Lecture # 9

Physics 7A

Summer Session II 2008

Evaluations First

• You get 15 minutes to do Evaluations• This Evaluation is for Cassandra the Lecturer• You will have a chance to evaluate the course

in DL (as well as your DL instructor).• The Evaluations are given to you at the

beginning of class so that you are encouraged to write comments during ‘my time’ and not to cut into ‘your time.’

Announcements• Review Sessions Start tomorrow in 266 Everson

– Amandeep: 11-1 Thursday– Yen: 1:40-3:40 Thursday– Yi: 8-10AM Friday– Swapno: 10-noon Friday

– You are encouraged to go to any (and as many) review sessions you like. You do not need to go to your TA’s review.

• There will be one more lecture on Monday, it will be a review and whatever doesn’t get covered today in lecture.

• Final is a week from Today

Last time…

After graduating from Davis, you decide you love it here so much that

you want to start a farm and live here forever..

You go to the state Fair to get your first animals. There are only two kinds for sale and you can only buy 1.

You go back the next day, and pick another one. What are your possible combinations?

What are the States? And what are the microstates of the system?

State = 1Cow and 1 Sheep State = 1Cow and 1Sheep

State = 2 Cows State = 2 Sheep

Microstate = CS Microstate = SC

Microstate = SSMicrostate = CC

Three States, but 4 microstates, which State is most probable?

If you leave it up to chance…

You are most likely to have a state of one cow and one sheep, because there are more microstates in that state!

Right… Sooo what does this have to do with Physics?Right… Sooo what does this have to do with Physics?

OK we’ll go back and talk about particles again…

I have 10 particles in a box with some KE

If I leave the particles alone for a long time, and then I come back at some random moment and take a picture of where the particles are at that instant, which of the two cases am I more likely to find?

Case A)Case B)

C) They are equally as likelyD) Can we go back to cows and Sheep?

I don’t believe you, it is s unlikely that we would see them all lined up neatly in a row. The two cases can’t be equally likely.

I don’t believe you, it is s unlikely that we would see them all lined up neatly in a row. The two cases can’t be equally likely.

Your first sentence is correct, but that doesn’t mean that the second one is true… let’s see why.

For kicks (no credit) tell me which of these cases I showed you before?

A)

B)

C)

D)

We have trouble discerning between cases that look ‘random’ to us.

Case A) Case B)

Case A is equally as likely as case B but there are more microstates that ‘look like’ Microstate. So even though all microstates have the same probability, the ‘state’ of looking random, is more likely than the ‘state’ of looking orderly.

Microstate A) Microstate B)

“Ordered microstate”

equally likely as a random

microstate

Are we likely to find the system in “ordered state” or “random state”?? …umm almost all the microstates look “random”

Microstates vs states

Only 10*9*8*7*6*5*4*3*2= 3.63x106 ways to do this.

But… 56*55*54*53*52*51*50*49*48*47=1.3x1017 total microstates so… 1/35607051480 chance of this happening. And this is only for 10

particles!!!!!

What is the probability that the particles fill the 10 specific boxes in case A?

Saying the particles can only occupy certain places is adding a ‘constraint’.

Let’s go back to the farm….

Now you want chickens

How can we add constraints to the system?

Adding a fence will force the chickens to be in a smaller area.

What if it’s really cold out?

Chickens will stay in the coop.

What if you forget to feed the chickens?

The chickens will not move.

These are all examples of limiting the position or velocity states of the chickens.

Chicken CoopChicken Coop

Microstates

ConstraintsConstraints StatesStatesMicrostatesMicrostates

Things we worry about:

Constraints:

States:

Tell us which microstates are allowed.Examples The volume of a box constraints the possible positions of gas atoms.The energy of the box constraints the possible speeds of gas atoms.

Groups of microstates that share some average properties,i.e. A collection of states that “look” the same macroscopically.Examplesgasses: P ~ average density, V~volume filled, T~average KE

Microstates

So far we have described systems using P, V, T, .....

Incomplete information about the system

A microstate is a particular configuration of atoms/molecules in the system.

For example,For a box of gas, you need to specify where all the atoms are i.e., position (x,y,z), and how fast they are moving i.e., velocity (vx, vy, vz).

In DL you learned…

Flipping a coin 3 times:

Microstates: all possible combinations of coin flips

Constraints: some combinations not possible (e.g. HHTHHH)

MicrostatesStates

States: total number of heads

Hypothesis: every microstate is equally likely. Hypothesis: every microstate is equally likely. The state that is most likely is the one with the most is the one with the most

microstatesmicrostates

Prob.

where kB is Boltzman’s constant

If our system is composed of two sub-systems A and B:

We can add the entropy of the subsystems to get the total entropy.

Entropy

Why split our system into subsystems?

Splitting 10 coin flips into the first 2 flips and the remaining 8 is perverse

ice (0ice (000 C) C)

Water (0Water (000 C) C)

But calculating heat needed to raise temperature of this system to 100 C we would split into subsystems:

the ice and the water

How to calculate entropy?

We should divide our box up into “atom-sized” chuncks. But how

big should our velocity microstates be?

ice (0ice (000 C)C)

Water (0Water (000 C) C)

How can we get a definite answer for the number of

microstates in this system?

Relating entropy to microstates is useful for conceptually understanding what entropy is.

At this level, it is not useful for calculating the change in entropy

For slow, reversible processes:

To get to entropy we can “turn this expression around”

If temperature is constant, then we can easily integrate:

This last equation is not generally true; as heat enters or leaves a system the temperature often changes.

(isothermal only!)

How to calculate entropy? Answered

Q

If the process is not slow or reversible, or it is very difficult you can use the fact entropy is a

state function

If you can find any process from the initial to final state, you can use this path to calculate ∆ S for the

process in question!

(As in calculation of enthalpy in DLM14)

How to calculate entropy? Answered 2

Summary

Lesson: some states are more likely than others.

But once the coins are flipped we know what they are going to be. Our system does not evolve in time.

Solids, liquids and gasses do not stay in the same microstate -- they change in time.

10 fair coin flips

e.g.First coin flip is T. So I have 49 H, 1T => state is 49Second coin flip is H. Still have 49 H => state is 49Third coin flip is T. Now state is 48 etc.......

What does our state look like as a function of “time”?

50 fair coin flips

“Short” example(400 flips)

with 50 coins

“Long” example(1500 flips)

with 50 coins

“Short” example(400 flips)

with 100 coins

“Long” example(1600 flips)

with 100 coins

“Interactions” (i.e. coin flips) take a certain amount of time each.

After many interactions, the systems settle down “close” to the most likely value. This is what is meant by equilibrium.

The system may depart from equilibrium, but large departures are rare and typically don’t last very long.

Equilibrium

• Equilibrium is the most likely state

• Each microstate is equally likely, so the equilibrium state has the most microstates.

• Therefore the equilibrium state has the highest entropy.

• For large (i.e. moles of atoms) systems, the system is (essentially) always evolving toward equilibrium. Therefore the total entropy never decreases:

What’s equilibrium to do with Entropy?

Second law of Thermodynamics