1.2 The Ideal Gas
3
be a truly daunting task. We shall therefore reduce our goal to that of learning
about and understanding mainly the subfield of Statistical Thermodynamics, which
has the derivation of the laws of thermodynamics from fundamental principles as
one of its main objectives. To see what it is that we are trying to accomplish, we
shall first consider a simple example that illustrates both the distinction between
the microscopic and macroscopic descriptions of matter and the procedures that we
wish to follow in the remainder of this book.
1.2 The Ideal Gas
The concept of an ideal gas is, in principle, an abstraction. It is traditionally
represented in terms of a collection of N noninteracting point particles confined
to a vessel of volume V , and in thermal equilibrium with the vessel walls (or
‘surroundings’). Thermal equilibrium is characterized by a common temperature T .
The bulk gas behaviour is then represented by an ‘equation of state’ that may also
be termed an ‘ideal gas law’ that relates the pressure, P , that the ideal gas particles
exert on the vessel walls to the number of particles (or the number of moles), the
gas volume, and the temperature of the system. We shall consider two models for
the ideal gas, one classical and the other quantum mechanical.
The most commonly encountered ideal gas is that of noninteracting point
particles that obey the laws of classical mechanics; we shall see that this model
gives rise to the canonical ideal gas law,
P V = Nk B T ,
in which k B ≡ 1.380649 × 10 −23 J K
−1 is the Boltzmann constant. Note that
this traditional representation of ideal gas particles ignores interparticle and/or
particle–wall interactions that must exist in order for the gas to achieve thermal
equilibrium with its surroundings, minimally via energy exchange between gas
particles and the container walls. This ideal gas model provides an increasingly
accurate description of gas behaviour as its (number) density n ≡ N/V decreases.
Indeed, this idealized equation of state has become so ubiquitous that it is commonly
referred to as the ideal gas equation of state. Let us note that the classical ideal gas
is described in terms of three independent variables, namely, pressure P , volume V ,
and temperature T .
Photons (which are quanta of electromagnetic field energy) can be considered
to be ideal quantum mechanical particles, if for no other reason than that they
cannot be described in terms of classical mechanics. As they have rest mass zero
and travel at the speed of light, they are also inherently relativistic. A thought
experiment along lines proposed by Leff [1] will allow us a means for envisaging
the construction of a photon gas from energy stored in the container walls. Consider
a cylindrical container closed at one end and fitted with a frictionless piston, with
the entire assembly surrounded by an ultrahigh vacuum chamber and maintained
3
be a truly daunting task. We shall therefore reduce our goal to that of learning
about and understanding mainly the subfield of Statistical Thermodynamics, which
has the derivation of the laws of thermodynamics from fundamental principles as
one of its main objectives. To see what it is that we are trying to accomplish, we
shall first consider a simple example that illustrates both the distinction between
the microscopic and macroscopic descriptions of matter and the procedures that we
wish to follow in the remainder of this book.
1.2 The Ideal Gas
The concept of an ideal gas is, in principle, an abstraction. It is traditionally
represented in terms of a collection of N noninteracting point particles confined
to a vessel of volume V , and in thermal equilibrium with the vessel walls (or
‘surroundings’). Thermal equilibrium is characterized by a common temperature T .
The bulk gas behaviour is then represented by an ‘equation of state’ that may also
be termed an ‘ideal gas law’ that relates the pressure, P , that the ideal gas particles
exert on the vessel walls to the number of particles (or the number of moles), the
gas volume, and the temperature of the system. We shall consider two models for
the ideal gas, one classical and the other quantum mechanical.
The most commonly encountered ideal gas is that of noninteracting point
particles that obey the laws of classical mechanics; we shall see that this model
gives rise to the canonical ideal gas law,
P V = Nk B T ,
in which k B ≡ 1.380649 × 10 −23 J K
−1 is the Boltzmann constant. Note that
this traditional representation of ideal gas particles ignores interparticle and/or
particle–wall interactions that must exist in order for the gas to achieve thermal
equilibrium with its surroundings, minimally via energy exchange between gas
particles and the container walls. This ideal gas model provides an increasingly
accurate description of gas behaviour as its (number) density n ≡ N/V decreases.
Indeed, this idealized equation of state has become so ubiquitous that it is commonly
referred to as the ideal gas equation of state. Let us note that the classical ideal gas
is described in terms of three independent variables, namely, pressure P , volume V ,
and temperature T .
Photons (which are quanta of electromagnetic field energy) can be considered
to be ideal quantum mechanical particles, if for no other reason than that they
cannot be described in terms of classical mechanics. As they have rest mass zero
and travel at the speed of light, they are also inherently relativistic. A thought
experiment along lines proposed by Leff [1] will allow us a means for envisaging
the construction of a photon gas from energy stored in the container walls. Consider
a cylindrical container closed at one end and fitted with a frictionless piston, with
the entire assembly surrounded by an ultrahigh vacuum chamber and maintained