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 = alpha x P (1.11)

where alpha 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 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 alpha

alpha = 273.16/ ptp

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

T = 273.16 (P/ ptp)

However, since the values of both pressures, p and ptp, depend in part on the amount of gas in the bulb, the value assigned 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 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.

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.

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