480 Dr. M. Wilde rmann on Real and Apparent 



the beaker, but keeps its particles thoroughly mixed with the 

 liquid. A very fine sensitive -Ol^theruiometer is of great 

 importance. I had a thermometer made for the purpose with 

 a long bulb, so as to have a large surface in contact with the 

 liquid; the glass of the bulb was made as thin as possible, 

 and was protected from the stirrer by a cage. I investigated 

 water, aqueous solutions, solutions of water and acetic acid. 

 From the obtained results, I find that the equation 



_, [log n (t 2 — t v) —log n (ti — toc) +log n {t —t^ 



— log n {t — 1 2 )] =C(z s —z 1 ) 



holds good, where t ov is the temperature to which the liquid 

 was overcooled, t is the freezing-point, t y and t 2 the tempe- 

 ratures of the liquid at the times z x and z 2 . From this we 

 obtain for the velocity of reaction the differential equation 



— =C(A> — t) (t — t ov ), i. e. the velocity ivith which an overcooled 

 dz 



liquid crystallizes, while approximating to its freezing-tempera- 

 ture, is, at the given time z, directly proportional to the remoteness 

 of its temperature from the freezing-temperature and to the total 

 surface of the solid in contact with the liquid. The equation, 

 it is clear, can hold good only for heterogeneous systems and 

 not for homogeneous, i.e. t — t ov cannot be put =0. The 

 value of C for t — t ov =l°, i. e. for one degree of overcooling, 

 is found to be about 14 or 15. 



V. On the Velocity of Melting of the Separated Ice in the 

 Warmer Water or Solution. 



The measurement of the velocity of melting of ice-cubes in 

 water and aqueous solutions enabled me to show (1) that 



the equation -j- =K 1 (f — t)f{t) holds good, i.e. the velocity 



of ice-melting is directly proportional to the remoteness from 

 the freezing -temperature and to the surface of the ice in contact 

 with the liquid: (2) since the ice-cubes melt in water and 

 dilute aqueous solutions with nearly the same velocity, pro- 

 vided the surface in contact with the liquid is the same, it 

 follows that the value of K x can be assumed to be in all cases 

 the same also when ice is separated from overcooled liquids. 

 The velocity of melting of separated ice cannot be measured 

 with any degree of accuracy, since in warming the liquid 

 an essential part of the fine ice disappears ; the value of 

 th also cannot be determined with accuracy. Though there 

 is no possibility of calculating the equation, we ai'e able to 



