Page 27 - Thermodynamics of Biochemical Reactions

2.2 Fundamental Equation for the Internal Energy 21

state of a mass of an ideal gas is specified by its temperature 7; pressure P, and
amount n because the volume V is given by V = nRT/P, where R is the gas
constant. Alternatively, I! 7; P or I! P, n, or I! 7; n could be specified. An ideal
gas is a special case, but three properties are sufficient to define the state of a
simple system (that is one substance in one phase) provided one property is
extensive. If work in addition to PV work is involved, more variables have to be
specified. These properties and others that characterize the state of a system are
referred to as state properties because they determine the state of the system. The
internal energy U is also a state property. It is appropriate to call them state
functions because they are independent of the path of change and can be
manipulated by the operations of algebra and calculus.
A thermodynamic property is said to be extensive if the magnitude of the
property is doubled when the size of the system is doubled. Examples of extensive
properties are volume V and amount of substance n. A thermodynamic property
is said to be intensive if the magnitude of the property does not change when the
size of the system is changed. Examples of intensive properties are temperature,
pressure, and the mole fractions of species. The ratio of two extensive properties
is an intensive property. For example, the ratio of the volume of a one-component
system to its amount is the molar volume: V, = V/n.
Experience shows that for a system that is a homogeneous mixture of N,
substances, N, + 2 properties have to be specified and at least one property must
be extensive. For example, we can specify 7; P, and amounts of each of the N,
substances or we can specify 7; P, and mole fractions x, of all but one substance,
plus the total amount in the system. Sometimes we are only interested in the
intensive state of a system, and that can be described by specifying N, + 1
intensive properties for a one-phase system. For example, the intensive state of a
solution involving two substances can be described by specifying 7; P, and the
mole fraction of one of substances.
When other kinds of work are involved, it is necessary to specify more
variables, but the point is that when a small number of properties are specified,
all the other properties of the system are fixed. This is in contrast with the very
large number of properties that have to be specified to describe the microscopic
state of a macroscopic system. In classical physics the complete description of a
mole of an ideal gas would require the specification of 3N, components in the
three directions of spatial coordinates and 3N, components of velocities of
molecules, where N, is the Avogadro constant.

2.2 FUNDAMENTAL EQUATION FOR THE INTERNAL
ENERGY

The first and second laws of thermodynamics for a homogeneous closed system
involving only PV work lead to the fundamental equation for the internal energy
U:

dU = TdS - PdV (2.2-1)
where T is the temperature, S is the entropy of the system, P is the pressure, and
V is the volume. The first law states that dU = dq - PdV when only pressure-
volume work is involved and the second law states that dS > dq/z where q is heat
and d indicates an inexact differential. The integral of an inexact differential
depends on the path, but the integral of an exact differential does not. The test
for exactness is given in Section 2.3. The greater than or equal sign indicates that
the difl'erential entropy is equal to dq/T for a reversible process and is greater than
dq/T in a spontaneous process. It is important to note that all five thermodynamic
properties in equation 2.2-1 have exact differentials and that the fundamental
equation for U is written in the notation of calculus. This means that these
properties behave like mathematical functions so that further relations can be