Page 32 - Physical Chemistry

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that of liquid mercury, which is the basis for the u temperature scale. However, in the Section 1.5
zero-pressure limit, the deviations from Charles’ law are the same for different gases. Ideal Gases
In the limit of zero pressure, all gases show the same temperature-versus-volume be-
havior at constant pressure.
Extrapolation of the N low-pressure V-versus-u curves in Fig. 1.8 to low temper-
2
atures shows that they all intersect the u axis at the same point, approximately 273°
on the mercury centigrade scale. Moreover, extrapolation of such curves for any gas,
not just N , shows they intersect the u axis at 273°. At this temperature, any ideal
2
gas is predicted to have zero volume. (Of course, the gas will liquefy before this tem-
perature is reached, and Charles’ law will no longer be obeyed.)
As noted, all gases have the same temperature-versus-volume behavior in the
zero-pressure limit. Therefore, to get a temperature scale that is independent of the
properties of any one substance, we shall define an ideal-gas temperature scale T by
the requirement that the T-versus-V behavior of a gas be exactly linear (that is, obey
Charles’ law exactly) in the limit of zero pressure. Moreover, because it seems likely
that the temperature at which an ideal gas is predicted to have zero volume might well
have fundamental significance, we shall take the zero of our ideal-gas temperature
scale to coincide with the zero-volume temperature. We therefore define the absolute
ideal-gas temperature T by the requirement that the relation T BV shall hold
exactly in the zero-pressure limit, where B is a constant for a fixed amount of gas at
constant P, and where V is the gas volume. Any gas can be used.
To complete the definition, we specify B by picking a fixed reference point and
assigning its temperature. In 1954 it was internationally agreed to use the triple point
(tr) of water as the reference point and to define the absolute temperature T at this
tr
triple point as exactly 273.16 K. The K stands for the unit of absolute temperature, the
kelvin, formerly called the degree Kelvin (°K). (The water triple point is the temper-
ature at which pure liquid water, ice, and water vapor are in mutual equilibrium.) At
the water triple point, we have 273.16 K T BV , and B (273.16 K)/V , where
tr tr tr
V is the gas volume at T . Therefore the equation T BV defining the absolute ideal-
tr tr
gas temperature scale becomes
V
T 1273.16 K2 lim const. P, m (1.15)
PS0 V tr

How is the limit P → 0 taken in (1.15)? One takes a fixed quantity of gas at some
pressure P, say 200 torr. This gas is put in thermal equilibrium with the body whose tem-
perature T is to be measured, keeping P constant at 200 torr and measuring the volume
V of the gas. The gas thermometer is then put in thermal equilibrium with a water triple-
point cell at 273.16 K, keeping P of the gas at 200 torr and measuring V . The ratio V/V
tr tr
is then calculated for P 200 torr. Next, the gas pressure is reduced to, say, 150 torr,
and the gas volume at this pressure is measured at temperature T and at 273.16 K; this
gives the ratio V/V at P 150 torr. The operations are repeated at successively lower
tr
pressures to give further ratios V/V . These ratios are then plotted against P, and the
tr
curve is extrapolated to P 0togive the limit of V/V (see Fig. 1.9). Multiplication of
tr
this limit by 273.16 K then gives the ideal-gas absolute temperature T of the body. In
practice, a constant-volume gas thermometer is easier to use than a constant-pressure
one; here, V/V at constant P in (1.15) is replaced by P/P at constant V.
tr tr Figure 1.9
Accurate measurement of a body’s temperature with an ideal-gas thermometer is
tedious, and this thermometer is not useful for day-to-day laboratory work. What is Constant-pressure gas thermometer
done instead is to use an ideal-gas thermometer to determine accurate values for sev- plots to measure the normal boiling
point (nbp) of H O. Extrapolation
eral fixed points that cover a wide temperature range. The fixed points are triple points 2
gives V nbp /V 1.36595 ,so T nbp
5
tr
and normal melting points of certain pure substances (for example, O ,Ar, Zn, Ag). The
2 1.36595 (273.16 K) 373.124 K
5
specified values for these fixed points, together with specified interpolation formulas 99.974°C.