Phys 233 Day 17, T1 & T2: Temperature & Ideal Gases 1

Phys 233 Day 17, T1 & T2: Temperature & Ideal Gases 1

Mon. 11/7

Tues. 11/8

Wed., 11/9

Thurs. 11/10

T2 Ideal Gas S3, S4

L9: Ideal Gas, LC9, LR 3, 7

T3 Gas Proc S2, S7

HW9: O5: 1,3; T1: S.4, S.7, S.10

RE-T2; Lab Notebook & Full Report

PL9; Quiz 7: O4, O5

RE-T3

Mon., 11/14 HW10: T2: S.4, S.5, S.6 ;T3: S.3, S.4, S.7

Equipment

o Ppt.

o HW to hand back

o Lab 8 handout

o Thermometers

o Hot pot, boiling water, beaker & room temp water, and metal slug on string

o Piston and masses

o Spring scale

o Molecules.exe with Brownian Motion

T1 Temperature

T1.1 Intro to the Unit

T1.2 Irreversible Processes

T1.3 The Paradigmatic Thermal Process

T1.4 Temperature and Equilibrium

T1.5 Thermometers

T1.6 Temperature and Thermal Energy

T2 Ideal Gases

T2.1 A Microscopic model of an Ideal Gas

T2.2 Temperature and Energy

T2.3 Molecular Speeds and Brownian Motion

T2.4 The thermal Energy of a Gas

T2.5 Solids and Liquids

T2.6 Conclusions

Questions?

T1.1 Intro to Unit T

The vast majority of systems studied by Physics are fundamentally ensembles of large

numbers of particles. This section of the course (and, then Phys 344 next Spring) focuses

on the two areas of Physics that directly address the properties unique to ensembles:

Statistical Mechanics and Thermodynamics. Starting with the Mechanics that describe

individual particles and employing Statistics to extrapolate from the few to the many,

Statistical Mechanics builds our understanding of ensembles from first principles.

Thermodynamics applies this understanding to describe how ensembles behave and do

work in both natural and man-made systems.

I’ve often said that there are two types of theoretical models in physics – ones that are

built to be ‘good enough’ for addressing a problem at hand and another is built to be

‘fundamental’ for understanding something in terms of the most universal principles.


o Thermodynamics was born of a need to address very practical problems; it is the

science of the industrial revolution – it’s a great example of a ‘good enough’

theory.

o Statistical mechanics was born of a need to understand thermodynamics in terms

of the most universal principles – it’s a great example of a ‘fundamental’ theory.

This course will then be a little schizophrenic – moving between the ‘good enough /

practical’ approach and the ‘fundamental understanding’ approach. That can be a little

disorienting if you aren’t expecting it, so this is your ‘heads-up.’ Let me tell you from

experience though, the alternative, of studying Thermodynamics separate from Statistical

Mechanics is extremely dissatisfying.

The “Great Idea” of this Unit is Boltzmann’s: The Irreversible behavior of a complex

system can be explained by the statistical consideration of the reversible interactions of

its molecules.

Thermo/Stat-mech books tend to get off to unsatisfying starts, and this text is no

different. The first chapter, intentionally poses many questions that won’t get answered

for some time, but it also presents things that themselves beg unspoken questions that this

chapter alone does not prepare you to answer. I’m happy to give you snippet/sound-

bite/preview answers today, but rest assured that all these questions will be satisfactorily

answered, in their own time.

T1.2 Irreversible Processes

PPT

T1T.1 – Characterize each of the following processes as being reversible (A) or

irreversible (B).

a. A living creature grows

b. A ball is dropped and falls freely downward

c. A ball rebounds elastically from a wall

d. A piece of hamburger meat cooks on a grill

e. A cube of ice melts in a glass

f. A bowling ball elastically scatters some bowling pins

This section points out that, while the individual laws of physics are time symmetric,

meaning that simple processes of individual particles are just as plausible forward as

backward, the behavior of large systems of particles is observably not reversible – a rock

doesn’t just jump out of a pond, a block doesn’t suddenly start sliding along a table and

cool down.

This section doesn’t give an explanation, but points out that one will come in later

chapters.

T1.3 the Paradigmatic Thermal Process (demo)

Ppt.


Objects in ‘thermal contact’ approach ‘thermal equilibrium.’ Put a hot block of

metal in cold water. As we all know, eventually the water and block will arrive at the

same, luke-warm temperature (and if we wait longer, it will settle at room temperature.)

o The block ‘heats’ the water – raising its temperature and energy content, and the

water ‘cools’ the block – lowering its temperature and energy content.

o Biq Questions: Some ‘big questions’ this raises.

What is Temperature? (measure of energy density / how readily an object

gives up internal energy)

What is Heat? (energy transfer prompted by temperature difference)

What is thermal equilibrium? (two objects having the same temperature)

Why is the process irreversible? (on average, the particles of the metal

are jiggling more energetically than are the water molecules, so when they

collide, on average, they tend to transfer energy to the water molecules)

T1.4 Temperature and Equilibrium

What is Temperature?

You walk barefoot across the beach on a summer day, and you sense that the sand is

‘hot’; you walk barefoot across a snow bank and you sense that the snow is ‘cold.’ It’s

that property that we call temperature. We start off with a very operational definition,

theoretically hollow and unsatisfying, but useful: what a thermometer measures.

In later chapters, we’ll learn how to relate this to more tangible, fundamental properties.

In broad strokes, it’s a measure of energy density, not per volume or mass, but per ways

of having energy, and that directly relates to how readily a system gives up energy when

it interacts with another system.

Zeroeth law of Thermodynamics

Thermometers work – the transitive property of temperature: if objects A and B are in

thermoequilibrium (same temperature) and objects B and C are as well, then A and C

must be as well. Dub object B a thermometer, and there you are, ‘thermometers work.’

Ppt.

T1T.4 – Imagine that we place an aluminum cylinder, a wooden block, and a Styrofoam

cup on a table and leave them there for several hours. We then come back into the

room and feel each of the objects

Which (if any) feels the coolest?

Which (if any) actually is coolest?

A. The aluminum cylinder

B. The wooden block

C. The Styrofoam cup

D. All are the same


T1.5 Thermometers

Of course, that bring us to the questions of what is a thermometer, and how does it work.

We can partly address those now.

Direct & indirect measurements. When you think about it, very few properties are

directly measurable, indeed, all we can directly measure are distance, direction, and time.

Mass – indirect. (demo: spring scale) For example, say you want to measure the mass

of something, you hang it from a spring scale and the needle attached to the spring

moves along some row of numbers. What are you directly measuring? The displacement

of that needle / the stretch of the spring. However, from that you can deduce the mass of

the hanging object.

Temperature – indirect. Similarly, you can’t directly measure temperature, but you can

observe a property that varies in response to temperature (just like the length of the

spring varies in response to the object hanging from it.)

o A wire’s resistance varies with temperature

o The color of light emitted by an object varies with temperature

o The volume of an object varies with temperature (holding pressure constant)

o The pressure of a gas varies with temperature (holding volume constant).

So, you can make a “thermometer” based on observing the variations in any of these

properties.

Liquid Thermometer – expansion (demo). The most common thermometer uses the

fact that volume of an object varies with temperature, a.k.a. thermal expansion. We met

this back in Phys 231, but in broad strokes, if, as I claim, temperature measures the

density of internal energy, then we’re saying that objects tend to expand when they’ve

got more internal energy. For a solid, ‘more internal energy’ means that the atoms are

jiggling more violently / higher energy level, and their new average separation is a little

greater (since there’s an asymmetry in their bonding – the bonds simply can’t get

compressed beyond 0, but can stretch and stretch, so the more energy, they stretch more

than they compress.)

Constant-Volume Gas Thermometer. (demo) The book singles out this one. It doesn’t

shed much light on the fundamental nature of temperature. Rather, it’s a nice example of

how one can practically go about creating a calibrated and sharable / reproducible

measurement technique.

o Pressure. First off, recall that pressure is the ratio of Force to the Area over

which it’s being applied.

A

FP the units are N/m

2 which defines the Pascal, Pa = N/m

2.

Pressure. Of course each collision is a little push which can be quantified as a force

that’s shared between the two colliders. In gases, we often speak, not of the force of a

single collision, but the density of forces getting exerted across an area due to a whole

slew of collisions. This defines the pressure.

o Pressure = force per area= “Energy Density” = Energy per volume. The

pressure at a point is the differential bit of force being applied perpendicularly

across a differential, flat area containing the point.

Units:

o In fundamental SI units, 2

1m

NPascal


o The relation of other common units: 1 atmosphere = 1.01×105Pa =

760 mmHg (a.ka. Torr) = 14.7 lb/in2 (a.k.a. psi).

The atmosphere is conveniently defined as the average

atmospheric pressure at sea level.

The Torr and psi are both very commonly used.

Not a Vector I: While force and area are both vector quantities, we are

specifying just the component of the force running perpendicularly across

the area, i.e., parallel to the area vector. So Pressure is not itself a vector.

Uniform. When a gas is in equilibrium, the pressure is equal throughout.

This makes sense if you think about its relation to force – if the force per

area weren’t uniform, there’d be a force gradient somewhere, i.e., greater

force pushing particles one direction than the reverse, so they’d up and

accelerate in that direction – not quite what you’d call “equilibrium.”

Not A Vector II: Omni-directional o What isn’t necessarily so obvious is that, at a point in a static gas

or liquid the pressure will be the same in all directions. Not just the

same to the left as to the right, but the same to the left as, say, up.

o Illustrative Examples

Pool Balls - To understand this, think first about breaking a

rack of balls – how the motion of each ball is transferred to its

neighbors at an angle – the initial head-on push of the cue ball

quickly gives over to pushes fanning out in all directions.

Water Balloon -Think about a water balloon. If you push

down on it, the water squishes in all other directions – in so

doing, it redirects your force so that an equal force is applied to

all areas of the balloon surface – equal pressure.

o So, if you have a piston full of gas and place some mass on its head to apply a

pressure, the piston head will be held in equilibiruim where the pressure from

above (due to the weight of the mass sitting on top) equals the pressure from

below (due to the gas particles knocking into it).

o This thermometer works because the hotter the gas is, the more forcefully the

particles bounce into the head, so the more mass you need to pile on top of the

piston head to keep it from rising. Now, that the pressure and temperature should

vary linearly is rather special; under what conditions and why this should be so,

we’ll get to next chapter, but for now, let’s see what we can do with this.

o So, consider a given thermometer

first dipped and equilibrated with water at one temperature, T.

TP Second dipped and equilibrated with water at another temperature, call it

TC for “comparison.”

cc TP

o Then

c

ccc

P

PTT

T

T

P

P


o Now, the task is selecting a Tc, Pc pair that is universally reproducible. Kind of

like when you want to talk about a mountain’s elevation, it’s always relative to

sea level because that’s more or less equally accessible everywhere on Earth, we

want a Tc, Pc pair that can be determined by anyone using any constant-volume

thermometer.

o Triple Point. As you’ll learn later, there’s only one pressure -temperature point

at which water can coexist in all three phases (solid, liquid, gas). This is known

as the ‘triple point’ for water: (P3, T3). See, the exact temperature at which water

melts varies with pressure, and so does the temperature at which it boils, but there

is only one pressure and temperature at which all three phases coexist. So, if we

take those to be our calibration point,

3

3P

PTT

Temperature Scales.

For better and for worse, there are there common temperature scales in use in the US – Farenheit

is what most of us use on a daily basis, Celsius gets a little play, and Kelvin gets used by

scientists. The one thing that they all share is that they’re linear scales, that is, if you made

Farenheit temperature along the x-axis and Celsius temperature along the y-axis, and then

marked on the plot the outside temperatures over the course of the day, they’d line up along a

diagonal line.

Ppt.

Of course, to fine the equation of a line, you can use two points on the

line to find the slope, and then one of those points to find the intercept.

For example, Table 1.1 tells us that Tungten melts at 3410C, 6170F and

room temperature is 22 C, 72 F.

So, F

C

F

C

FF

CC

T

Tslope

F

C

9

5

6098

3388

726170

223410

Then again,

CTT

CFCTslopeTTslopeT

FFC

C

FC

FCFC

8.17

8.1761703410interceptintercept

95

95

T1.6 Temperature and Thermal Energy

Cast your memory back to Phys 231, there you should have encountered that changing the

internal energy of a system (say, vigorously stirring a liquid) changed the system’s

temperature proportionally,

Tc

TF


dT

dU

mcmcdTdU

1

The exact value of the proportionality constant, the “specific heat capacity” depends on the

material, but we can define it via the derivative of internal energy with respect to

temperature. So when you have two (or more) objets come into thermal contact, we know

that a) they’ll eventually come into thermal equilibrium and b) their internal energies will

vary along with their temperatures.

...... 333222111321 TcmTcmTcmUUUU system

whereif TTT .1.11

and ffff TTTT ....3.2.1

when thermal equilibrium is eventually

reached.

If this is an isolated system with no other form of energy chaging (say, kinetic), then

......0 333222111321 TcmTcmTcmUUUU system

Ppt.

T1T.8 – Imagine that we place a 100-gram aluminum block with an initial temperature

of 100°C in a Styrofoam cup containing a 100-gram sample of water at 0°C. (The

specific heat of aluminum and water are 900 J kg-1

K-1

and 4186 J kg-1

K-1

,

respectively.) The final temperature of the system will be closest to

A. 0°C

B. 20°C

C. 50°C

D. 80°C

E. 100°C

T2

In this section, we begin to develop the particle picture of a gas and use that to motivate the Ideal

Gas law. At this point, the law is only motivated, not derived, and the necessary

approximations/idealizations are spelled out.

A little T2.3

Particle Picture background

History / Perspective: It’s hard to imagine, but the particle picture of matter didn’t get

nailed down until the turn of the 20th

Century. To get us there, theoretical work was done

by some of the giants of science including familiar names like Maxwell and Einstein.

o Brownian Motion

V. Lab 2: Kinetic Theory 1: Brownian Motion (open Molecules.exe, and from the file menu

open Brownian Motion.

o Einstein used the particle gas theory to model Brownian Motion – something that

could only be described by this model – and thus ensured the place of particle gas

theory. Under the microscope, or even in a patch of sun in a dusty room, you see

dots of dust randomly moving around, changing direction seemingly

spontaneously. The only way to accurately model their motion is by saying that

they are being buffeted by tinier, unseen particles which are also randomly


moving around, colliding with each other as well as the dust – these are the

molecules of the gas.

(In Brownian Motion, to reveal the hidden particles, select the Misc tab and change their color

to something other than the background)

Ppt.

Elements of the Ideal Gas Model

1. A gas consists of a huge number of tiny atoms/molecules

2. Individual atoms/molecules are tiny compared to their average separation

3. The atoms/molecules obey Newton’s laws of motion

4. The atoms/molecules don’t interact with each other

5. Collisions between the atoms/molecules and the container walls are elastic

6. The motion of the molecules is entirely random

Some T2.1 and T2.3 Pressure and Temperature

Speaking of Collisions…we can derive a relationship between macroscopic pressure

and microscopic particle properties.

Ppt.

T2T.3 – The average x component of the velocity of molecules in a container of

gas at rest is zero (T or F).

T2T.1 – If the speed of a molecule in a container doubles (other things remaining

the same), the average pressure that this molecule exerts on the container wall

A. Remains the same

B. Doubles

C. Quadruples

D. It is impossible to say

Particle Picture of Pressure. o Since pressure is uniform throughout a gas, we derive a relationship for it at

some place convenient, like the wall of a gas container, and know that our

results apply everywhere throughout the gas.

o If you could resolve the individual air molecules zipping around inside a tire,

you’d see something like a buzzing swarm of knats. Looking just at one patch

of wall of the tire, you’d see gazillions of little knats ricocheting off the patch

in all directions. It’s the cumulative force of all those collisions, divided by

the area of the patch, that we’d call the pressure on the tire wall. We’ll start

by focusing on just one knat/particle’s contribution to that pressure, and then

scale up the N particles. We’ll find that the pressure on the wall relates to the

energy of the particle.

o Let’s consider the pressure on a wall of area A perpendicular to the x-axis.

x

x

A

FP

So, what is the average force exerted on the wall by the particle

slamming into it? px

py ip

fp


Well Newton’s 3rd

Law says that it is equal and opposite to

the force the wall exerts on it, ( atomwallWallatom FF ) and that

is easier to analyze.

What is the force exerted on the particle by the wall?

Say this is the only big force on the atom, so

t

pFF netatomwall

.

What is the change in momentum of the atom when it collides with

the wall?

Say the atom is going somewhat y-ish, somewhat z-ish, and

somewhat x-ish. When it collides with the wall in the x-

direction, it ricochets with unchanged y and z motion, but

reversed x motion, so

xxxxif mvpppppp 22

. (note: this

follows from the collision being elastic and the wall’s not

moving – so the ball must have as much momentum and kinetic

energy before as after)

What is the time over which this force is applied?

Well, it’s probably actually quite brief, but since the atom

repeatedly bounces back and forth between this and the

opposite wall, and since we’re eventually going to be

considering a continuous bombardment of particles, it’s safe to

find the average force per this atom’s bombardment, i.e., use

the time between two successive collisions: xv

xt

2, x is

the width of the box in the x direction, the factor of 2 is

because, starting from the moment of one collision, the particle

would make a round trip to the opposite wall and back, go 2

x, before hitting the wall again.

Put this all together to see what we’ve got so far.

2

1

22

1

1

2

2121

)(211

x

x

x

x

x

x

xx

xatomwallwallatom

mvVP

V

mv

AL

mv

v

L

mv

At

mv

AP

t

vm

At

mv

AA

F

A

FP

The subscript is to remind us that this is the contribution of just

one particle.

Relate x-component of the velocity to total speed?

Well, on average, the particle goes just as much in the x, y, and

z directions, so,

x


o

___2

___2

___2

___2

___2

___2

___2

___2

___2

3 x

zyx

zyx

vv

vvvv

vvv

Then we have

___

2

13

1vmVP

Now’s a good time to scale up to handle a gas of N particles. The

pressure due to N particles is just N times that due to one average

particle.

___

2

13

1vNmVNPPV

Here’s a good place to pause and highlight a result, the average speed

(of sorts) of a gas particle is

P

vv rms 3___

2

o Where V

Nm

V

Mis the mass density.

Another way to package this information is

N

VPEK

EKNvmNvNmPV

trans

trans

23

_____

_____

32

___2

21

32

___2

31

..

..

The average translational energy is simply related to the pressure.

Using one of these relations

Ex: Speed of ‘air molecules’, N2. Given that air is at atmospheric pressure, 1 atm = 105 Pa.

And that it has a density of 1.29 kg/m3, what is the average (rms) speed of an ‘air molecule’?

o sm

m

kgrms

PaPv 494

29.1

1033

3

5

; not coincidentally, a tad over the speed

of sound in air.

Transition to Thermal and Temperature

Now, in T1, temperature is defined to be proportional to pressure while we’re holding

the volume and number of particles constant.

Without actually deriving things yet, it seems reasonable that it’s really pressure per

density of particle that’s proportional to temperature (think of it this way – increase

the density of particles at a constant temperature and the pressure builds). So we have

TN

PV.

o We can make this proportionality an equality if we simply introduce a

proportionality constant, k.


o TNkPVTkN

PVBB Viola! The Ideal Gas Law.

o Boltzmann’s Constant and Kelvin Temperature

So, what is this constant and what is this temperature scale.

Kelvin / Absolute

The Scale is known as Kelvin or, sometimes, Absolute. What’s

absolute about it is that, for an ideal gas, there is absolutely no

pressure when T = 0 in the Kelvin scale.

Its gradation is historic / arbitrary, and that arbitrayness is adsorbed

in the proportionality constant.

Boltzmann’s Constant

k = 1.38×10-23

J/K.

Note: practically everywhere in physics that you see T for

temperature, you’ll see it multiplied by this constant. It’s always

got to be there to adsorb the arbitraryness of the Kelvin gradation –

if physicists wanted to be radical, they could clean this up by

defining a new temperature scale that adsorbs the constant.

o Ideal Gas Law. The idealization that appears in this derivation is essentially that

the particles interact elastically, in other words, there’s no change in potential

energy. Of course, in real gases, made of real molecules, the particles can get

polarized and thus share potential energies associated with moderately long-range

interactions. So, one could say that the Ideal gas law becomes a better and better

approximation if you decrease and decrease the gas’s density (so particles are far

enough apart they don’t feel each other’s charges.

Ppt.

T2T.5 - Two identical rooms, A and B, are sealed except for an open doorway

between them. Room A is warmer than room B. Which room has the

greater number of molecules? (Hint: What things must be the same in the

two rooms?)

A. Room A

B. Room B

C. Both have the same number of molecules

Ppt.

T2S.7 – Imagine that we add 250 J of thermal energy to 1 liter (1000 cm3) of

air initially at room temperature and normal pressure. By how much does

its pressure increase if its volume is held constant?

Some T2.2 Temperature, Energy, and Speed.

o So, we’ve related Temperature to Pressure, and before that we related Pressure to

Energy and speed. Now we can relate Temperature to Energy and speed.

o On the one hand, we have the Ideal Gas Law:

o TNkPV B (1)

o On the other hand, we have


o transEKNPV_____

32 .. (2)

o where ___

2

21

_____

.. vmEK trans (3).

o Putting (1) & (2) together, we have

TkEK

EKTk

Btrans

transB

23

_____

_____

32

..

.. (note to self: for equipartition, back this out a bit

and show that there is ½ kT for each ½ mv2 component.)

Qualitative Meaning

This tells us, as we may intuitively suspect, temperature is

close akin to energy. It’s tempting to say that temperature is a

measure of the particle’s average energy. That’s not quite it,

but temperature is indeed fundamentally linked to particle

energy. We’ll see the connection later.

o Adding (3) to the mix, we have

m

Tkvv

m

Tkv

vmTk

Brms

B

B

3

3

___2

___2

___2

21

32

Qualitative Meaning

This tells us, not too surprisingly, the higher the temperature,

the faster particles whiz about.

Question: what’s the mass dependence?

o Perhaps a little more surprisingly, since its energy, and

not speed that directly relates to temperature, more

massive particles, at the same temperature, move

slower.

Ppt. Summary

Ppt.

What is the rms speed of sodium (Na) atoms at room temperature? (Note: Sodium is very

reactive, so it won’t be remain a monatomic gas!)

Suppose a gas of sodium atoms is cooled with laser (in an apparatus called a magneto-

optical trap or MOT) to 240 μK. What is the rms speed of an atom in this case?

Two gasses Two masses (blue is more massive than red)

Some T2.4 Equipartition of Energy

Intro. / Transition from just translational kinetic energy

o We’ve performed the thought experiment of placing a single atom in a box, and

from that we derived the Pressure – Velocity relationship, which quickly lead us

to the Pressure – Kinetic Energy relationship, and finally the Temperature –


Kinetic Energy relationship:

kTkTkTkTvmvmvmEK zyxtrans 2

3

2

1

2

1

2

12

2

12

2

12

2

1.. . Each of the

three factors of ½ kT follows directly from each of the three velocity terms. More

generally…

Equipartition Theorem: At temperature T, the average energy of any quadratic degree

of freedom is ½ kT.

o The quadratic criteria might seem a bit odd, but it figures into the math of

calculating the average. We’ll see this in more depth later.

o The quadratic criteria might seem a bit restrictive, but consider how vibration and

rotation are accounted for in the energy relation: ½I x2, ½k ( x)

2.

o Monatomic degrees (translation)

A single atom can translate in the x, y, or z. Classically, you might think

of it as a solid ball, thus able to spin about each of these axes as well. It

would be better to remember that it’s a collection of electrons about a

collection of protons & neutrons. Quantum mechanics governs their

allowed states of motion, and the Equipartition Theorem is a classical

result – it only works if the allowed states are relatively close in energy.

o Diatomic degrees (rotation and vibration)

A diatomic atom on the other hand is fairly easily rotated. Thus there are

two degrees of (meaningful) rotational freedom. This can contribute 2 (½

kT).

Additionally, the bond that holds the two atoms together can be vibrated

like a spring. Since each vibration both moves the atoms relative to each

other (kinetic) and flexes the bond (potential) it can contribute 2 (½ kT).

For that matter, the bond can twist and bend in addition to stretch, so each

of these flexings can contribute factors of ½ kT.

o Tri (or more) atomic degrees (a third rotation and more vibration)

Except in linear molecules, adding a third (fourth, fifth…) atom breaks the

symmetry and makes rotation about all three axes meaningful: 3 (½ kT).

The addition of more atoms also multiplies the types of relative motions

possible, thus allowing for more bond-flexing terms.

o Degree of Freedom: Not just kinds of motion that the object can have, but kinds

of motions that the object is free to change.

Quantum Effects – Freezing out Freedoms

o Electronic Transitions – not degrees of freedom near room temp. In quantum

we learn that motions are quantized. For example, an electron can be in the

bottom 1S state or in the 2P state, but it can’t remain ½ - way between. It turns

out that the energy kick needed to get the electron from 1S to 2P is so much more

than the average translational kinetic energy of a gas particle anywhere shy of

stellar temperatures, that this does not represent a degree of freedom at common

temperatures. We say that this degree of freedom is frozen out.

o Bond flexing – not degrees of freedom near room temp. Similarly, the energy

necessary to change a mode of bond flexing is not available near room

temperature. So in most cases you’ll encounter, diatoms only have 5/2 kT and

larger molecules only have 3 kT of average kinetic energy associated with degrees

of freedom.

Ppt.


o Average thermal energy of an ideal gas

o If you have N particles, each with f (quadratically represented) degrees of

freedom, then the total average thermal energy of your gas is

kTNfU thermal 2

1

Note: this is just the accessible energy, it does not include the potential

energy associated with maintaining bonds or with the rest mass. As long

as we’re nowhere near a phase change however, this is the only kind of

energy that is free to respond to changes in temperature. So,

ernalthermal UUTkNf int21

Some T2.5 Average Thermal Temperature in a Solid

o In a solid, each atom is bound to its neighbors, so there’s no rotation or translation

to speak of, just vibration. Each atom has three perpendicular directions in which

to vibrate, so per atom there are f = 6 degrees of freedom (one kinetic and one

potential term per direction of motion.)

Example: 1.25 List all the degrees of freedom, or as many as you can, for a molecule of water.

Translation along 3 axes, Rotation about 3 axes (since not linearly symmetric), Vibration

in 3 modes – each H can move independently toward and away from the O, also the angle made

by the H-O-H bonds can open and close. Each vibration mode counts as 2 degrees as it varies

both kinetic and potential energy, so they count as a total of 6 degrees of freedom.

Documents

Phys 233 Day 17, T1 & T2: Temperature & Ideal Gases 1