Intro Physics II
Administrivia
Intro Physics II Physics 11b
! !
Web quizzes start Monday (none this week) Regular office hours start this week !
Temperature Maxwell Distribution
!
!
Bonus office hours this week (no section) !
!
!
! !
! ! ! !
TF’s taking OH during regular section times ! See website for locations ADF OH Wed @ 8 Lyman 238 ! There will be food
HW #1 this week due at 4 PM (typo on HW page); 4 PM in future as on syllabus Should be completing course sectioning on website I will be outside door in Science Center after class !
Survey
ADF Tue, Wed @ 1 See web for TF office hours
Hard to hang around here—class in B @ 11:30
What we learned last time
Most of you are biochem Most sophomores and juniors Many taking MCAT Several mention PowerPoint concerns !
!
0th law of thermodynamics !
! !
Please speak up if lecture ! Goes by too fast ! Is just reading from the slides ! Becomes “as boring as hay”
“If two bodies are each in equilibrium with a third, they will be in equilibrium with one another”
Pressure / Density What is temperature? !
Molecular model of ideal gas ! ! !
Negligible volume occupancy Negligible interactions Elastic collisions
p=
Goals for today ! ! !
What is temperature? Ideal gas thermometer Maxwell Distribution
mρ v 2 3
On Averages and Order
!
x =
∑ xi
In general 2
2 i
xi x ≠ x = ∑ x =∑ N ! In same way, in general 2
2
∫
[ f ( x )] dx ≠
x
x2
-3
9
3
9
2
<x>=0 <x>2=0
<x2>=9
2
[∫
f ( x )dx
]
1
Ideal Gas Aside p=
Two volumes of ideal gases
mρ v 2
2 2N m v pV = 3 2
3
!Let us leave a two different dilute noble gases (say, He and Ne) in an isolating containers for a long time, isolated from one another by a wall !
Ought to look a little familiar! !Take it apart !
p=
mN v 2 3V
Looks Looks like like n KE (i.e. T) !
Ideal Gas law is Physics! !
!
We would find that
!
!
Will see later today how to turn this into IGL
!
Gas Volumes in Contact !Now raise the separating wall and wait for things to settle down !
!
!
The average speed of
Atoms zoom around, bounce into one another, zoom around… What happens to the average speed of He and Ne atoms?
He atoms remains the same Ne atoms remains the same But He and Ne speeds may still be—and in general are– different from each other
! ! !
Three cases ! !
Nothing He speeds up, Ne slows down Ne speeds up, He slows down
!Since total KE is conserved in collisions, the following are not possible ! ! ! !
!This is known as “thermal equilibrium”
Perhaps the average speed
!
! !
He atoms increases Ne atoms decreases
Total kinetic energy content of He rises, Ne falls !Energy had to flow from Ne to He !
!
Need not mean that total energy of Ne was larger than He to start with!
Or, if
Third objects !Let’s go back to case 1 !Instead of lowering the barrier, let’s open the door and insert an object onto He side !
! !
Ne atoms increases He atoms decreases
Energy flowed from He to Ne
Popsicle, Chunky® Soup, whatever
Three cases for He atomic speeds !
! !
!
!
Putting the two gas volumes into contact did not change the energy content of either gas
!
Both speed up Both slow down One speeds up, the other constant One slows down, the other constant
Cases 2 & 3
Average atomic speed in each gas is constant over time Average atomic speeds may be different in the two gases
Case 1: Nothing
!
!
An average speed of atoms exists and could (in principle) be calculated from measurements Similarly, a total kinetic energy of all atoms may be calculated
!
Stay the same Speed up Slow down
Remarkable fact:
!
If you had put the object on the Ne side, the result would have been the same
2
0th law of thermodynamics ! !
This fact is known as the 0th law of thermodynamics Often restated as !
!
!Let us have two volumes of He gas with different average speeds !Remove partition !After some time, the new system has settled down and has a new average speed !This average must be between the average speeds of the two original gas volumes
“If two bodies are each individually in thermal equilibrium with a third body, they are in thermal equilibrium with each other”
It is an experimental, not theoretical, truth !
!
!
Two Containers of He
If two iron filings attract one another, they do not each attract a third NFL (in regulation): Buffalo tied San Diego tied Patriots tied Seattle ! But Buffalo and Seattle did not tie!
!
As the third is arbitrary, it is universal ! !
All objects in equilibrium with our He gas are somehow “the same” We can assign each group of objects in equilibrium a label (called “T”)
!
m
(N
1
2 2
v + N 2 v + N 1∆
)
2 1 KE 2 = N 2 KE 2 = N 2m v 2
)
= (N 1 + N 2 )
m
2 2
( v )+ N
= (N 1 + N 2 ) KE 2 + N KE tot =
KE tot N1 N 2
= KE 2 +
N1 m ∆ N1 N 2 2
> KE2
= KE 2 +
1
m 1
m
∆
!
∆
= KE1 −
N1
N2 m ∆ N2 2
< KE1
! !
!
!
!
!
It has ice, it has water, it has vapor
Let us find a volume of He whose average speed is such that when we insert this glass container into it Nothing happens to the average speed of the He
We label this volume of He with the number “273.16”
More concretely 0
273.16
Observing average speeds of atoms in the gas
We also can recognize objects in equilibrium with ideal gas volumes !
!
Label 0: ideal gas volume in which atomic speeds are 0 Let us wander around the universe until we find a closed glass container in isolation with the following property !
N1 + N 2 m N2 m ∆ ∆− 1 N1 N 2 2 N N2 2
We can recognize ideal gas volumes in equilibrium with one another !
!
!
!
Temperature: The Argument !
Exists and is single-valued Is ordered ! Is continuous It may be mapped onto the real numbers !
(
KE tot =
We take it as self-evident that if a set of labels T !
2 1 2 KE1 = N 1 KE1 = N 1m v1 = 1 mN 1 v 2 + ∆ 2 2
!
2
1
v12 ≡ v 22 + ∆
This implies an ordering to our label T
Temperature: the ideal gas thermometer
2 Volumes of He Label 1,2 v1>v2
Mathematical proof following page
If we re-separate the gas atoms, the faster has slowed down and the slower sped up !
Again, observing average speed of atom in gas
0th law: we can form collections of objections with the same label T
Kinetic energy considerations in ideal gases suggest labels ordered Define label T=0 as belonging to the collection of ideal gas volumes which have no atomic motion, and objects in equilibrium with these volumes Define the label T=273.16 as belonging to the collection of ideal gases & objects in equilibrium with a closed glass tube containing water, ice, and steam
3
Is this Physics? !
!
Phases of H2O? Properties of He atoms and gases of noble elements? Sounds an awfully lot like !
!
!
!
!
!
!
!
!
Pressure = mρ/3 Pressure at T=0 is 0 Experimental fact !
!
Pressure related to speed Pressure well-defined at T=0
The water vapor inside any magic glass tube is always the same Pressure at our special P=4.58 torr
T point is also special
!
The Concept
Must do so in a way that does not depend on the particular gas we use Use three observations we’ve made so far
!
! ! !
Readily and unambiguously measurable
Choose your favorite gas and make it very dilute Choose a volume of it whose temperature label is 273.16 Note the gas pressure on the walls of the container ≡p3
Ideal Gas Thermometer p = patm + ρgh
The pressure on the walls will be reduced
Let the two volumes come into equilibrium with one another Label each other volume as follows !
!
No motion is no motion
Carry your gas around to other gas volumes which are labeled neither 0 nor 273.16 As your reference comes into contact !
!
!
Suggest that pressure may play a useful role
Temperature: the ideal gas thermometer !
Useful observations
!
How do we label temperature points in between? !
He
0th law tells us this
Absolute 0 is also independent of gas composition !
!
Ne
Must also describe how to label other volumes of He gas Must demonstrate that it did not essentially depend on our use of He rather than Ne, Ar, or any other monatomic gas
If I didn’t need to mention this for the argument, then obviously the argument doesn’t depend on whether my gas has 2 protons, 10 protons, or weighs 4 Daltons or 20. Conclusions about ordering, etc. are unaltered We know that we could use any ideal gas to define 273.16 !
!
!
I was raised by wolves on a tropical Pacific island Our periodic table looks like this
Chemistry
Universality
!
!
The next step is to demonstrate that our labeling is universal, translatable, and applicable to any object in the universe !
!
Did I mention this?
+h
T=273.16(p/p3)
Does this depend on the gas you use?
4
Does it depend on the gas? ! ! !
p 273.16 p3
In fact, yes, very slightly Modify procedure Use limit of a series of gas volumes !
373.5 Nitrogen
Each more dilute than the one before
!
Measure assigned temperature as a function of concentration Extrapolate to 0 concentration
!
The extrapolation is independent of gas used!
!
Density-Label: Steam @ 1 atm
!
373.25
373.00
We can recognize ideal gas volumes in equilibrium with one another !
!
!
!
! !
!
Observing average speeds of atoms in the gas
We also can recognize objects in equilibrium with ideal gas volumes 0th
Again, observing average speed of atom in gas
law: we can form collections of objections with the same label T
Kinetic energy considerations in ideal gases suggest labels ordered Define label T=0 as belonging to the collection of ideal gas volumes which have no atomic motion, and objects in equilibrium with these volumes Define the label T=273.16 as belonging to the collection of ideal gases & objects in equilibrium with a closed glass tube containing water, ice, and steam
40
!We did glide over one subtle point
!
There exists a single average speed for the ideal gas volume However, not every molecule is at this speed
!
We can recognize ideal gas volumes in equilibrium with one another
!
We also can recognize objects in equilibrium with ideal gas volumes
!
!
Glancing collisions would cause some to slow down /stop Subsequent collisions between stopped & moving atoms would slow down to intermediate values But average kinetic energy remains same
Observe average speeds of atoms in the gas
!
Again, observe average speed of atom in gas
!
! ! !
!
!
0th law: we can form collections of objections with the same label T Kinetic energy considerations in ideal gases suggest labels ordered Define label T=0 as belonging to the collection of ideal gas volumes which have no atomic motion, and objects in equilibrium with these volumes (although actually, there are none of these!) Define the label T=273.16 as belonging to the collection of ideal gases & objects in equilibrium with a closed glass tube containing water, ice, and steam ! Recognize that pressure correlates well to temperature
We define all other labels using an ideal-gas thermometer T=273.16lim(p/p3) Definition independent of gas we use in the thermometer
! !
Review of Mechanics
v
!
KE: Before = After (‘)
!
1
Total Momentum Before = After (‘)
!
1
v 2
Stopped
2
mv
1
1
+ mv
2 2
1
2
1
= mv ' + mv ' 1
2
r
2
r
1
!E.g. even if they all started that way !
120 p (kPa) 3
80
Temperature: The Argument
More on ideal gases: Maxwell !
Hydrogen
A unique, ordered label for EVERY OBJECT IN THE UNIVERSE
Temperature: The Argument !
Helium
Unique T independent of gas
1 2
v 2
!
1 1x 2 1 2 y2 1 1 x 2 1 1 y 2 mv + mv = mv ' + mv ' 2 2 2 2 2 2 1 1 + mv 2 x ' + mv 2 y '
Solved if v2x’=v2y’=0
1
r
'
2
'
In x
r
1
!
r
2
r
1
'
r
2
'
In y
r m v2
r r m v1 '+ mv2 '
5
Probability Distributions !End up with a distribution of velocities in the gas !Of course, average kinetic energy cannot change (energy conserved) !Define a distribution P(v) !
!Derivation beyond scope of the course
∫ P (v )dv = 1
3 2
− Mv M 2 2 RT P (v ) = 4π v e 2πRT
2 may not equal !Average KE is given by !
! !
!
=< ½ mv2>= ½ m Not ½ m2
Result ! !
=3/2 RT m/M = 3/2 kBT ! k =Rm/M=R/N B A
(vmax)2= 2 RT M
!
v v+dv
∂v
Proportional to temperature!
!
* Some texts use f(v) = NP(v); f(v)dv is the
vmax
number, not fraction, between v and v+dv
!Average value of v may not equal vmax
Long tail to right skews average high
2
!Can find maximum (most probable speed) by setting ∂P = 0
Some comments on distribution !
P(v)
Result is important
!
The product P(v)dv is the fraction of molecules with speed between v and v+dv
Normalization condition
There’s a reason Maxwell got a Nobel prize
!
P(v)
!Total fraction of all molecules must be one* !
Maxwellian Distribution
3/2
2
v
∫
∫v e 4
−C
v2 2
C
2
=3
RT
v 2
Temperature !
dv
!
dv
!This is a hard but doable integral !Do need answer today
v
!
2
4 − C ve v P (v )dv 4π 2π =∫ = ∫ 1 P ( v )dv 2
Summary
!
!
Scale defined using triple point of water, absolute 0, and pressure Use ideal gas thermometer to label temperature of everything else We’ll never need to think about T again! ! “There are no subtleties”
p T = 273.16 lim ρ →0 p 3
Maxwell Distribution !
Describes probability of different molecular speeds in the gas
3 2
− Mv
2
M 2 2 RT P (v ) = 4π v e 2 RT 2
v
=3
RT
6
Intro Physics II Physics 11b
! !
Web quizzes start Monday (none this week) Regular office hours start this week !
Temperature Maxwell Distribution
!
!
Bonus office hours this week (no section) !
!
!
! !
! ! ! !
TF’s taking OH during regular section times ! See website for locations ADF OH Wed @ 8 Lyman 238 ! There will be food
HW #1 this week due at 4 PM (typo on HW page); 4 PM in future as on syllabus Should be completing course sectioning on website I will be outside door in Science Center after class !
Survey
ADF Tue, Wed @ 1 See web for TF office hours
Hard to hang around here—class in B @ 11:30
What we learned last time
Most of you are biochem Most sophomores and juniors Many taking MCAT Several mention PowerPoint concerns !
!
0th law of thermodynamics !
! !
Please speak up if lecture ! Goes by too fast ! Is just reading from the slides ! Becomes “as boring as hay”
“If two bodies are each in equilibrium with a third, they will be in equilibrium with one another”
Pressure / Density What is temperature? !
Molecular model of ideal gas ! ! !
Negligible volume occupancy Negligible interactions Elastic collisions
p=
Goals for today ! ! !
What is temperature? Ideal gas thermometer Maxwell Distribution
mρ v 2 3
On Averages and Order
!
x =
∑ xi
In general 2
2 i
xi x ≠ x = ∑ x =∑ N ! In same way, in general 2
2
∫
[ f ( x )] dx ≠
x
x2
-3
9
3
9
2
<x>=0 <x>2=0
<x2>=9
2
[∫
f ( x )dx
]
1
Ideal Gas Aside p=
Two volumes of ideal gases
mρ v 2
2 2N m v pV = 3 2
3
!Let us leave a two different dilute noble gases (say, He and Ne) in an isolating containers for a long time, isolated from one another by a wall !
Ought to look a little familiar! !Take it apart !
p=
mN v 2 3V
Looks Looks like like n KE (i.e. T) !
Ideal Gas law is Physics! !
!
We would find that
!
!
Will see later today how to turn this into IGL
!
Gas Volumes in Contact !Now raise the separating wall and wait for things to settle down !
!
!
The average speed of
Atoms zoom around, bounce into one another, zoom around… What happens to the average speed of He and Ne atoms?
He atoms remains the same Ne atoms remains the same But He and Ne speeds may still be—and in general are– different from each other
! ! !
Three cases ! !
Nothing He speeds up, Ne slows down Ne speeds up, He slows down
!Since total KE is conserved in collisions, the following are not possible ! ! ! !
!This is known as “thermal equilibrium”
Perhaps the average speed
!
! !
He atoms increases Ne atoms decreases
Total kinetic energy content of He rises, Ne falls !Energy had to flow from Ne to He !
!
Need not mean that total energy of Ne was larger than He to start with!
Or, if
Third objects !Let’s go back to case 1 !Instead of lowering the barrier, let’s open the door and insert an object onto He side !
! !
Ne atoms increases He atoms decreases
Energy flowed from He to Ne
Popsicle, Chunky® Soup, whatever
Three cases for He atomic speeds !
! !
!
!
Putting the two gas volumes into contact did not change the energy content of either gas
!
Both speed up Both slow down One speeds up, the other constant One slows down, the other constant
Cases 2 & 3
Average atomic speed in each gas is constant over time Average atomic speeds may be different in the two gases
Case 1: Nothing
!
!
An average speed of atoms exists and could (in principle) be calculated from measurements Similarly, a total kinetic energy of all atoms may be calculated
!
Stay the same Speed up Slow down
Remarkable fact:
!
If you had put the object on the Ne side, the result would have been the same
2
0th law of thermodynamics ! !
This fact is known as the 0th law of thermodynamics Often restated as !
!
!Let us have two volumes of He gas with different average speeds !Remove partition !After some time, the new system has settled down and has a new average speed !This average must be between the average speeds of the two original gas volumes
“If two bodies are each individually in thermal equilibrium with a third body, they are in thermal equilibrium with each other”
It is an experimental, not theoretical, truth !
!
!
Two Containers of He
If two iron filings attract one another, they do not each attract a third NFL (in regulation): Buffalo tied San Diego tied Patriots tied Seattle ! But Buffalo and Seattle did not tie!
!
As the third is arbitrary, it is universal ! !
All objects in equilibrium with our He gas are somehow “the same” We can assign each group of objects in equilibrium a label (called “T”)
!
m
(N
1
2 2
v + N 2 v + N 1∆
)
2 1 KE 2 = N 2 KE 2 = N 2m v 2
)
= (N 1 + N 2 )
m
2 2
( v )+ N
= (N 1 + N 2 ) KE 2 + N KE tot =
KE tot N1 N 2
= KE 2 +
N1 m ∆ N1 N 2 2
> KE2
= KE 2 +
1
m 1
m
∆
!
∆
= KE1 −
N1
N2 m ∆ N2 2
< KE1
! !
!
!
!
!
It has ice, it has water, it has vapor
Let us find a volume of He whose average speed is such that when we insert this glass container into it Nothing happens to the average speed of the He
We label this volume of He with the number “273.16”
More concretely 0
273.16
Observing average speeds of atoms in the gas
We also can recognize objects in equilibrium with ideal gas volumes !
!
Label 0: ideal gas volume in which atomic speeds are 0 Let us wander around the universe until we find a closed glass container in isolation with the following property !
N1 + N 2 m N2 m ∆ ∆− 1 N1 N 2 2 N N2 2
We can recognize ideal gas volumes in equilibrium with one another !
!
!
!
Temperature: The Argument !
Exists and is single-valued Is ordered ! Is continuous It may be mapped onto the real numbers !
(
KE tot =
We take it as self-evident that if a set of labels T !
2 1 2 KE1 = N 1 KE1 = N 1m v1 = 1 mN 1 v 2 + ∆ 2 2
!
2
1
v12 ≡ v 22 + ∆
This implies an ordering to our label T
Temperature: the ideal gas thermometer
2 Volumes of He Label 1,2 v1>v2
Mathematical proof following page
If we re-separate the gas atoms, the faster has slowed down and the slower sped up !
Again, observing average speed of atom in gas
0th law: we can form collections of objections with the same label T
Kinetic energy considerations in ideal gases suggest labels ordered Define label T=0 as belonging to the collection of ideal gas volumes which have no atomic motion, and objects in equilibrium with these volumes Define the label T=273.16 as belonging to the collection of ideal gases & objects in equilibrium with a closed glass tube containing water, ice, and steam
3
Is this Physics? !
!
Phases of H2O? Properties of He atoms and gases of noble elements? Sounds an awfully lot like !
!
!
!
!
!
!
!
!
Pressure = mρ/3 Pressure at T=0 is 0 Experimental fact !
!
Pressure related to speed Pressure well-defined at T=0
The water vapor inside any magic glass tube is always the same Pressure at our special P=4.58 torr
T point is also special
!
The Concept
Must do so in a way that does not depend on the particular gas we use Use three observations we’ve made so far
!
! ! !
Readily and unambiguously measurable
Choose your favorite gas and make it very dilute Choose a volume of it whose temperature label is 273.16 Note the gas pressure on the walls of the container ≡p3
Ideal Gas Thermometer p = patm + ρgh
The pressure on the walls will be reduced
Let the two volumes come into equilibrium with one another Label each other volume as follows !
!
No motion is no motion
Carry your gas around to other gas volumes which are labeled neither 0 nor 273.16 As your reference comes into contact !
!
!
Suggest that pressure may play a useful role
Temperature: the ideal gas thermometer !
Useful observations
!
How do we label temperature points in between? !
He
0th law tells us this
Absolute 0 is also independent of gas composition !
!
Ne
Must also describe how to label other volumes of He gas Must demonstrate that it did not essentially depend on our use of He rather than Ne, Ar, or any other monatomic gas
If I didn’t need to mention this for the argument, then obviously the argument doesn’t depend on whether my gas has 2 protons, 10 protons, or weighs 4 Daltons or 20. Conclusions about ordering, etc. are unaltered We know that we could use any ideal gas to define 273.16 !
!
!
I was raised by wolves on a tropical Pacific island Our periodic table looks like this
Chemistry
Universality
!
!
The next step is to demonstrate that our labeling is universal, translatable, and applicable to any object in the universe !
!
Did I mention this?
+h
T=273.16(p/p3)
Does this depend on the gas you use?
4
Does it depend on the gas? ! ! !
p 273.16 p3
In fact, yes, very slightly Modify procedure Use limit of a series of gas volumes !
373.5 Nitrogen
Each more dilute than the one before
!
Measure assigned temperature as a function of concentration Extrapolate to 0 concentration
!
The extrapolation is independent of gas used!
!
Density-Label: Steam @ 1 atm
!
373.25
373.00
We can recognize ideal gas volumes in equilibrium with one another !
!
!
!
! !
!
Observing average speeds of atoms in the gas
We also can recognize objects in equilibrium with ideal gas volumes 0th
Again, observing average speed of atom in gas
law: we can form collections of objections with the same label T
Kinetic energy considerations in ideal gases suggest labels ordered Define label T=0 as belonging to the collection of ideal gas volumes which have no atomic motion, and objects in equilibrium with these volumes Define the label T=273.16 as belonging to the collection of ideal gases & objects in equilibrium with a closed glass tube containing water, ice, and steam
40
!We did glide over one subtle point
!
There exists a single average speed for the ideal gas volume However, not every molecule is at this speed
!
We can recognize ideal gas volumes in equilibrium with one another
!
We also can recognize objects in equilibrium with ideal gas volumes
!
!
Glancing collisions would cause some to slow down /stop Subsequent collisions between stopped & moving atoms would slow down to intermediate values But average kinetic energy remains same
Observe average speeds of atoms in the gas
!
Again, observe average speed of atom in gas
!
! ! !
!
!
0th law: we can form collections of objections with the same label T Kinetic energy considerations in ideal gases suggest labels ordered Define label T=0 as belonging to the collection of ideal gas volumes which have no atomic motion, and objects in equilibrium with these volumes (although actually, there are none of these!) Define the label T=273.16 as belonging to the collection of ideal gases & objects in equilibrium with a closed glass tube containing water, ice, and steam ! Recognize that pressure correlates well to temperature
We define all other labels using an ideal-gas thermometer T=273.16lim(p/p3) Definition independent of gas we use in the thermometer
! !
Review of Mechanics
v
!
KE: Before = After (‘)
!
1
Total Momentum Before = After (‘)
!
1
v 2
Stopped
2
mv
1
1
+ mv
2 2
1
2
1
= mv ' + mv ' 1
2
r
2
r
1
!E.g. even if they all started that way !
120 p (kPa) 3
80
Temperature: The Argument
More on ideal gases: Maxwell !
Hydrogen
A unique, ordered label for EVERY OBJECT IN THE UNIVERSE
Temperature: The Argument !
Helium
Unique T independent of gas
1 2
v 2
!
1 1x 2 1 2 y2 1 1 x 2 1 1 y 2 mv + mv = mv ' + mv ' 2 2 2 2 2 2 1 1 + mv 2 x ' + mv 2 y '
Solved if v2x’=v2y’=0
1
r
'
2
'
In x
r
1
!
r
2
r
1
'
r
2
'
In y
r m v2
r r m v1 '+ mv2 '
5
Probability Distributions !End up with a distribution of velocities in the gas !Of course, average kinetic energy cannot change (energy conserved) !Define a distribution P(v) !
!Derivation beyond scope of the course
∫ P (v )dv = 1
3 2
− Mv M 2 2 RT P (v ) = 4π v e 2πRT
2 may not equal !Average KE is given by !
! !
!
=< ½ mv2>= ½ m Not ½ m2
Result ! !
=3/2 RT m/M = 3/2 kBT ! k =Rm/M=R/N B A
(vmax)2= 2 RT M
!
v v+dv
∂v
Proportional to temperature!
!
* Some texts use f(v) = NP(v); f(v)dv is the
vmax
number, not fraction, between v and v+dv
!Average value of v may not equal vmax
Long tail to right skews average high
2
!Can find maximum (most probable speed) by setting ∂P = 0
Some comments on distribution !
P(v)
Result is important
!
The product P(v)dv is the fraction of molecules with speed between v and v+dv
Normalization condition
There’s a reason Maxwell got a Nobel prize
!
P(v)
!Total fraction of all molecules must be one* !
Maxwellian Distribution
3/2
2
v
∫
∫v e 4
−C
v2 2
C
2
=3
RT
v 2
Temperature !
dv
!
dv
!This is a hard but doable integral !Do need answer today
v
!
2
4 − C ve v P (v )dv 4π 2π =∫ = ∫ 1 P ( v )dv 2
Summary
!
!
Scale defined using triple point of water, absolute 0, and pressure Use ideal gas thermometer to label temperature of everything else We’ll never need to think about T again! ! “There are no subtleties”
p T = 273.16 lim ρ →0 p 3
Maxwell Distribution !
Describes probability of different molecular speeds in the gas
3 2
− Mv
2
M 2 2 RT P (v ) = 4π v e 2 RT 2
v
=3
RT
6
