Download Lecture 4 Thermal Equilibrium, Temperature, and Entropy

Lecture 4
Thermal Equilibrium, Temperature, and
Entropy
Recall that, since the systems 1 and 2 are independent of each other, the total multiplicity
for fixed (E, V, N ) is just:
X
Ω(E, V, N, α)
Ω(E, V, N ) =
α
X
=
Ω(E, V, N, E1 , V1 , N1 )
(4.1)
E1 ,V1 ,N1
X
=
Ω1 (E1 , V1 , N1 )Ω2 (E2 , V2 , N2 )
E1 ,V1 ,N1
where the summations are performed over all possible macrostates of the combined system.
If we let N1 and V1 be fixed, then what does the total multiplicity look like as a function of
E1 ?
Ω (E ) Ω (E )
1
1
1
1
0
E*
1
E
1
Figure 4.1: Resultant multiplicity for two systems brought in contact which achieve thermal
equlibrium at E1∗
• Note that this is a “sharp” function for large systems (refer to Prob. 2.21 in Tutorial
1) and only a small number of macrostates dominate the statistics of the combined
system.
• The most probable macrostate (configuration) occurs at E1∗ which corresponds to thermal equilibrium. This is the macrostate which has the greatest multiplicity.
• Since the fluctuations are small, we can replace our averages, hXi over all macrostates
by that of an average only over the most probable configuration. This replacement
should have minimal error.
4.1
Thermal Equilibrium
Let us return to Eq. 4.1 but, as mentioned above, fix N1 , V1 and, since E, V, N, V2 , N2 were
also fixed, thus E1 is the only independent variable. Thus, we have
X
X
Ω(E) =
Ω1 (E1 )Ω2 (E2 ) =
Ω1 (E1 )Ω2 (E − E1 )
(4.2)
E1
E1
By our Third Postulate, we expect that the most probable macrostate corresponding to E1∗
is such that,
¯
µ
¶
µ
¶
dΩ(E) ¯¯
∂Ω1
∂Ω2
dE2
dE1
+ Ω1
= 01
=
Ω2
¯
dE E ∗
∂E1 V1 ,N1 dE1
∂E2 V2 ,N2 dE1
1
µ
∂ ln Ω1
∂E1
¶
V1 ,N1
dE2
dE1
= −1, therefore, dividing by Ω1 Ω2 we find,
µ
¶
µ
¶
1 ∂Ω1
1 ∂Ω2
=
Ω1 ∂E1 V1 ,N1
Ω2 ∂E2 V2 ,N2
µ
¶
∂ ln Ω2
=
∂E2 V2 ,N2
Recall that E2 = E − E1 and, thus,
We define a quantity σ, called the entropy as: the measure of the disorder of a system in a given macrosta
σ(E, V, N ) ≡ ln Ω(E, V, N )
(4.3)
which is just a pure number (dimensionless). Using this definition, we arrive at the fundamental condition for thermal equilibrium for two systems in thermal contact,
µ
∂σ1
∂E1
¶
µ
=
V1 ,N1
∂σ2
∂E2
¶
(4.4)
V2 ,N2
• At thermal equilibrium, there is no substantial heat transfer (although there will be
small temperature fluctuations).
1
When performing thermodynamic derivatives, such as these, we must be careful to clearly label the
macrostate parameters, such as N1 , V1 , which are held fixed while differentiating
• If there is no heat transfer, then the two systems must be at the same temperature,
i.e.,
τ1 = τ2 .
• From Eq. 4.4, this implies that τ is a function of ∂σ/∂E. In particular, we define the
fundamental temperautre, τ , as
¶
µ
1
∂σ
.
(4.5)
=
τ
∂E V,N
• Now the “standard” entropy S, which you are most likely familiar with, is Bolzmann’s
Definitions,
S = kB σ = kB ln Ω(E, V, N )
(4.6)
where kB = 1.381 × 10−23 J/K is Boltzmann’s contant.
• With this definition, we can express the “standard” temperature as,
1
=
T
µ
∂S
∂E
¶
(4.7)
V,N
Some observations about Entropy:
1. Entropy is additive, which can be seen from the definition of the total multiplicity for
two systems in contact,
Ω(E, V, N ) = Ω1 (E1 , V1 , N1 )Ω2 (E2 , V2 , N2 )
∴, S(E, V, N ) = S1 (E1 , V1 , N1 ) + S2 (E2 , V2 , N2 ).
This implies that entropy is an extensive quantity.
Definitions:
Extensive Quantity: proportional to the size of the system → double the size, double
the quantity. Intensive Quantity: independent of the size of the system e.g. temperature, pressure.
2. Entropy is increased when constraints are removed:
↓ Constraints →↑ no. of accessible microstates of the system →↑ no. of degrees of freedom →↑ S
(a)
(b)
Figure 4.2: By removing the partition, a significant constraint on the system is lifted allowing
for a greater number of accessible microstates and, hence, an increase in entropy
Picture: Note: Even though we said at the beginning of the course that a configuration like
(a) (but without the barrier) was not impossible to observe, although highly improbable,
we would never see the system go from this maximum probable configuration (b) to the
initial one (a) (without the barrier). That is, this is an irreversible process, even though the
equations of motion are reversible in time.
The Second Law of Thermodynamics or The Law of Increase of Entropy (Multiplicity): The entropy of a closed system either remains constant or increases when a
constraint is removed.
Consider our two systems in thermal contact and evaluate the “rate of change of entropy”:
dS
∂S dE1
∂S dE2
=
+
dt
∂E1 dt
∂E2 dt
or
Clausius’ Principle:
dS
=
dt
µ
1
1
−
T1 T2
¶
dE1
>0
dt
(4.8)
1
If T1 < T2 , dE
> 0, since heat flows from the subsystem at higher temperature to that at
dt
lower temperature.
Picture:
T1 < T2
1
2
Heat Flow
E1
E2
Entropy
S(E)
Energy
E
E2
E1
t
t
Figure 4.3: A demonstration of Clausius’ Principle: since T1 < T2 , heat will flow from 2 → 1
and the entropy will increase until the two systems are at the same temperature and the
entropy reaches a constant value (maximum threshold for a given total energy E)
Let us apply these to a system composed of two Einstein solids in thermal contact.
Example 4.1 Two Einstein Solids in Thermal Contact
• Consider two Einstein solids, A and B, where NA = 4 and NB = 5 and they contain a
total of 10 units of energy, qtotal = qA + qB = 10.
• When brought into thermal contact, the two systems will exchange energy until they
reach thermal equilibrium → which corresponds to the macrostate (of the combined system) which has the greatest multiplicity and, hence, the greatest probability of occuring
• Calculate the multiplicities for all of the possible macrostates of the combined system?
Which macrostate has the greatest probability of occuring?
Table 4.1: Different possible macrostates and multiplicities for a system of two Einstein solids,
one containing four oscillators, the other having five oscillators, and both sharing a total of ten
units of energy
qA
0
1
2
3
4
5
6
7
8
9
10
ΩA
1
4
10
20
35
56
84
120
165
220
286
qB
10
9
8
7
6
5
4
3
2
1
0
ΩB
1001
715
495
330
210
126
70
35
15
5
1
Ωtotal = ΩA ΩB
1001
2860
4950
6600
7350
7056
5880
4200
2475
1100
286
8000
Ωtot
6000
4000
2000
0
0
2
4
qA
6
8
10
Figure 4.4: Resultant multiplicity for two Einstein solids brought in contact which achieve
thermal equlibrium at qA = 4
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