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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