4
1 Basic Background Material
at temperature T , and with the piston face initially in contact with the back wall
of the cylinder. We shall also stipulate that all interior walls of the container be
ultrasmooth. This initial configuration thus represents the zero-volume state for our
photon gas, with N = 0 photons present. Upon moving the piston very slowly away
from the end wall of the cylinder, the container walls (including the piston face)
will emit photons into the space created by the receding piston. Some of these
photons will scatter elastically off the ultrasmooth walls of the container, some
will be absorbed by the walls, and new photons will be emitted by the walls. A
dynamic equilibrium will be established once the average absorption and emission
rates balance. 1 This construct gives us a fully quantum mechanical photon gas having
volume V and temperature T as independent variables. As photons can be emitted
and absorbed, the number of photons is not an independent variable but depends
upon both V and T , i.e., N ≡ N(T , V ).
1.2.1 Classical Ideal Gas Equation of State: Microscopic
Derivation
As we are all aware, the ideal gas law relates the pressure, volume, and temperature
of an ideal gas (a gas in which interparticle interactions can be ignored). You will
also remember that this law is utilized to calculate one of the three variables in terms
of known values of the other two. Perhaps the most difficult of the three variables
to interpret is the pressure of a gas. At a qualitative level, we can view it either
macroscopically as a force per unit area, or microscopically as due to myriads of
molecules impinging upon a surface, thereby causing both the molecules and the
surface to recoil.
Let us introduce a model for explaining gas pressure. We shall attempt to carry
out a quantitative description of gas pressure based upon a model (which is idealized
in order to dispense with irrelevant details and to emphasize the essential elements).
Incidentally, this is typical of scientific procedure as a first step in the study of a
phenomenon. For our model, let us consider a frictionless piston at one end of a
rigid cylinder (see Fig. 1.1). Let the area of the piston face be A. We shall fill the
piston between the rigid end wall and the piston with atoms of some gas. Let us
represent these atoms as rigid (hard) spheres in constant motion. The atoms, in
moving around, will collide with one another, the walls of the chamber, and the
piston face. If we have a vacuum to the right of the piston (see Fig. 1.1), then the
piston will be pushed to the right, and to maintain the piston at position x, a force
F will have to be applied from the right. The relevant question at this point is ‘How
much force?’. Let us define the pressure in terms of force and area as
1 In this sense, it would seem that Aristotle may inadvertently have been correct with his postulate
‘Horror vacui’ (commonly expressed as ‘Nature abhors a vacuum’).
1 Basic Background Material
at temperature T , and with the piston face initially in contact with the back wall
of the cylinder. We shall also stipulate that all interior walls of the container be
ultrasmooth. This initial configuration thus represents the zero-volume state for our
photon gas, with N = 0 photons present. Upon moving the piston very slowly away
from the end wall of the cylinder, the container walls (including the piston face)
will emit photons into the space created by the receding piston. Some of these
photons will scatter elastically off the ultrasmooth walls of the container, some
will be absorbed by the walls, and new photons will be emitted by the walls. A
dynamic equilibrium will be established once the average absorption and emission
rates balance. 1 This construct gives us a fully quantum mechanical photon gas having
volume V and temperature T as independent variables. As photons can be emitted
and absorbed, the number of photons is not an independent variable but depends
upon both V and T , i.e., N ≡ N(T , V ).
1.2.1 Classical Ideal Gas Equation of State: Microscopic
Derivation
As we are all aware, the ideal gas law relates the pressure, volume, and temperature
of an ideal gas (a gas in which interparticle interactions can be ignored). You will
also remember that this law is utilized to calculate one of the three variables in terms
of known values of the other two. Perhaps the most difficult of the three variables
to interpret is the pressure of a gas. At a qualitative level, we can view it either
macroscopically as a force per unit area, or microscopically as due to myriads of
molecules impinging upon a surface, thereby causing both the molecules and the
surface to recoil.
Let us introduce a model for explaining gas pressure. We shall attempt to carry
out a quantitative description of gas pressure based upon a model (which is idealized
in order to dispense with irrelevant details and to emphasize the essential elements).
Incidentally, this is typical of scientific procedure as a first step in the study of a
phenomenon. For our model, let us consider a frictionless piston at one end of a
rigid cylinder (see Fig. 1.1). Let the area of the piston face be A. We shall fill the
piston between the rigid end wall and the piston with atoms of some gas. Let us
represent these atoms as rigid (hard) spheres in constant motion. The atoms, in
moving around, will collide with one another, the walls of the chamber, and the
piston face. If we have a vacuum to the right of the piston (see Fig. 1.1), then the
piston will be pushed to the right, and to maintain the piston at position x, a force
F will have to be applied from the right. The relevant question at this point is ‘How
much force?’. Let us define the pressure in terms of force and area as
1 In this sense, it would seem that Aristotle may inadvertently have been correct with his postulate
‘Horror vacui’ (commonly expressed as ‘Nature abhors a vacuum’).