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Entropy

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Physics 202

Professor Lee Carkner

Ed by CJV

Lecture -last

- 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!

- Adding x joules of heat will produce a temperature increase of y degrees
- 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)

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