Download Entropy and The Second Law of Thermodynamics

LECTURE 10
ENTROPY AND THE SECOND LAW
OF THERMODYNAMICS
Text Sections
21.2, 21.3
Sample Problems
21.1, 21.2
Suggested Questions
3, 4
Suggested Problems
29P
Summary
Entropy of an equilibrium state
Quasistatic (reversible) processes
Changes in entropy during processes
Entropy as a state function
The Second Law of Thermodynamics
Specific objectives
•
•
•
Explain the difference between quasistatic (reversible) and non-quasistatic
(irreversible) processes
Calculate the change in entropy for quasistatic (reversible) processes and for nonquasistatic (irreversible) processes using the entropy change for quasistatic
processes
Explain the Second Law of Thermodynamics and its relation to entropy
LECTURE 10
ENTROPY AND THE SECOND LAW
OF THERMODYNAMICS
Entropy of an equilibrium state
In the last lecture we defined entropy in terms of the probability of various microscopic
configurations, i.e. in terms of the microscopic disorder or complexity of the system.
Historically entropy was first introduced in (macroscopic) thermodynamics in order to
describe the behaviour of steam engines. We will follow this macroscopic approach in
the last three lectures.
The Zeroth Law indicates that it is possible to associate a thermodynamic parameter,
temperature, with each equilibrium state.
The First Law indicates that it is possible to associate another thermodynamic parameter,
internal energy, with each equilibrium state. The Law is a statement of conservation of
energy.
As we have seen there are processes which are consistent with the First Law, i.e. which
conserve energy, but which do not occur naturally. The Second Law and entropy are
required to determine which processes can occur.
Entropy, like temperature and energy, is associated with equilibrium states. Unlike
energy however it is not conserved in real processes occurring in isolated (closed)
systems when an internal constraint is removed. As we saw last lecture, for example,
entropy increases in a fee expansion of a gas. The process is macroscopically
irreversible.
***
Changes in entropy during processes
In Lecture 5 we discussed the fact that real processes of gas systems cannot be plotted on
a p – V diagram because all states other than the initial and final states are not
equilibrium states. Only these two states can be plotted (see Fig 21-2). If the process is
carried out very slowly in a large number of small steps then the system is never far from
equilibrium and the process can be approximated by a line on a p – V diagram joining the
end points. The idealised process represented by the line is called a quasistatic process
(i.e. almost static). (Some textbooks including HRW call it a reversible process. This
can lead to confusion as reversible is also used with a different meaning.) Quasistatic
processes are reversible in the sense that the line can be traversed in either direction.
Examples of non-quasistatic (irrreversible) processes include:
free expansion of a gas (see Fig 21-1); and
two systems with different initial temperatures coming to thermal equilibrium.
Similar considerations to the above apply to other types of thermodynamic system
described by thermodynamic parameters analogous to p and V. Some examples of such
systems were given in Lecture 5.
The change in entropy of a system in a quasistatic process between an initial ( i ) and a
final ( f ) equilibrium state is defined by
f
∆ S = Sf − Si =
∫
i
dQ
T
where dQ is the infinitesimal amount of heat transferred to the system and T is the
temperature (in kelvin!) at that stage of the process.
If we know the entropy at any one equilibrium state then this equation can be used to find
the entropy at all other equilibrium states of the system. Even if we don’t know the
entropy at any equilibrium state we can find the entropy at all equilibrium states except
for an additive constant (c.f. potential energy).
Once this is done the change in entropy during any process (whether it is quasistatic or
not) between two equilibrium states can be found.
***
Entropy as a state function
Guided by the insight provided by the microscopic description we have asserted that
entropy is a property of an equilibrium state. In a purely macroscopic description this has
to be determined by careful experiments on many different systems (c.f. internal energy).
It is possible however to verify that entropy is a state function in the special case of an
ideal gas.
Consider a quasistatic process from an initial equilibrium state (i ) to a final equilibrium
state (f ). The First Law applies to each of the infinitesimal sub-processes, i.e.
d Eint = dQ − dW
.
Hence
dQ
d E int
dW
=
+
T
T
T
n cV d T
pd V
dT
dV
=
+
= n cV
+ nR
T
T
T
V
by using the Ideal Gas Law.
Therefore for the full process
∆ S = S − Si
f
=
∫
i
dQ
= n cV
T
Tf
∫
Ti
dT
+ nR
T
Vf
∫
Vi
dV
V
.
This type of integral was evaluated in Lecture 6 (see also derivation of Eqn 20-10 of
HRW). Therefore
T 
V 
S f − Si = n cV l n  f  + n R l n  f 
 Ti 
 Vi 
.
The right-hand side of the equation is independent of the thermodynamic parameters for
the intermediate states of the process. The difference in entropy therefore depends only
on the initial and final equilibrium states and does not depend on the particular quasistatic
process between them. For example, the entropy change is the same for the two
processes shown in Figs 21-1 (non-quasistatic) and 21-3 (quasistatic).
The equation can also be used to find the change in entropy during a non-quasistatic
process between the two equilibrium states.
If the initial and final temperatures are equal, i.e. Tf = Ti , then the first term in the
above equation is zero and
V 
∆ S = Sf − Si = n R ln  f 
 Vi 
.
This includes the special case where the process is isothermal, i.e. T = Tf = Ti
.
For a free expansion process in which the final volume is twice the initial volume
∆ S = n R ln ( 2 )
in agreement with the microscopic result.
If the initial and final volumes are equal, i.e. Vf = Vi , then the second term in the above
equation is zero and
 T 
∆ S = Sf − Si = n cV ln  f 
 Ti 
.
This includes the special case where the process is isochoric, i.e. V = Vf = Vi
.
The Second Law of Thermodynamics
This Law can be stated in many ways. For example:
The total entropy of all systems taking part in a process never decreases. It remains the
same only if the process is quasistatic.
The entropy of an isolated (closed) system can never decrease. It remains the same only
if all internal processes are quasistatic.
Note that real processes are never exactly quasistatic. Total entropy (often grandly called
the entropy of the Universe even though only a few parts of the Universe are involved)
always increases if any change occurs.
The first statement of the Law above emphasises that it is necessary to consider the
changes in entropy of all systems taking part in the process. The entropy of one or more
systems can decrease but only if the entropy of one or more other systems increase
sufficiently for the total entropy to either remain the same or increase.
For example, in the quasistatic isothermal expansion of an ideal gas system, a positive
amount of heat Q is transferred to the gas at a constant temperature T . During the
process the entropy of the gas increases by Q / T . The heat is extracted from a heat
reservoir at temperature T . This is a quasistatic isothermal process for that system also.
The entropy of the heat reservoir decreases by Q / T . The total entropy of the two
systems (gas and reservoir) does not change.
If the gas system is compressed, its entropy decreases and that of the reservoir increases
by the same amount.
Alternatively the gas system and the reservoir can be regarded as together forming an
isolated system. Since all internal processes are quasistatic the entropy of that isolated
system does not change.
Entropy is sometimes called the “Arrow of Time” since it indicates whether a process or
its time-reversed form will occur in nature. Only one of these two real processes can
occur spontaneously.