h/A two-atom system has
the highest number of available
states when the energy is equally
divided. Equal energy division is
therefore the most likely possibil-
ity at any given moment in time.
i/When two systems of 10
atoms each interact, the graph of
the number of possible states is
narrower than with only one atom
in each system.
already been defined in a fundamental manner, so we can take this
as a definition of temperature:
1
T=
dS
dQ,
where dSrepresents the increase in the system’s entropy from adding
heat dQto it.Examples with small numbers of atoms
Let’s see how this applies to an ideal, monoatomic gas with a
small number of atoms. To start with, consider the phase space
available to one atom. Since we assume the atoms in an ideal gas
are noninteracting, their positions relative to each other are really ir-
relevant. We can therefore enumerate the number of states available
to each atom just by considering the number of momentum vectors
it can have, without considering its possible locations. The relation-
ship between momentum and kinetic energy isE= (p^2 x+p^2 y+p^2 z)/ 2 m,
so if for a fixed value of its energy, we arrange all of an atom’s possi-
ble momentum vectors with their tails at the origin, their tips all lie
on the surface of a sphere in phase space with radius|p|=√
2 mE.
The number of possible states for that atom is proportional to the
sphere’s surface area, which in turn is proportional to the square of
the sphere’s radius,|p|^2 = 2mE.
Now consider two atoms. For any given way of sharing the en-
ergy between the atoms, E = E 1 +E 2 , the number of possible
combinations of states is proportional toE 1 E 2. The result is shown
in figure h. The greatest number of combinations occurs when we
divide the energy equally, so an equal division gives maximum en-
tropy.
By increasing the number of atoms, we get a graph whose peak
is narrower, i. With more than one atom in each system, the total
energy isE= (p^2 x,1+p^2 y,1+p^2 z,1+p^2 x,2+p^2 y,2+p^2 z,2+ ...)/ 2 m. Withn
atoms, a total of 3nmomentum coordinates are needed in order to
specify their state, and such a set of numbers is like a single point
in a 3n-dimensional space (which is impossible to visualize). For a
given total energyE, the possible states are like the surface of a
3 n-dimensional sphere, with a surface area proportional top^3 n−^1 ,
orE(3n−1)/^2. The graph in figure i, for example, was calculated
according to the formulaE^291 /^2 E^292 /^2 =E 129 /^2 (E−E 1 )^29 /^2.
Since graph i is narrower than graph h, the fluctuations in energy
sharing are smaller. If we inspect the system at a random moment in
time, the energy sharing is very unlikely to be more lopsided than
a 40-60 split. Now suppose that, instead of 10 atoms interacting
with 10 atoms, we had a 10^23 atoms interacting with 10^23 atoms.
The graph would be extremely narrow, and it would be a statistical
certainty that the energy sharing would be nearly perfectly equal.
This is why we never observe a cold glass of water to change itself330 Chapter 5 Thermodynamics