Transcript Refrigerators and Entropy

Entropy
Physics 202
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)
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):
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
State Function
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
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:
DSbox = (-Q/Tbox)
DSres = (+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:
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
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 I
1 microstate
Probability = (1/16)
Configuration II
4 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:
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
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