Today’s Lecture
Reversibility
Entropy a State Variable
Irreversibility
Where do we find irreversible processes?...
Pretty much everywhere, damn it!..
And we are not getting any younger either!..
You can’t possibly run
that movie back…
Losing, breaking, destroying,
saying stupid things….
Seriously.
Three common scenarios of irreversibility in thermodynamics.
1) Mixing and loosing structural order in general. Two molecularly
mixed fluids never “unmix”.
http://mutuslab.cs.uwindsor.ca/schurko/animations/irreversibility/happy.htm
A broken vase never repairs itself…
2) Conversion of mechanical energy into internal energy (dissipation
into heat).
Ordered motion of an object is converted into disordered motion of
its molecules. Never coming back…
http://mutuslab.cs.uwindsor.ca/schurko/animations/secondlaw/bounce.htm
3) Heat transfer from a hotter to a cooler object – never goes in the
opposite direction.
Irreversibly lost
opportunities...
#1 Expanding gas…
On the way from a to b the gas could
be harnessed to do some mechanical
work at expense of its internal energy…
W = −ΔU
Q=0
Instead of that we have
W =0
Maxwell’s
demon
ΔU = 0
#2 Two systems with different temperatures reaching
equilibrium…
There was an opportunity for a spontaneous
process – heat flow from Th to Tc.
It could be used to run a heat engine
between the two reservoirs (hot and cold).
Maxwell’s demon: high speed molecules
go to the right, low speed – to the left.
Maxwell distribution after thermal
equilibrium is established… Order is lost!
There is no way the molecules would spontaneously break
into two groups – with high and low temperatures.
Entropy
Entropy provides a quantitative measure of disorder.
Consider the isothermal expansion of an ideal gas. If we add heat dQ and let
the gas expand just enough to keep the temperature constant, then from the
first law:
The gas becomes more disordered because there is a larger volume and hence
more randomness in the position of the molecules.
We define the infinitesimal entropy change dS during an
infinitesimal process as:
For an isothermal process the change in entropy is ΔS = Q/T. Higher temperature
implies greater randomness of motion. If the substance is initially cold then adding heat
causes a substantial fractional increase in molecular motion (and randomness). But if
the substance is already hot then adding the same quantity heat adds relatively little
molecular motion that what was already present. Hence Q/T characterizes the increase
in randomness (or disorder) when heat flows into a system.
Entropy
Consider the Carnot
Cycle where we found
If we change the definition of Qc so that it is
the heat added vs heat rejected then
Any closed reversible cycle can be made
up of incrementally small isothermal and
adiabatic cycles. This leads to
This means that the integral
representing the change in entropy,
is path
independent!
What is the change in entropy for an
arbitrary reversible cyclic process?
Entropy
If we take a system around a path that is not closed then its entropy does
change. Since the change in ΔS is path independent, entropy is a state
property, like temperature, and is independent of the path between state 1
and state 2.
Consider mixing 1kg of 0oC with 1kg of 100oC water, an irreversible process.
To find the change in entropy, assume two reversible processes, (i) slowly
heating the cold water to 50oC while (ii) slowly cooling the hot water to 50oC.
This example is an irreversible processes. However entropy is a state
property which is independent of how you got to that state. Hence this result
was obtained by reaching the final state via two reversible processes.
Note that ΔS is greater than zero!
Entropy
Consider mixing .5kg of Cu at 300oC with 2kg of 20oC water, an irreversible
process. To find the change in entropy assume two reversible processes,
(i) slowly heating the water to the equilibrium temperature and (ii) slowly cooling
the Cu to the equilibrium temperature.
First we must determine the equilibrium temperature, Teq, obtained via
the two reversible processes:
The change in entropy is given by the integral of dQ/T for these two processes:
Again, this result was obtained by reaching the final state via two
reversible processes. Again ΔS is greater than zero!
Entropy
This definition of entropy is only meaningful for reversible processes. An
irreversible process takes a system out of equilibrium, this means that T
may not be well defined. But entropy is a state variable, it doesn’t care
how you got to a certain state only what the state is!
Consider adiabatic free expansion, an irreversible process, where the
gas is discharged into a vacuum chamber. Energy is conserved, hence
neither the temperature nor internal energy of the gas changes. From
the ideal gas law in an isothermal expansion:
The energy that became unavailable to do work is:
All of these examples were irreversible processes. Remember entropy is
a state property and as such is independent of how you get to that state.
Hence the results were obtained by reaching the final states via
reversible processes.
Adiabatic Free Expansion
Since entropy is a state variable we can
find the change in entropy with alternate
paths. Again consider adiabatic free
expansion in which the volume increases
by a factor of 4. Since the temperature is
unchanged we get from state A to state C
via a reversible isothermal expansion.
What about a two step process, (i) reversible adiabatic expansion (isentropic
process) followed by (ii) heating via a constant volume process?
As a state variable the change in entropy is path independent! To what
temperature must a gas be cooled after a free expansion from Vi to Vf in order
for the change in entropy to be zero?
Entropy
The examples on the previous slide described systems that evolved into systems of
higher disorder. In free expansion and coming to thermal equilibrium, the ability to do
work was lost as the systems evolved into states of higher disorder.
The mixing of colored ink starts from a state of
relative order to a state that is more disordered. A
state of higher entropy. Spontaneous unmixing, a
net decrease in entropy, is never observed!
In general, “When all systems taking part
are included, the change in entropy is
greater than or equal to zero”.
Entropy
In general, “When all systems taking part are
included, the change in entropy is greater than
or equal to zero”.
As an example consider 4 coins that lie on a table
in random orientations.
There is only 1 way that they have either 4 heads
or 4 tails. These are the orientations of highest
order. There are 4 ways that they have either 3
heads and 1 tail or 3 tails and 1 head. These are
the orientations of next highest order. Finally there
are 6 ways in which there are 2 heads and 2 tails.
This is the state of highest disorder!
For any system, the most probable state is the one
with the greatest number of corresponding
microscopic states, which is also the macroscopic
state with the greatest disorder and largest entropy!
Summary of the Second law of Thermodynamics
Cyclic processes:
Second Law of Thermodynamics:
It is impossible to construct a heat engine that extracts
heat from a reservoir and delivers an equal amount of heat.
Irreversible processes decrease the organization of a system. This
is due to statistical reasons.
Efficiency of a reversible
heat engine is
Entropy measures the relative disorder of a system and corresponds to
decreasing energy quality. The entropy difference between two states is
This integral may be done assuming reversible
processes as entropy is a state variable!