Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Boltzmann Distribution Law, Slides of Physics

number density nV (E ) is called distribution function

Typology: Slides

2020/2021

Uploaded on 05/24/2021

millyx
millyx 🇺🇸

4.7

(9)

249 documents

1 / 29

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Boltzmann Distribution Law
The motion of molecules is extremely chaotic
Any individual molecule is colliding with others at an
enormous rate
Typically at a rate of a billion times per second
We introduce the number density nV (E )
This is called a distribution function
It is defined so that nV (E ) dE is the number of
molecules per unit volume with energy between E
and E + dE
From statistical mechanics, the number density is
nV (E ) = n0e E /kBT Boltzmann distribution law
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff
pf12
pf13
pf14
pf15
pf16
pf17
pf18
pf19
pf1a
pf1b
pf1c
pf1d

Partial preview of the text

Download Boltzmann Distribution Law and more Slides Physics in PDF only on Docsity!

Boltzmann Distribution Law

  The motion of molecules is extremely chaotic

  enormous rateAny individual molecule is colliding with others at an  Typically at a rate of a billion times per second

  We introduce the

(^) number density

(^) n V (^) ( E (^) )

 This is called a distribution function

 It is defined so that

(^) n V (^) ( E (^) ) (^) dE

(^) is the number of

molecules per unit volume with energy between

E

and

E

(^) dE

 

n From statistical mechanics, the number density is

V

E

n

0 e

(^) – E (^) / k B T

Boltzmann distribution law

  divided byenergy state varies exponentially as the energyprobability of finding the molecule in a particularThe Boltzmann distribution law states that the

(^) k B T

  equilibrium is shown at rightof gas molecules in thermalThe observed speed distribution

  P ( v ) is called the

(^) Maxwell-

functionBoltzmann speed distribution

P ( v )

P ( v )

  The peak shifts to the right as

T

(^) increases

 with increasing temperatureThis shows that the average speed increases

  possible speed is 0 and the highest is infinityThe asymmetric shape occurs because the lowest

N

v=NP

v ) 

  P ( v ) is a probability distribution function, it gives

intervalthe fraction of molecules whose speeds lie in the

(^) dv

(^) centered on the speed

(^) v .

  on the mass and on temperatureThe distribution of molecular speeds depends both

  liquids than in gasesof gases even though the speeds are smaller inThe speed distribution for liquids is similar to that

  where atoms can still move around.helium, which is known to be a “quantum solid”anymore, they vibrate. The only exception is solidIn solids, atoms do not have translational energy

P ( v ) dv

(^) = (^1)

(^0) 



Example: What is the

(^) rms

(^) speed of hydrogen at

T

=300 K? How much slower are O

2 (^) molecules

compared to H

2 ? 

M (H (^2) ) = 2.016 g/mole

v rms (^) = 1930 m/s at

(^) T (^) = 300 K

v rms (^) =

3 M RT

v rms (^) (O 2 )

v rms (^) (H 2 ) = M H 2

M O 2

v rms (^) (O 2 )

v rms

(^) (H

2 ) =

(^322) (^) =

4 1

O

2 (^) is 4 times slower than H

2 

H (^2) 

Kinetic Theory of the Gases, Part 2

The Mean Free Path

  fashionmolecules in a randoma gas collides with otherA molecule moving through

  referred to as aThis behavior is sometimes

(^) random-walk

process

  The

(^) mean free path

decreasesmolecules per unit volume increases as the number of

12/08/



Lecture 6



  The mean free path,

l , equals the average distance

v  t traveled in a time interval

t divided by the

number of collisions that occur in that time interval:

  collision frequency The number of collisions per unit time is the

  collisionThe inverse of the collision frequency is the

(^) mean free time

l

v (^)  t

 (^) d 2 v (^)  t

( ) n v = 1

 (^) d (^2) N / V

f

d

2

vN / V

(units: particles s

) 

  of an ideal gaschange the temperatureSeveral processes can

  Since

T

(^) is the same for

each process,

E

int (^) is also

the same

  the work (1st law)the different paths, as isThe heat is different for

  temperature isparticular change inassociated with aThe heat and work

(^) not

(^) unique

Molar Specific Heat

  An ideal monatomic gas contains non-interacting atoms

  increasing the translational kinetic energy of the gaswith a fixed volume, all of the energy goes intoWhen energy is added to a monatomic gas in a container  There is no other way to store energy in such a gas

  Therefore,

E

int (^) is a function of

T

(^) only:

E

int (^) = 3/

(^) nRT

 The exact relationship depends on the type of gas

  At constant volume,

Q
E

int (^) = (^) nC

V (^)  T



onesThis applies to all ideal gases, not just monatomic

  Solving for

C

V (^) gives

C

V (^) = 3/

R

(^) = 12.5 J/mol

(^). K^

 For all monatomic gases

 results for monatomic gasesThis is in good agreement with experimental

  In a constant-pressure process,

E

int (^) = (^) Q

(^) + (^) W

(^) and

C

P (^) – (^) C

V (^) = (^) R



C This also applies to any ideal gas P (^) = 5/

R

(^) = 20.8 J/mol

(^). K^

  We can also define



(^) C P

C V

= (^5) R / (^) 2

3 R / (^) 2 (^) =

3 5 (^) = (^1)

. 67

Molar Specific Heats of Other Materials

 

vibrational motions of the moleculesinclude contributions from the rotational andThe internal energy of more complex gases must

 

the thermal expansion is small andconstant pressure, very little work is done sinceIn the cases of solids and liquids heated at

C

P

and

C

V

are

approximately equal

Equipartition of Energy

  must be taken into accountcontributions to internal energyWith complex molecules, other

  center of masstranslational motion of theOne possible energy is the

  various axes also contributesRotational motion about the  around theWe can neglect the rotation

(^) y (^) axis in a

compared to thediatomic since it is negligible

(^) x

(^) and

(^) z (^) axes

Linear



molecule



  freedom (each degree = 1/2The translational motion adds three degrees of

R

  degrees of freedom (each degree = 1/2The rotational motion in a linear molecule adds two

R

  more degree of freedom (each degree = 1The vibrational motion for a diatomic adds one

R

  Therefore,

E

int (^) = 7/

(^) nRT

(^) and

C

V (^) = 7/

R

  the molar specific heat is a function of temperatureThis is inconsistent with experimental results as

  monatomic gasAt low temperatures, a diatomic gas acts like a   C

V (^) = 3/

R

  At about room temp. the value increases to

C

V (^) = 5/

R

 not vibrational energyThis is consistent with adding rotational energy but

  At high temperatures, the value increases to

C

V (^) = 7/

R

 and translationalThis includes vibrational energy as well as rotational

  are more complex (3For molecules with more than two atoms, the vibrations

N

atoms

-6(5) degrees of freedom)

  The number of degrees of freedom becomes larger

  the more “ways” there are to store energyThe more degrees of freedom available to a molecule, 

This results in a higher molar specific heat