2.1 Basic Thermodynamic Definitions
29
To illustrate more precisely what we mean by extensive and intensive thermodynamic properties, we shall examine the specific example of a single-component
system, made up of a macroscopic sample of a pure chemical substance which, for
simplicity, we shall take to be an ideal gas. The relevant thermodynamic equation
of state is then the ideal gas law, P V = Nk B T . The mass of the system is given
by M = Nm, with m the mass of an individual ideal gas particle. Because the
number of moles, n, in the system is given by n = N/N 0 , with N 0 the Avogadro
number, it is clear that n is an extensive property of the system, as it depends directly
upon the amount of the substance. Recall that the temperature and pressure of a
system are known from experiment to be intensive properties. It will then be clear
that the volume, V , of this ideal gas system will be proportional to N , so that V
is an extensive property of this system. We have seen that the internal energy of
an ideal gas is given by U =
3
2 Nk B T , so that U is also an extensive property
of this system. These are special cases of a more general behaviour associated
with what are known in mathematics as homogeneous functions (see Sect. B.1.2).
It is by now well-established that all extensive thermodynamic state functions are
homogeneous functions of first order in the extensive variables and that all intensive
thermodynamic state functions are homogeneous functions of zeroth order in the
extensive variables.
Intensive properties may be further employed to characterize a system as
spatially homogeneous if they are continuous functions of position throughout the
region of space occupied by the system or as spatially heterogeneous if they are
not. In particular, a heterogeneous system will typically consist of two or more
distinct homogeneous regions separated by surfaces of discontinuity; each distinct
homogeneous region in a heterogeneous system is termed a phase.
A system that is simultaneously in a state of mechanical equilibrium (in which no
unbalanced force exists either in the interior of the system or between the system and
its surroundings) and a state of chemical equilibrium (i.e., suffering no spontaneous
change in its internal composition or transfer of matter from one part of the system
to another, however, slowly) is said to be in a state of thermal equilibrium should
there be no spontaneous change in the thermodynamic properties characterizing its
mechanical and chemical equilibrium when it is separated from its surroundings and
isolated. We note in particular that all parts of a system in thermal equilibrium are
said to be at the same temperature.
A system is said to be in thermodynamic equilibrium should all requirements
for mechanical, chemical, and thermal equilibrium of the system be satisfied
simultaneously. Moreover, if two closed equilibrium systems are placed in thermal
contact, each system will either remain in equilibrium upon contact or it will
not. Temperature is the property of a system that determines whether or not it
will remain in equilibrium upon thermal contact with another system. Should a
system not remain in equilibrium once thermal contact with another system has
been established, we say that the temperatures of the two systems are unequal.
When two closed systems at different temperatures, say T 1 and T 2 , are brought into
thermal contact, both systems undergo changes that lead to final states of thermal
equilibrium that correspond to a common temperature that lies intermediate between
29
To illustrate more precisely what we mean by extensive and intensive thermodynamic properties, we shall examine the specific example of a single-component
system, made up of a macroscopic sample of a pure chemical substance which, for
simplicity, we shall take to be an ideal gas. The relevant thermodynamic equation
of state is then the ideal gas law, P V = Nk B T . The mass of the system is given
by M = Nm, with m the mass of an individual ideal gas particle. Because the
number of moles, n, in the system is given by n = N/N 0 , with N 0 the Avogadro
number, it is clear that n is an extensive property of the system, as it depends directly
upon the amount of the substance. Recall that the temperature and pressure of a
system are known from experiment to be intensive properties. It will then be clear
that the volume, V , of this ideal gas system will be proportional to N , so that V
is an extensive property of this system. We have seen that the internal energy of
an ideal gas is given by U =
3
2 Nk B T , so that U is also an extensive property
of this system. These are special cases of a more general behaviour associated
with what are known in mathematics as homogeneous functions (see Sect. B.1.2).
It is by now well-established that all extensive thermodynamic state functions are
homogeneous functions of first order in the extensive variables and that all intensive
thermodynamic state functions are homogeneous functions of zeroth order in the
extensive variables.
Intensive properties may be further employed to characterize a system as
spatially homogeneous if they are continuous functions of position throughout the
region of space occupied by the system or as spatially heterogeneous if they are
not. In particular, a heterogeneous system will typically consist of two or more
distinct homogeneous regions separated by surfaces of discontinuity; each distinct
homogeneous region in a heterogeneous system is termed a phase.
A system that is simultaneously in a state of mechanical equilibrium (in which no
unbalanced force exists either in the interior of the system or between the system and
its surroundings) and a state of chemical equilibrium (i.e., suffering no spontaneous
change in its internal composition or transfer of matter from one part of the system
to another, however, slowly) is said to be in a state of thermal equilibrium should
there be no spontaneous change in the thermodynamic properties characterizing its
mechanical and chemical equilibrium when it is separated from its surroundings and
isolated. We note in particular that all parts of a system in thermal equilibrium are
said to be at the same temperature.
A system is said to be in thermodynamic equilibrium should all requirements
for mechanical, chemical, and thermal equilibrium of the system be satisfied
simultaneously. Moreover, if two closed equilibrium systems are placed in thermal
contact, each system will either remain in equilibrium upon contact or it will
not. Temperature is the property of a system that determines whether or not it
will remain in equilibrium upon thermal contact with another system. Should a
system not remain in equilibrium once thermal contact with another system has
been established, we say that the temperatures of the two systems are unequal.
When two closed systems at different temperatures, say T 1 and T 2 , are brought into
thermal contact, both systems undergo changes that lead to final states of thermal
equilibrium that correspond to a common temperature that lies intermediate between