Entropy

Physics 202Professor Lee

CarknerEd by CJVLecture -last

Entropy

What do irreversible processes have in common? They all progress towards more randomness

The degree of randomness of system is called entropy For an irreversible process, entropy always

increases In any thermodynamic process that

proceeds from an initial to a final point, the change in entropy depends on the heat and temperature, specifically:

S = Sf –Si = ∫ (dQ/T)

Isothermal Entropy In practice, the integral may be hard to compute

Need to know Q as a function of T Let us consider the simplest case where the

process is isothermal (T is constant):S = (1/T) ∫ dQ

S = Q/T This is also approximately true for situations

where temperature changes are very small Like heating something up by 1 degree

State Function

Entropy is a property of system Like pressure, temperature and volume

Can relate S to Q and thus to Eint & W and thus to P, T and V

S = nRln(Vf/Vi) + nCVln(Tf/Ti) Change in entropy depends only on the

net system change Not how the system changes

ln 1 = 0, so if V or T do not change, its term drops out

Entropy Change Imagine now a simple idealized system

consisting of a box of gas in contact with a heat reservoir Something that does not change

temperature (like a lake) If the system loses heat –Q to the

reservoir and the reservoir gains heat +Q from the system isothermally:Sbox = (-Q/Tbox) Sres = (+Q/Tres)

Second Law of Thermodynamics

(Entropy) If we try to do this for real we find that the positive term is always a little larger than the negative term, so:

S>0 This is also the second law of thermodynamics Entropy always increases Why?

Because the more random states are more probable

The 2nd law is based on statistics

Reversible If you see a film of shards of ceramic

forming themselves into a plate you know that the film is running backwards Why?

The smashing plate is an example of an irreversible process, one that only happens in one direction

Examples: A drop of ink tints water Perfume diffuses throughout a room Heat transfer

Randomness Classical thermodynamics is deterministic

Adding x joules of heat will produce a temperature increase of y degrees Every time!

But the real world is probabilistic Adding x joules of heat will make some

molecules move faster but many will still be slow

It is possible that you could add heat to a system and the temperature could go down If all the molecules collided in just the right way

The universe only seems deterministic because the number of molecules is so large that the chance of an improbable event happening is absurdly low

Statistical Mechanics Statistical mechanics uses microscopic

properties to explain macroscopic properties

We will use statistical mechanics to explore the reason why gas diffuses throughout a container

Consider a box with a right and left half of equal area

The box contains 4 indistinguishable molecules

Molecules in a Box There are 16 ways that the molecules can

be distributed in the box Each way is a microstate

Since the molecules are indistinguishable there are only 5 configurations Example: all the microstates with 3 in one side

and 1 in the other are one configuration If all microstates are equally probable than

the configuration with equal distribution is the most probable

Configurations and Microstates

Configuration I1 microstate

Probability = (1/16)

Configuration II4 microstates

Probability = (4/16)

Probability

There are more microstates for the configurations with roughly equal distributions

The equal distribution configurations are thus more probable

Gas diffuses throughout a room because the probability of a configuration where all of the molecules bunch up is low

Multiplicity The multiplicity of a configuration is the number

of microstates it has and is represented by: = N! /(nL! nR!)

Where N is the total number of molecules and nL and nR are the number in the right or left half

n! = n(n-1)(n-2)(n-3) … (1) Configurations with large W are more probable

For large N (N>100) the probability of the equal distribution configurations is enormous

Microstate Probabilities

Entropy and Multiplicity The more random configurations are most

probable They also have the highest entropy

We can express the entropy with Boltzmann’s entropy equation as:

S = k ln W Where k is the Boltzmann constant (1.38 X 10-23

J/K) Sometimes it helps to use the Stirling

approximation:ln N! = N (ln N) - N

Irreversibility Irreversible processes move from a low

probability state to a high probability one Because of probability, they will not move

back on their own

All real processes are irreversible, so entropy will always increases

Entropy (and much of modern physics) is based on statistics The universe is stochastic

Engines and Refrigerators An engine consists of a hot reservoir, a

cold reservoir, and a device to do work Heat from the hot reservoir is transformed

into work (+ heat to cold reservoir) A refrigerator also consists of a hot

reservoir, a cold reservoir, and a device to do work By an application of work, heat is moved

from the cold to the hot reservoir

Refrigerator as a Thermodynamic System

We provide the work (by plugging the compressor in) and we want heat removed from the cold area, so the coefficient of performance is:

K = QL/W Energy is conserved (first law of thermodynamics), so the

heat in (QL) plus the work in (W) must equal the heat out (|QH|):

|QH| = QL + W

W = |QH| - QL

This is the work needed to move QL out of the cold area

Refrigerators and Entropy We can rewrite K as:

K = QL/(QH-QL) From the 2nd law (for a reversible, isothermal

process):QH/TH = QL/TL

So K becomes:KC = TL/(TH-TL)

This the the coefficient for an ideal or Carnot refrigerator Refrigerators are most efficient if they are not kept

very cold and if the difference in temperature between the room and the refrigerator is small