to determine collision parameters; Assumptions: • • • • •

The gas is assumed to be composed of individual particles; The dimensions of the particles are small compared to the distances between them; The particles are in constant motion, therefore they have kinetic energy; The collisions between the particles are considered perfectly elastic; Neither attractive, nor repulsive forces act between the particles;

2

Kinetic Energy– molecular interpretation 3 & ( 2 ' ) ) ) According to the equipartition principle each degree of freedom contributes equally: !" = * +,-* + * +,.* + * +,/* Translational kinetic energy of a monoatomic particle:

!" =

Diatomic (linear) molecule has not only translational but also rotational kinetic energy: !" = !12345 + !271 + !89: ≠0 ≈0

1 1 1 & ( & ( & ( 2 ' 2 ' 2 '

3 5 2 !" = !12345 + !271 = &' ( + &' ( = &' ( 2 2 2

(3 translational and 2 rotational degrees of freedom) 1 & ( 2 ' 1 & ( 2 ' Polyatomic nonlinear molecule has 3 translational and 3 rotational degrees of freedom: 3 3 !" = !12345 + !271 = &' ( + &' ( = 3&' ( 2 2

1 & ( 2 '

1 & ( 2 '

1 & ( 2 ' 3

Kinetic theory of ideal gases The particle with m mass collides with the wall in the yz plane of the unit cell.

z

its speed: v = (vx2+ vy2+ vz2)1/2 v m x y

At the moment of the collision the particle exerts Fw force on the wall. This force is balanced according to Newton’s third law by −"w force exerted by the wall on the particle: "# =

%& %+ %((+) =( = () = %' %' %'

The particle travels 2x distance (round trip) before colliding with the same wall ./ Select x direction only for now: Number of collisions in unit time: 01 Velocity before the collision: −vx. ; Change of linear momentum in unit time (force): velocity after the collision: vx. ; . 5.6 2. 23/ = "1 = ( 24/ =(2mvx)(01/ )= / 24 1 change of velocity: 2vx change of linear momentum: Δpx = 2mvx Pressure: (+10 (+10 " " = = 71 = = < 8 9: ;9:

4

Pressure of a gas derived from Kinetic Theory

m m

m m

m m

Fraction of particles

Now we consider N particles instead of only one. Still consider movement along the x direction only. The particles have different velocities due to collisions (Maxwell distribution).

z

m

x

y

Most probable speed (vmp) Average speed (vavg ) Root-mean-square speed (vrms) Area under the curve = total number of particles (N)

speed speed We have to introduce a parameter that is describing the speed of the entire population of N particles → vrms N number of particles at vrms will have the same kinetic energy as the N number of particles with their individual speeds. We established on the previous slide that Px ∝ vx2 for one particle. ( "#$%,' =

( ( ( ( + "(,' + ",,' +. . +".,' "*,' /

(

=

. "1,' ∑12*

→

For N particles:

4' =

( /5"#$%,' 6

Pressure of one dimensional gas

/ 5

Pressure of a gas in terms of root-mean-square speed In reality particles can move to any direction. All directions have equal probabilities:

z

) ) ) ) = %&'(,+ %&'(,/ + %&'(,0 %&'(

The pressure then: x y

!=

1 3

) ) ) ) = %&'(,/ = %&'(,0 = %&'( thus, %&'(,-

) #$%&'( 3+

or

!+ =

1 ) #$%&'( 3

Note that it is a form of Boyle’s law: PV = constant ) If $%&'( ∝ T the above equation is consistent with Gay-Lussac’s law

https://opentextbc.ca/chemistry/chapter/9-5-the-kinetic-molecular-theory/

6

Pressure of a gas in terms of root-mean-square speed In reality particles can move to any direction. All directions have equal probabilities: 1 ) ) ) ) ) ) ) ) = %&'(,+ %&'(,/ + %&'(,0 = %&'(,/ = %&'(,0 %&'( = %&'( thus, %&'(,-

z

The pressure then:

!=

or

!+ =

1 ) #$%&'( 3

3

Note that it is a form of Boyle’s law: PV = constant

x y

) #$%&'( 3+

) If $%&'( ∝ T the above equation is consistent with Gay-Lussac’s law

1 ) 345 = #$%&'( 3

Equation of the state for ideal gas: !+ = 345

since

# =6 3

1 ) 6$%&'( 3 M 345 = 7

45 = ) %&'(

%&'( =

345 7

7 https://opentextbc.ca/chemistry/chapter/9-5-the-kinetic-molecular-theory/

Pressure of ideal gas mixtures ,

Dalton’s law (1801):

!"#"$% = !' + !) + !* + ⋯ + !, = - !.

where, P1 , P2 , …Pn are the partial pressures of component 1, 2,…n

./'

Partial pressure is the pressure exerted by the individual component as if they would alone occupy the same container at the same temperature. Dalton’s discovery:

!. = 0. !"#"$% mole fraction of the ith component: 0. =

1. 1' + 1) + ⋯ + 1,

!"#"$% = 0' !"#"$% + 0) !"#"$% + ⋯ + 0, !"#"$% !"#"$% =

,

1, 23 1' 23 1) 23 23 +⋯+ + = - 1. 4 4 4 4 ./'

Temperature dependence of kinetic energy 1 2

/ Average kinetic energy per one particle: !",$%& = *+,-.

We also determined: Combining these two equations: If P= const then V ∝ N, Ek,avg

nA

01 =

2 5! 3 "

and

1 / 3*+,-. 3 2 1 01 = 32!",$%& = 3!",$%& 3 3 01 =

01 =

2 56!",$%& 3

Total kinetic energy per mole gas: Ek

01 = 578

2 5! = 578 3 " 3 !" = 78 for 1 mole 2

Boltzmann constant: 1.38 x 10-23 J/K Universal gas constant: 8.314 J/molK 7 9: = 6 Avogadro’s number: 6.022 x 1023 mol-1 3 !" = 9: 8 for 1 particle 9 2

Particle collisions We are interested in: • Collision numbers • Collisions per unit of time (collision frequency) • Collision density • Distance traveled by the particle between collisions (mean free path) Assumptions: • Particles behave as rigid spheres; • Collisions between them are perfectly elastic; • Assume particle A and B and allow first only A to move. 10

Particle collisions – the collision cylinder Average distance traveled by particle A at unit time = vA,avg collision B x dA

Volume swept out by the imaginary sphere: & '($,*+, ! = #

%$xA

dAB

Notations: dA : diameter of particle A dB : diameter of particle B dAB: collision diameter dAB = (dA +dB )/2

dB

B x

B x

B x collision

limit of collision

no collision

Collision will occur if the center-to-center distance between A and any B becomes smaller than dAB or collision will occur if the center of B is within V volume. 11

Collision frequency, collision density Let us have NB number of stationary particle B in the system besides the one moving A Their number in the unit volume: NB/V The number of centers of B swept out by the cylinder: !" =

' (",*+,-& $%"& .

[s-1] This is the number of collisions per unit time or collision frequency.

Let us have NA number of moving particle A (not one as above) besides the NB number of particle B Their number in the unit volume: NA/V The total number of A-B collisions (if only As are allowed to move): !"& =

' (",*+,- " -& $%"& .'

[m-3 s-1]

This is collision density.

If only A type of particles are present in the system then the total number of A-A collisions: !""

1 ' $%" (",*+,- "' =2 .'

[m-3 s-1]

factor of ½ is needed to avoid double count: A1–A2 collision is the same as A2–A1 12

Collision frequency, collision density cont. Finally, we allow B to move as well. The average speeds of A and B are different since

0" ≠ 0&

but

' ' + (&,*+, For collisions we need the average relative speed: ("&,*+, = (",*+, collision frequency:

' ' ' (",*+, $%"& + (&,*+, .&

!" =

for A-B collisions and

/

!" =

2$%"' (",*+,. /

collision density :

("",*+, = 2(",*+,

for A-A collisions if only As present

/=

' ' ' (",*+, + (&,*+, $%"&

!"& =

"

." .& ",*+,

/'

for A-B collisions and !"" =

0" = 0"

345 and 6

3=

." 78

' 2$%"& (",*+,."' 2/ ' for A-A collisions if only As present 13

Mean free path (!) The average distance particles travel between two collisions !=

#$%&'()* &+',*-*# $( .($& &$/* (./0*+ 12 )1--$%$1(% $( .($& &$/*

for pure A: !=

!=

!=

,3,567 ; 29#3: ,3,567 3 <

<

and ( = ;3 @A ?

29#3:;3 => 29#3: ?@A 14...