164 



PHENOMENA, ATOMS, AND MOLECULES 



The two methods give analogous, but quantitatively slightly different 

 results. Smoluchowski ^ considers that the accuracy of experimental results 

 has not yet been sufficient to distinguish between them, notwithstanding 

 Knudsen's opinion to the contrary, and that in all probability the truth 

 is intermediate between the two. 



The formulas developed by Smoluchowski are only adapted to be 

 applied to small differences of temperature and to heat conduction between 

 surfaces whose radius of curvature is large compared to the mean free 

 path of the molecules. In the present experiments, however, we are deal- 

 ing with very small wires and with temperature drops of sometimes a 

 thousand degrees or more, and we shall therefore need to derive the 

 equations independently. We shall choose a simpler, although less rigorous 

 method, based upon the same principles as those used by Smoluchowski. 

 In this way we shall gain a clearer insight into the phenomena occurring 

 around the wire and will be in a better position to apply similar principles 

 to the problem of calculating the degree of dissociation of the gas. 



Theory of Heat Conduction from Small Wires. — Let us consider a 

 small heated wire A (Fig. i ) placed in the axis of a large cylindrical bulb B 

 of diameter h, containing gas at such low pressure that the effects of con- 

 vection can be neglected. Let T2 be the 

 temperature of the wire and Ti that of 

 the bulb. If A, represents the length of 

 the mean free path, then the distance 

 which a molecule leaving the wire travels 

 before colliding with other molecules is 

 approximately X.^ On the other hand, 

 the molecules striking the wire come 

 from an average distance "k. Let us draw a 

 circle, C (in Fig. i), of diameter d -|-2X, 

 around the wire, whose diameter is d and 

 a similar circle D of diameter h — 2X. 

 We may now look upon the heat conduction from the wire A to the 

 cylinder B as taking place in the following steps: 



I. Molecules of gas leave the wire and travel to C, where they are 



^Ann. Phys., 35, 983 (1911). 



^ Knudsen, Ann. Phys., 34, 654 (1911), shows that the temperature of the mole- 

 cules striking a plane surface corresponds to the temperature of the gas at a distance 

 equal to 0.95 times the mean free path of the molecules. Using more recent data than 

 that available to Knudsen, I calculate from his formulas that this coefficient should 

 be about 1.2. Undoubtedly the coefficient would be somewhat different for very small 

 wires from what it is for plane surfaces, but in any case its order of magnitude is close 

 to unity. Fairly large errors in the value of this coefficient make only very slight differ- 

 ences in the results that are to be obtained by its aid. 



