32
2 Macroscopic Thermodynamics
arbitrarily either the final temperature T f or the final volume V f , but not both, as to
do so would entail a violation of the relation (2.2.3). Although we have determined
this restriction by considering a rather special type of system, it happens that it
extends beyond the ideal gas; indeed, it applies to reversible adiabatic expansions of
any gas.
As it is known that the number of independent thermodynamic state functions for
a system having a fixed mass equals the number of work terms plus one [6], then for
a system that possesses only pressure–volume work, there will be two independent
state functions, which we would normally have chosen as T and V . However, we
have just seen for an adiabatic change that the constraint (2.2.3) only allows us to
choose the final value either of T or of V , but not of both T and V : this restriction
therefore implies that there must be some other thermodynamic state function for
the system that happens not to change value during a reversible adiabatic expansion.
We shall label this (at present) unknown state function by S, so that its change, S,
for a reversible adiabatic process is given by
S = S f − S i ≡ 0 .
This thermodynamic state function has been given the name entropy; as a consequence, adiabatic processes are therefore also referred to as isentropic. Entropy is an
inherently difficult property to understand and appreciate, especially from a strictly
thermodynamic perspective, as there is no known means for actually performing its
direct measurement. We shall, however, be able to arrive at a deeper understanding
and develop a more comprehensive appreciation of this seemingly mysterious, yet
very important, thermodynamic state function once we have had an opportunity to
examine it from the perspective of statistical mechanics.
The three laws of thermodynamics may be expressed in a number of different
ways. For our present purposes, the most appropriate formulation involves the
thermodynamic energy, entropy, and temperature, in terms of which the first law is a
statement of the conservation of energy, the second law is a statement of the increase
of the total entropy (of the universe), while the third law states that the absolute
temperature for a thermodynamic system remains positive. Thermodynamics is thus
concerned with the discovery and nature of universal laws that govern the utilization
and conservation of energy on a macroscopic level. It will therefore be of no surprise
that these laws should be recovered naturally from a microscopic description of
natural systems, such as that provided by statistical mechanics.
The thermodynamic formulation of the first law of thermodynamics couples the
conservation of energy with the knowledge that for a macroscopic system there are
two means available for the transfer of energy between subsystems of an isolated
system, namely work done and heat exchanged. If we split the universe into two
subsystems, one of which is the thermodynamic system of interest, the other the
rest of the universe, and labelled as the ‘surroundings’, then work provides a means
for energy transfer due to unbalanced forces between the thermodynamic system
and its surroundings. By convention, the amount of work done on a system (by its
surroundings) increases the internal energy U of the system by an amount U
2 Macroscopic Thermodynamics
arbitrarily either the final temperature T f or the final volume V f , but not both, as to
do so would entail a violation of the relation (2.2.3). Although we have determined
this restriction by considering a rather special type of system, it happens that it
extends beyond the ideal gas; indeed, it applies to reversible adiabatic expansions of
any gas.
As it is known that the number of independent thermodynamic state functions for
a system having a fixed mass equals the number of work terms plus one [6], then for
a system that possesses only pressure–volume work, there will be two independent
state functions, which we would normally have chosen as T and V . However, we
have just seen for an adiabatic change that the constraint (2.2.3) only allows us to
choose the final value either of T or of V , but not of both T and V : this restriction
therefore implies that there must be some other thermodynamic state function for
the system that happens not to change value during a reversible adiabatic expansion.
We shall label this (at present) unknown state function by S, so that its change, S,
for a reversible adiabatic process is given by
S = S f − S i ≡ 0 .
This thermodynamic state function has been given the name entropy; as a consequence, adiabatic processes are therefore also referred to as isentropic. Entropy is an
inherently difficult property to understand and appreciate, especially from a strictly
thermodynamic perspective, as there is no known means for actually performing its
direct measurement. We shall, however, be able to arrive at a deeper understanding
and develop a more comprehensive appreciation of this seemingly mysterious, yet
very important, thermodynamic state function once we have had an opportunity to
examine it from the perspective of statistical mechanics.
The three laws of thermodynamics may be expressed in a number of different
ways. For our present purposes, the most appropriate formulation involves the
thermodynamic energy, entropy, and temperature, in terms of which the first law is a
statement of the conservation of energy, the second law is a statement of the increase
of the total entropy (of the universe), while the third law states that the absolute
temperature for a thermodynamic system remains positive. Thermodynamics is thus
concerned with the discovery and nature of universal laws that govern the utilization
and conservation of energy on a macroscopic level. It will therefore be of no surprise
that these laws should be recovered naturally from a microscopic description of
natural systems, such as that provided by statistical mechanics.
The thermodynamic formulation of the first law of thermodynamics couples the
conservation of energy with the knowledge that for a macroscopic system there are
two means available for the transfer of energy between subsystems of an isolated
system, namely work done and heat exchanged. If we split the universe into two
subsystems, one of which is the thermodynamic system of interest, the other the
rest of the universe, and labelled as the ‘surroundings’, then work provides a means
for energy transfer due to unbalanced forces between the thermodynamic system
and its surroundings. By convention, the amount of work done on a system (by its
surroundings) increases the internal energy U of the system by an amount U