29
So, now let’s consider the reversible isothermal expansion of an ideal gas.
Let the volume double in this expansion.
Δ
E = q
rev
+ w
rev
= 0, so q
rev
= w
rev
= +RT ln (V
f
/V
i
)
Now, let’s define
Δ
S = q
rev
/T, so
Δ
S = +R ln (V
f
/V
i
). ]
In this particular case,
Δ
S = +R ln 2
So, we see that the entropy change is
positive
for this expansion.
We claim that this
expansion is thermodynamically
favorable
.
Note that we must calculate the entropy change using the heat transferred
reversibly
.
So, if we want to calculate entropy changes, we need to be sure that we know the heat
transfer for a reversible
process that takes us from the initial to the final state.
Before we examine the implications of this idea, let’s compare this results with the
statistical definition of entropy.
According to Boltzmann,
Δ
S =
S
f
– S
i
= k ln
Ω
f
 k ln
Ω
i
If we use our simple model for configurational entropy, there is only one microstate of
the system in which all N particle are on the left side of the chamber.
Ω
i = 1. S
i
= 0.
If
we have Avogadro’s number of molecules distributed over both
halves of the box in the
final state, then
Ω
f
=
.
So, S
f
= k ln
= kN
0
ln 2 = R ln 2.
This is exactly the result we got from the thermodynamic definition
Δ
S = q
rev
/T.
The thermodynamic definition is actually the most fundamental
definition of
the entropy
change.
When is a process irreversible?
Reversibility and Irreversibility
The quantity q
irrev
/T
has no meaning
.
The Second Law of Thermodynamics says that a process is irreversible if the entropy
change
for the Universe (system + surroundings)
is positive.
So, we need to
calculate the entropy change for the system and the surrounding to know whether a
process is spontaneous or not.
P
gas
P
gas
P
ext
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Let’s go back to our expansion of an ideal gas from V to 2V.
Let’s compare reversible
and irreversible
processes.
We saw that the Carnot
cycle, comprised of reversible processes, was a very efficient way to convert heat
into work.
When we compared the Carnot cycle with a simpler “square” cycle
operating under similar conditions, we found that the efficiency of converting heat
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 Spring '10
 farrar
 Entropy, entropy change, K1 mol1, Pgas Pext

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