BASIC DEFINITIONS AND CONCEPTS OF THERMODYNAMICS

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27 Οκτ 2013 (πριν από 3 χρόνια και 10 μήνες)

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BASIC DEFINITIONS AND CONCEPTS OF THERMODYNAMICS


We begin with reviewing some definitions of terms. Thermodynamics is a very logical
subject, and words must be precisely defined.

A. System and Surroundings


The types of systems are defined by constraints
imposed by the wall, and are:

1. open system: free exchange of matter and
energy with the surroundings
2. closed system: no matter exchange
3. adiabatic system: no heat exchange
4. isolated system: no exchange of any type




The isolated system is particularly important. Note that isolated system = system +
surroundings (E&C calls this "isolated enclosure")





B. State Functions (Properties)

The state of a system is described by variables known as STATE FUNCTIONS (or
FUNCTIONS OF STATE or PROPERTIES). These are variables that depend only on the
state of the system and not on its history or how the system was brought to its present
state. T, P, V and the thermodynamic functions E (energy), H (enthalpy), S (entropy) and
G (Gibbs free energy) are state functions. State functions characterize the state of the
system, and have the property that changes in the function are independent of the means
by which the change was produced.

It is a remarkable experimental result that the state of a simple system can be completely
described by specification of just two properties of the system and the number of moles
of each component. Therefore, in principle, any one property of the system can be
determined from two other properties and the mole numbers. That is, any property is a
function of two others and the mole numbers as independent variables. The choice of the
two properties is made on the basis of convenience. The following are considered “natural
variables” for the particular property (we will see later why this is so):


P = P(V,T, n
1
, n
2
…n
i
) H = H(S, P, n
1
, n
2
, ..n
i
) E = E(S, V, n
1
, n
2
, ..n
i
)

S = S(E,V, n
1
, n
2
, ..n
i
) G = G(T, P, n
1
, n
2
, ..n
i
)

1
The first equation is called an “equation of state” that relates P,V and T. For a gas the
equation is P = nRT/V. The other equations are known as “fundamental equations” and
provide complete knowledge for computing equilibrium conditions of a system. Of course
we would need a real functional relationship to do this. We will obtain the functional form
for E and G later.

Thermodynamic state space. Since the number of
variables needed to specify a property is only 2 for a fixed
composition, we can visualize equilibrium states on a simple
plot of 3 dimensions (if composition were a variable, we
would need more dimensions). Because any property
depends only on the present state of the system, and not on
how it was created, changes in state functions do not
depend on the path taken, i.e., for either path shown from
state 1 to state 2:

∆V = V
2
– V
1
, ∆P = P
2
– P
1
, ∆T = T
2
– T
1



Exact differentials (sections 1-6 and 1-7 in E&C). The total differentials of state
functions (properties) are called “exact” differentials and have important mathematical
properties. For example, for any property a particular functional relationship exists,
namely,

F = F(x
1
, x
2
, n
1
, n
2
, ..n
i
) The total differential is

i
nxx
i
i
nxnx
dn
n
F
+ dx
)
x
F
( + dx
)
x
F
= dF
ij
ii
















,,
2
,
2
1
,
1
21
12
(
(1)


−=
2
1
12
FFdF
(2)

Equation (2) makes it clear that the change in a property is independent of the path by
which the change was brought about. That is, the value of the integral is F
2
-F
1
, no matter
how the change from 1 to 2 was brought about.

It follows from (2) that



= 0dF
(3)

where the circle in the integral sign indicates integration over a closed path, that is, one
that starts and ends at the same state. None of the properties expressed by equations
(1)-(3) hold for variables that are not state functions of the system.
Not all variables are state functions. As a simple example, consider the cities LA
and San Diego. Suppose we seek a property of San Diego. Consider the distance, s,
2
from LA. Does this characterize San Diego? No, because it depends on the path we take,
and there are an infinite number of paths. The distance must be computed over a specific
path. No specific functional relationship exits between the distance and the variables x,
y on the earth's surface, and ds is not an exact differential. Functions that are not state
functions have a differential known as an inexact differential.
The population of San Diego, on the other hand, characterizes the city, and is
independent on how the population came to be (either by immigration or birth). It is a
property of San Diego.
The reason that state functions are so important in thermodynamics is as follows:
suppose a system undergoes a change in state from 1 to 2, and we wish to compute the
change in the function F. The actual change may be hopelessly complex, involving many
variables, possibly unknown. However, because F is a state function we can devise a
simple, hypothetical path between the same final and initial states along which we can
compute the change, and claim with confidence that ∆F is the same for the both paths.
This allows us to ignore the great complexity of the change and compute

F with the
simple, hypothetical path:

F(actual) =

F(hypothetical).

C. Equilibrium

A system is said to be at equilibrium when all of its macroscopic properties are
time-independent and remain so when the system is isolated from its surroundings. The
latter condition is needed to exclude the steady-state.

D. Intensive and Extensive Properties

Intensive: properties whose values are the same for the entire system or any small part
of it at equilibrium, i.e., they do not depend on the size of the system. Examples are T,
P, density, viscosity.

Extensive: properties that depend on the extent, or size, of the system. Examples are the
volume, E, H, S, and G. If you double the size of the system, you double the volume, etc.
All of the extensive thermodynamic properties are homogeneous of order 1 in the
number of moles of substances (See Euler’s theorem in “useful mathematical
relationships” handout),

E. Work and Heat (section 1-4, E&C)

Work and heat are means by which the state of a system can be changed.

Work:
Work is defined in mechanics as the product of the force (F) in the surroundings
acting on the system and the displacement caused by the action of that force:


dxFdW
ext
=

3

Note: Work in general is not a state function, and the differential dW is not exact. Some
texts emphasize this fact by a "slash" across the d symbol. We cannot directly integrate
this expression to obtain the work without knowing the path followed by the force during
the process. That is, we must know F as a function of x. Work is an algebraic quantity, and
has a sign. We adopt the convention that work done on the system by the
surroundings is positive.

There are many types of work that can be done on (or by) a system. For example, a gas
may be compressed (or expanded) under the influence of an external pressure. This is
the familiar pressure-volume work, and is given by (derived on page 16 of E&C):


dVPdW
ext
−=


where P
ext
is the pressure exerted on the gas by the surroundings, and dV is the volume
change of the system. The sign convention is chosen so that work done on the system
by the surroundings is positive. The volume dV is negative for compression; thus the
work in compression is positive, as it should be. Problem Set #1 will give you the
opportunity to refresh your memory on calculations of work. The emphasis on
pressure-volume work in thermodynamics is simply that reactions occurring under the
constant pressure of the atmosphere may change volume. Thus, PV work is a universal
consideration.

Other kinds of work we will consider (Table 1-3, E&C):

gravitational work,


dmhhgdW )(
0
−=
(equivalent to dW = mgdh, used in E&C)

where g is the acceleration due to gravity, h is the height from a reference point at h
o
. The
quantity g(h-h
0
) is called the gravitational potential (relative to the reference). dW is the
work attending the lifting of a mass dm from height h
0
to h in the earths gravitational field.

units: (SI)
g is 9.8 m/sec
2

h is in meters
m is in kg
W is in Joules

electrical work,


dqdW )(
0
Φ−Φ=
(equivalent to dW=qd
Φ
)

where Φ is the electric potential,
Φ
0
is the electric potential at a reference point, and dq is
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an element of electrical charge. The work corresponds to moving the charge dq from a
point where the electric potential is
Φ
0
volts to one where it is
Φ
volts.

units: (SI)
Φ
is in volts ( a volt is a joule/coulomb)
q is in coulombs (F= 96,500 coulombs/mole of charge)
W is in Joules

surface work,


dAdW
γ
=


where
γ
is the surface tension and A is the area of the surface. This is the work involved
in increasing the area of an interface with interfacial tension
γ
by an amount dA (a problem
involving calculations with surface work is on problem set 1). In analogy with the other
cases,
γ
could be called the potential corresponding to the surface work.

units: (SI) (CGS)

γ
is in joules/m
2
erg/cm
2

A is in m
2
cm
2

W is in Joules W is in ergs

The general formulation of these expressions is that work = (intensive variable)(extensive
variable). The intensive variable, the "potential" corresponding to the force, and is just the
work/unit extensive variable. It may not be immediately obvious, but each of these
expressions is a force x displacement. We will return later to explain the form of the
equations for gravitational and electrical work and how they correspond to the force x
displacement definition.

Although in general work is not a state function, dW is in fact an exact differential for the
simple mechanical work as long as there are no frictional forces involved that dissipate
some of the work as heat. Work becomes a non-state function when heat interactions with
the surroundings are involved in the process (for example frictional losses).

Heat (Q):
Heat, like work, is an algebraic quantity and has a sign. We adopt the convention
that heat absorbed by the system from the surroundings is positive. The common
unit of heat is the Calorie.


F. Irreversible and reversible processes

We have so far discussed states of a system and the means by which changes in state
can be produced (heat and work). We now look at the processes (paths) by which heat
and work interactions take place. We consider two types, reversible and irreversible
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paths. Such paths exist for any process, physical or chemical. The ideas are illustrated
with a simple physical process although they are completely general.

The irreversible process.

Consider a gas in a cylinder fitted with a piston. The gas
and piston are the system, and all else constitutes the
surroundings. Initially, the internal pressure of the gas
(P
int
) exceeds the external pressure (P
ext
), and the
piston is held in place with a lock of some kind. The
system is in a constrained equilibrium (constrained by
the lock). How can we carry out an expansion of the
gas from one state to another? One means would be to
release the lock. In the absence of this constraint, the
piston would then move outward until P
int
=P
ext
. In this
process, the piston moves at a finite velocity, and there
will be frictional losses in the piston generating heat (of unknown amount). In addition, the
gas will be turbulent during the process, and pressure will vary from one point to the next.
This is a very complex process. Nonetheless, we can calculate the work as


dVPdW
ext
−=


)-(- = dV
VVPP
- = W
12extext
V
V
2
1




Notice that here the work appears as a state function, since it depends only on the initial
and final states. Why is this? This is so because we have defined the path as a constant
external pressure expansion. It depends only on the final and initial states and has a
definite value as long as we follow the path defined. Note also that the amount of work
does depend on the path we define; it is linear in P
ext
. The process we have just described
is called a real, actual, or spontaneous process, because it will actually occur. It is also
known as an irreversible process for reasons that will be clear shortly.

The reversible process.

Consider again the expansion of the gas, but this time by a pathway in which we let the
external pressure always be only infinitesimally less than the internal pressure. We need
no lock to constrain the piston, since P
int
= P
ext
within an infinitesimal amount. The piston
will move outward at an infinitesimally slow rate, and the system is essentially in
equilibrium at all times. This process cannot actually be carried out, since it would take an
infinitely long time. Thus, it is not a real or actual process, but is called a reversible
process or a non-spontaneous process. The name “reversible” has its origin in the fact
that an infinitesimal increase in the external pressure will cause the process to reverse.
That is not true for the real process, and hence its name irreversible. The work attending
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