Measuring temperature with the gas thermometer- the gas scale

It is instructive to consider how numerical values are associated with levels of temperature by the gas thermometer shown in Fig. 1.10. Let p stand for the pressure in a
constant-volume gas thermometer in thermal equilibrium with a bath. A value can be
assigned to the bath temperature very simply by a linear relation

T= apwhere is an arbitrary constant. The linear relationship is an arbitrary choice; other
selections for the correspondence between pressure and temperature could also be
made. The value of may be determined by inserting the thermometer into another bath
maintained at a standard fixed point: the triple point of water (Sec. 3.2) and measuring
the pressure, call it ptp, of the confined gas at the triple point temperature, 273.16 K.
Substituting values into Eq. 1.16 and solving for 

The temperature of the original bath, at which the pressure of the confined gas is p, is then

 

However, since the values of both pressures, p and ptp, depend in part on the
amount of gas in the bulb, the value assigned by Eq. 1.17 to the bath temperature
varies with the amount of gas in the thermometer. This difficulty is overcome in precision thermometry by repeating the measurements (in the original bath and the reference bath) several times with less gas in the bulb in each successive attempt. For
each trial the ratio p ptp is calculated from Eq. 1.17 and plotted versus the corresponding reference pressure ptp of the gas at the triple point temperature. When several
such points have been plotted, the resulting curve is extrapolated to the ordinate where
ptp 0. This is illustrated in Fig. 1.11 for constant-volume thermometers with a number of different gases.
Inspection of Fig. 1.11 shows an important result. At each nonzero value of the reference pressure, the p ptp values differ with the gas employed in the thermometer. However, as pressure decreases, the p ptp values from thermometers with different gases
approach one another, and in the limit as pressure tends to zero, the same value for
p ptp is obtained for each gas. Based on these general results, the gas temperature scale
is defined by the relationship
(1.13)
 

where “lim” means that both p and ptp tend to zero. It should be evident that the determination of temperatures by this means requires extraordinarily careful and elaborate experimental procedures.
Although the temperature scale of Eq. 1.18 is independent of the properties of any
one gas, it still depends on the properties of gases in general. Accordingly, the measurement of low temperatures requires a gas that does not condense at these temperatures, and this imposes a limit on the range of temperatures that can be measured by
a gas thermometer. The lowest temperature that can be measured with such an instrument is about 1 K, obtained with helium. At high temperatures gases dissociate, and
therefore these temperatures also cannot be determined by a gas thermometer. Other
empirical means, utilizing the properties of other substances, must be employed to
measure temperature in ranges where the gas thermometer is inadequate