Modern Engineering Thermodynamics

748 CHAPTER 18: Introduction to Statistical Thermodynamics
where W(A) is the number of ways that macrostate A can occur (i.e., the number of microstates per macrostate A),
and ∑ i W(i) is the total number of macrostates possible.
Then, the condition we call thermodynamic equilibrium is simply the macrostate that has the largest value of P.
A macrostate is an overview of a complex situation, whereas a microstate describes the details of how each element
of the system functions. As an analogy, consider a nationalpresidentialelection.Themacrostatesarethe
various possible winners of the election, and the microstates are the various combinations of ways in which the
voters may cast their ballots.
Late in the 19th century, electrical discharge experiments in various gases produced light emission spectra that
were very unusual. Instead of being an emission with a continuous color frequency (like white light), the emissions
consisted of discrete spectral lines located at fixed wavelengths. Figure 18.7 shows the emission spectrum
of atomic hydrogen in the visible region of the electromagnetic spectrum. From these emission spectra, it was
clear that, if the emission phenomenon is attributed to photon ejection by electrons as they move from an
atom’s outer orbit to an inner orbit, then the electrons must occupy discrete orbits and consequently are not
simply clustered around the nucleus in a random manner.
In 1913, Niels Bohr hypothesized that the electron orbits of an atom were quantitized (i.e., made discrete)
according to the value of the electron’s angular momentum as
mω = mVr = n ħ
2π
where ω is the angular velocity of the electron, V is the electron’s orbital velocity, r is the radius of the orbit, ħ is
Planck’s constant, and n = 1, 2, 3, … is the (primary) quantum number. Therefore, the radius of an electron’s
orbit is given by
r =
ħ
2πmV n
As the years passed, other quantum numbers had to be introduced to account for such things as the elliptical
shape of the orbit (this accounted for the finite width of the emission lines and was called the azimuthal quantum
number), the splitting of the spectral lines in a strong magnetic field (the magnetic quantum number), the
magnetic moment associated with the direction of electron spin (the electron spin quantum number), and so
forth.
The continual modification of the original Bohr model required to make it conform to experimental observations
started physicists looking for a new model. In 1924, Louis Victor Pierre Raymond de Broglie (1892–1987)
used an analogy between classical mechanics and geometric optics to formulate a dual particle-wave model for
matter. He argued that since the energy ε of a photon is given by
ε = ħv
where ħ is Planck’s constantandv is the photon’s frequency, and since Einstein’s mass-energy relation for the
photon is
ε = mc 2
where m is its mass and c is the velocity of light, then the linear momentum p of the photon can be written as
p = mc = ħv
c = ħ λ
Color
Violet Blue Green Red
389
397
410
434
486
656
Wavelength (nm)
FIGURE 18.7
The emission spectrum of atomic hydrogen.