Page 33 - Physical Chemistry

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Chapter 1 that use platinum resistance thermometers for temperatures between the fixed points,
Thermodynamics constitute the International Temperature Scale of 1990 (ITS-90). The ITS-90 scale is
designed to reproduce the ideal-gas absolute scale within experimental error and is used
to calibrate laboratory thermometers. Details of ITS-90 are given in B. W. Mangum,
J. Res. Natl. Inst. Stand. Technol., 95, 69 (1990); Quinn, sec. 2-12 and appendix II.
Since the ideal-gas temperature scale is independent of the properties of any one
substance, it is superior to the mercury centigrade scale defined in Sec. 1.3. However,
the ideal-gas scale still depends on the limiting properties of gases. The thermody-
namic temperature scale, defined in Sec. 3.6, is independent of the properties of any
particular kind of matter. For now we shall use the ideal-gas scale.
The present definition of the Celsius (centigrade) scale t is in terms of the ideal-
gas absolute temperature scale T as follows:

t>°C T>K 273.15 (1.16)*
Forthewatertriple-pointCelsiustemperaturet ,wehavet /°C (273.16K)/K 273.15
tr
tr
0.01, so t is exactly 0.01°C. On the present Celsius and Kelvin scales, the ice and
tr
steam points (Sec. 1.3) are not fixed but are determined by experiment, and there is no
guarantee that these points will be at 0°C and 100°C. However, the value 273.16 K for the
water triple point and the number 273.15 in (1.16) were chosen to give good agreement
with the old centigrade scale, so we expect the ice and steam points to be little changed
from their old values. Experiment gives 0.00009°C for the ice point and for the steam
point gives 99.984°C on the thermodynamic scale and 99.974°C on the ITS-90 scale.
Since the absolute ideal-gas temperature scale is based on the properties of a gen-
eral class of substances (gases in the zero-pressure limit, where intermolecular forces
vanish), one might suspect that this scale has fundamental significance. This is true,
and we shall see in Eqs. (14.14) and (14.15) that the average kinetic energy of motion
of molecules through space in a gas is directly proportional to the absolute tempera-
ture T. Moreover, the absolute temperature T appears in a simple way in the law that
governs the distribution of molecules among energy levels; see Eq. (21.69), the
Boltzmann distribution law.
From Eq. (1.15), at constant P and m we have V/T V /T . This equation holds
tr
tr
exactly only in the limit of zero pressure but is pretty accurate provided the pressure
is not too high. Since V /T is a constant for a fixed amount of gas at fixed P, we have
tr
tr
V>T K const. P, m
where K is a constant. This is Charles’ law. However, logically speaking, this equation
is not a law of nature but simply embodies the definition of the ideal-gas absolute tem-
perature scale T. After defining the thermodynamic temperature scale, we can once
again view V/T K as a law of nature.

The General Ideal-Gas Equation
Boyle’s and Charles’ laws apply when T and m or P and m are held fixed. Now con-
sider a more general change in state of an ideal gas, in which the pressure, volume, and
temperature all change, going from P , V , T to P , V , T , with m unchanged. To apply
1 1 1 2 2 2
Boyle’s and Charles’ laws, we imagine this process to be carried out in two steps:
1a2 1b2
P , V , T ¡ P , V , T ¡ P , V , T 2
2
1
1
1
a
2
2
1
Since T and m are constant in step (a), Boyle’s law applies and P V k P V ;
1 1 2 a
hence V P V /P . Use of Charles’ law for step (b) gives V /T V /T . Substitution
a 1 1 2 a 1 2 2
of V P V /P into this equation gives P V /P T V /T , and
a 1 1 2 1 1 2 1 2 2
P V >T P V >T const. m, ideal gas (1.17)
1
1 1
2
2 2