As (δε1/δv1)T is positive, we shall have to take for the maximum cooling such a pressure that the product pv decreases with v, viz. a pressure larger than that at which pv has the minimum value. By means of the equation of state mentioned already, we find for the value of the specific volume that gives the greatest cooling the formula
| RT1b | = | 2a | , |
| (v1 - b)² | v1² |
and for the value of the pressure
| p1 = 27 pc[1 - √ | 4 | T1 | ] [3 √ | 4 | T1 | - 1]. |
| 27 | Tc | 27 | Tc |
If we take the value 2Tc for T1, as we may approximately for air when we begin to work with the apparatus, we find for p1 about 8pc, or more than 300 atmospheres. If we take T1 = Tc, as we may at the end of the process, we find p1 = 2.5pc, or 100 atmospheres. The constant pressure which has been found the most favourable in Linde’s apparatus is a mean of the two calculated pressures. In a theoretically perfect apparatus we ought, therefore, to be able to regulate p1 according to the temperature in the inner spiral.
The critical temperatures and pressures of the permanent gases are given in the following table, the former being expressed on the absolute scale and the latter in atmospheres:—
| Tc | pc | Tc | pc | ||
| CH4 | 191.2° | 55 | CO | 133.5° | 35.5 |
| NO | 179.5° | 71.2 | N2 | 127° | 35 |
| O2 | 155° | 50 | Air | 133° | 39 |
| Argon | 152° | 50.6 | H2 | 32° | 15 |
The values of Tc and pc for hydrogen are those of Dewar. They are in approximate accordance with those given by K. Olszewski. Liquid hydrogen was first collected by J. Dewar in 1898. Apparatus for obtaining moderate and small quantities have been described by M. W. Travers and K. Olszewski. H. Kamerlingh Onnes at Leiden has brought about a circulation yielding more than 3 litres per hour, and has made use of it to keep baths of 1.5 litre capacity at all temperatures between 20.2° and 13.7° absolute, the temperatures remaining constant within 0.01°. (See also [Liquid Gases].)
(J. D. v. d. W.)