CHAPTER II
THERMODYNAMIC QUANTITIES
9. Temperature
describe macroscopic
physical quantities are those which
some which have both athermodynamic and a
Thermodynamic
of bodies. They include
states
mechanical significance, such as energy
and yolume. There are also,
purely
quantities of another kind,
which appear as a result of purely
however,
when applied to non-macroscopic
statistical laws and have no meaning
entropy.
systems, for example thermo
follows we shall define a number of relations between
In what
quantities which hold good whatever the particular bodies to which
dynamic
called thermodynamic relations.
these quantities relate. These are
discussed, the negligible fluctuations
When thermodynamic quantities are we shall
usually of no interest. Accordingly,
to which they are subject"are
regard the thermodynamic quantities as
entirely ignore such fluctuations, and
of the body.
varying only with the macroscopic stateequilibrium with each other, forming
Let us consider two bodies in thermal value
S of this system has its maximum
a closed system. Then the entropy energies
The energy E is the sum of the
(1or a given energy E of the system).
same applies to the entropy S
Ej and E, of the two bodies:E= Ej+Eg. The
body is a function of its energy:
of the system, and the entropy of each
being a constant, S is really a
S= Si(E)+S(E). Since E, = E-E1, E
Tünction of one independent variable, and
the necessary condition for a
maximum may be written
dS dSi dS, dE,
+
dE, dE, dE, dE1
dSi dS? = 0,
dE dE,
whence
dSi/dE, = dSy/dEg.
Ihis conclusion can easily be generalised to any number of bodies in equi
Iibrium with one another.
tFluctuations of thermodynamic quantities will be discussed in a separate chapter
(Chapter XIT).
33
Ihermodmamic Quantities
Thus, if a NVtem is in astate tthermodynamie cquilibrium, the derivA
ot the entropy with respetlo the cnegy is thec Nanme for
ix vonstant thoughout thesstem Aquantuty which is theevery part of it,
reciprocal of
derivative of the entropy So a body with reseet to its encrgy
the holte temperatwre T(or siniply the temyerature) of the body: Eis calle
(9
The temperatures ot bodies in cquulibrium with
onc another are theref
equal: Ti
Like the entropy, the temperature is seen to be a
purely statistical quanta.
which has meaning only for macroscopic bodies.
Let us next consider (wo bodies forming a closed
librium with cach other. Their temperatures Ti andsystem 7y are
but not in cau
In the course of time, cquilibrium will be then ditteren
established
tand their temperatures will gradually become cqual. between the bode
heir total entropy S During this proce
S +S must incrcase, i.c. its time derivative is poi
tive:
ds dS dS,
dr dr dr
dS, dEI +
dS, dEa
0.
dE, dr. dE, dr
Since the total energy is conserved, dE/dt +dEs/dr =0,
and so
dS dEi dE;
dr dE, dE,) dr ( dr
0.
Let the temperature of the second body be greater than that
(T, > T). Then dEi/de 0, and dEa/dr < 0. In other words, of the ist
of the second body decreases and that of the first increases. the energy
of the temperature may be formulated as follows: energy passes This from
property
at higher temperature to bodies at lower temperature. bodics
The entropy S is a dimensionless quantity. The definition (9.1) theretore
shows that the temperature has the dimensions of energy, and so can be
measured in energy units, for example ergs. In ordinary circumstances,
however, the erg is too large a quantity, and in practice the
temperature
customarily measured in its own units, called degrees Kelvin or simply
The conversion factor between ergs and degrees, i.e. the number degree
per degree, is called Boltzmann's constant and is usually denoted by olk; ets1s
value is!
k 1.38 X10 1 erg/deg.
T For reference, we may also give the conversion coctlicient between degrees
Aols: and electon
IeV 11,606 deg.
Macroscopic Motion 35
formulae the temperature will be assumed measured in
ative In all subsequent
units.Toconvert tothe temperature measured in degrees, in numerical
energy
calculations, we need
nd only replace T by kT. The continual use of the factor k,
the is to indicate the conventional units of temperature
allea whose only purpose
o86urement, would merely complicate the formulae.
9.1 f the temperature 1S in degrees, the factor k is usually included in the
definition of entropy:
fore S=klog 4I, (9.2)
tity, insteadof(7.7), in order to avoid the appearance of k in the general relations
ofthermodynamics. Then forrmula (9.1) defining the temperature, and there
qui fore allthe general thermodynamic relations derived subsequently in this
ent, chapter, are unaffected by the change to degrees.
Aies, Thus the rule for conversion to degrees is to substitute in all formulae
cess, T ’ kT, S ’S/k. (9.3)
Osi
