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Entropy
Physics 202
Professor Lee Carkner
Lecture 17
PAL #16 Internal Energy
 3 moles of gas, temperature raised from 300 to 400 K
 He gas, isochorically
 Q = nCVDT, CV = (f/2)R = (3/2) R
 Q = (3)(3/2)R(100) = 3740 J
 # 4 for heat, all in translational motion
 He gas, isobarically
 Q = nCPDT, CP = CV + R = (5/2) R
 Q = (3)(5/2)R(100) = 6333 J
 # 2 for heat, energy in translational and work
 H2 gas, isochorically
 Q = nCVDT, CV = (5/2) R, f = 5 for diatomic
 Q = (3)(5/2)R(100) = 6333 J
 # 2 for heat, energy into translational and rotational motion
 H2 gas, isobarically
 Q = nCPDT, CP = CV + R = (7/2) R
 Q = (3)(7/2)R(100) = 8725 J
 # 1 for heat, energy, into translation, rotation and work
Randomness
 Classical thermodynamics is deterministic

Every time!
 But the real world is probabilistic

 It is possible that you could add heat to a system and the
temperature could go down

 The universe only seems deterministic because the
number of molecules is so large that the chance of an
improbable event happening is absurdly low
Reversible

Why?
The smashing plate is an example of an
irreversible process, one that only happens in
one direction
Examples:


Heat transfer
Entropy
What do irreversible processes have in
common?

The degree of randomness of system is called
entropy


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

Let us consider the simplest case where the
process is isothermal (T is constant):
DS = (1/T) ∫ dQ
DS = Q/T

Like heating something up by 1 degree
Entropy Change
Imagine now a simple idealized system
consisting of a box of gas in contact
with a heat reservoir

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)

DS>0
This is also the second law of thermodynamics
Entropy always increases
Why?

The 2nd law is based on statistics
State Function
Entropy is a property of system

Can relate S to Q and W and thus P, T and V
DS = nRln(Vf/Vi) + nCVln(Tf/Ti)

Not how the system changes
ln 1 = 0, so if V or T do not change, its term
drops out
Statistical Mechanics

We will use statistical mechanics to explore
the reason why gas diffuses throughout a
container

The box contains 4 indistinguishable
molecules
Molecules in a Box
There are 16 ways that the molecules can be
distributed in the box

Since the molecules are indistinguishable there are
only 5 configurations

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

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!)

n! = n(n-1)(n-2)(n-3) … (1)

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

We can express the entropy with Boltzmann’s
entropy equation as:
S = k ln W

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

All real processes are irreversible, so entropy
will always increases

The universe is stochastic
Arrows of Time
Three arrows of time:
Thermodynamic

Psychological

Cosmological
Direction of increasing expansion of the
universe
Entropy and Memory

Memory requires energy dissipation as
heat

Psychological arrow of time is related
to the thermodynamic
Synchronized Arrows
Why do all the arrows go in the same direction?

Can life exist with a backwards arrow of time?

Does life only exist because we have a universe
with a forward thermodynamic arrow? (anthropic
principle)
Fate of the Universe
If the universe has enough mass, its
expansion will reverse

Cosmological arrow will go backwards

Universe seems to be open

Heat Death
Entropy keeps increasing

Stars burn out

Can live off of compact objects, but
eventually will convert them all to heat

Next Time
Read: 20.5-20.7
Homework: Ch 20, P: 6, 7, 21, 22