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# Entropy - PowerPoint PPT Presentation

Physics 202 Professor Lee Carkner Ed by CJV Lecture -last. Entropy. 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

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Professor Lee Carkner

Ed by CJV

Lecture -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:

DS = Sf –Si = ∫ (dQ/T)

• 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):

DS = (1/T) ∫ dQ

DS = Q/T

• This is also approximately true for situations where temperature changes are very small

• Like heating something up by 1 degree

• Entropy is a property of system

• Like pressure, temperature and volume

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

DS = 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

• 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:

DSbox = (-Q/Tbox) DSres = (+Q/Tres)

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

DS>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

• 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

• 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 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

• 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

Configuration I

1 microstate

Probability = (1/16)

Configuration II

4 microstates

Probability = (4/16)

• 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

• The multiplicity of a configuration is the number of microstates it has and is represented by:

W = 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

• 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

• 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

• 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

• 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

• 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