Although born in Bordeaux, France,

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1 Revisiting Black s Experients on the Latent Heats of Water J. Güéez Departaento de Física Aplicada Universidad de Cantabria E Santander Spain; gueezj@unican.es C. Fiolhais and M. Fiolhais Departaento de Física and Centro de Física Coputacional Universidade de Coibra P Coibra Portugal; tcarlos@teor.fis.uc.pt; tanuel@teor.fis.uc.pt Although born in Bordeaux France Joseph Black was purely Scottish by blood. He began studying edicine at the University of Glasgow where he first becae interested in cheistry. After copleting his edical studies in Edinburgh Black returned to Glasgow in 1756 to becoe a professor of edicine. His ain interest however reained cheistry and physics and his lecture courses eventually grew to attract students fro all over Europe and even Aerica. Featured in his lectures were vivid descriptions of his own discoveries and experients and in 1766 when he accepted a chair at the University of Edinburgh he was followed by any of his students. As a professor at Glasgow Black becae interested in calorietry and perfored experients on the specific heat and latent heats of water. He was the first to distinguish between teperature and heat. 1 The notion of specific heat has been attributed to hi 2 and the experients to deterine the specific heat (using the ixture ethod 3 ) which are coon in introductory physics laboratories go back to Black. On the other hand Black pointed out that previous ideas on elting and freezing e.g. that the fusion of ice is instantaneous and requires alost no heat 2 were wrong. He introduced the ter latent heat to describe the energy that sees to hide when elting occurs reappearing in freezing and also to hide in evaporation reappearing in condensation. 4 Black s calorietric experients coe to us in reports written by his students. One such report refers to the deterination of the latent heat of elting ice: Joseph Black (NBS Archives courtesy AIP Eilio Segrè Visual Archives) In order to understand this absorption of heat into the elting ice and concealent of it in the water ore distinctly I ade (aong others) the following experient. I put a lup of ice into an equal quantity of water heated to the teperature [80 C] and the result was that when the ice was all elted the fluid was no hotter than water just ready to freeze. 5 A second experient refers to the latent heat of vaporization of water: I procured soe cylindrical tin-plate vessels about 4 or 5 inches diaeter and flat bottoed. Putting a sall quantity 26

2 Fig. 1. Modern setups for the two experients described by Black. (a) Setup for deterining the water latent heat of fusion; (b) setup for deterining the water latent heat of evaporation. of water into the of the teperature [10 C] I set the upon a red-hot stove taking care that the fire should be pretty regular. After four inutes the water began sensibly to boil and in twenty inutes ore it was all boiled off. 6 The reproduction of historical experients of therodynaics leading to the sae results obtained by the pioneers of the subject ay help students to gain deeper insights on that subject. On the other hand in doing soe old experients one often coes to the conclusion that they are not copletely appropriate to accoplish their purposes. This is especially true in therodynaics where heat losses are always present. One ay wonder whether Black s experients do in fact give accurate results. In order to answer this question and since we think that odern analysis of historical experients has pedagogical interest we carried out these experients with the equipent shown in Fig. 1 which can be found in any introductory physics laboratory. Black s experient to find the latent heat of ice leads to an accurate value (other experients to deterine the latent heat of fusion of ice have been recently reported in the literature 7 ). However Black s experient to deterine the latent heat of stea if naively interpreted gives a result that differs fro the accepted value by 25%. In order to get ore accurate values cooling by radiation has to be considered. Latent Heat of Fusion A ass w of water at initial teperature t i is ixed with a ass i I of ice at 0 C in a calorieter [a glass vessel with a Styrofoa wrapping see Fig. 1(a)]. The teperature of the water is first raised to a few degrees above the desired initial teperature t i. The syste then starts to cool down and when the teperature t i is reached the water is rapidly poured onto the ice inside the calorieter (kept as the ice at 0 C). The ice-water ixture is vigorously stirred with a glass stirring rod until its teperature stabilizes at a value t f. It is iportant to keep ixing during the approxiately three inutes (the frictional work is negligible) required. Finally the reaining ice f I if any is reoved and weighted. Then I i I f I is the ass of the elted ice. Table I shows Black s easureents as well as the data obtained in our own experient for different quantities of water and ice. For a pure substance that undergoes both a teperature change and a phase transition due to contact with a heat source the heat absorbed is Q + c t with its ass the latent heat of the phase transition c the specific heat 27

3 Table I. Data for ixtures of ice at 0ºC and hot water. stands for the ass of liquid water. The initial ass of ice is i I and the final ass of ice (not elted) is f I. The water initial teperature and the final ixture teperature are t i and t f respectively. Run (g) i I (g) f I (g) t i ( C) t f ( C) The calorieter has been odeled by an (equivalent) aount of water C which absorbs the sae heat Q C as the calorieter when the ixture takes place. 3 Fro Eq. (2) we have Black (at constant pressure) assued to be constant and t t f t i the teperature variation. For the liquid water Q W t i ) (the specific heat of water is considered to be constant between 0 C and 80 C) and for the elted ice Q I I + I 0) where is the latent heat of fusion of ice. When the final teperature is t f 0 C the calorieter does not gain or lose heat and Q I Q W. Thus t i. (1) W I In the original experient by Black w I t i 80 C (and t f 0 C) leading to 80. Using 4.18 Jg -1 C -1 (a value later found by Joule) one finds 334 Jg -1 which is close to the current value Jg However when the final teperature of water differs fro 0 C heating of the calorieter has to be taken into account in the heat balance and Eq. (1) does not apply. A calorieter is an adiabatic syste (there is no heat exchange with its surroundings). For water ixed with ice in the calorieter we have Q I + Q W + Q C I + I 0) + t i )+ c 0) 0. (2) t i W t f + + I C. (3) I The equivalent in water of the calorieter depends on the therodynaical process but we perfored a single experient to estiate C (note that all runs in Table I were done in siilar conditions). We ixed equal asses of hot and cold water (approxiately 170 g each as in Table I) at 92.7 C and 2.7 C respectively (as in the experients of Table I). After stirring during three inutes the teperature 41.9 C was reached. Fro an equation like Eq. (2) but applied to hot water cold water and the calorieter we found C 57 4 g. Using the C in Eq. (3) we obtain values for which are given in the third colun of Table IV. These values are all pretty close to the current value. Latent Heat of Vaporization We placed water in a copper vessel with a flat botto [see Fig. 1(b)]. The teperature was raised to the boiling point 100 C at a pressure of bar (atospheric pressure) and then the water was totally converted to stea. An electric hotplate served as the heat source. Let P be the constant power supplied during the process 1 be the tie for water to reach the boiling point starting fro the teperature t i (roo teperature) and 2 be the tie for the water to evaporate copletely. One should consider that the container is always radiating to the environent since its teperature is higher than that of the air. Thus in the heating and evaporation processes P 1 (100 t i )+Q heating P 2 + Q boiling (4) I 28

4 where is the ass of water the latent heat of vaporization and Q heating and Q boiling are energy exchanges with the environent (heat losses) in the heating and boiling processes respectively. Following Ref. 9 not taking into account heat losses (i.e. Q heating Q boiling 0) with t i 10 C 1 4in and 2 20 in as in Black s work 2 c W (100 t i ) (5) 1 one obtains Jg -1 very different fro the current value Jg -1. In Table II we show Black s easureents and our own data collected in six runs of the sae experient using different quantities of water. The fourth colun of Table IV shows our own results for the latent heat of evaporation using Eq. (5). Since the values are too high we have to take into account heat losses. The effect of radiation can be dealt with using Newton s law of cooling. 10 To estiate the energy loss of the water in the cooling process we carried out the following experient. We heated a ass of water in the copper vessel up to the boiling point. Then we let it cool down to roo teperature t R in order to easure the cooling rate of water. We perfored this cooling experient twice with 100 g and 300 g. In Table III we report the elapsed ties and teperatures t. The power lost to the environent is Q dt W. (6) d Using Newton s law of cooling 10 in its differential for dt k d (t t R ) (7) where k is the cooling paraeter (the index stresses that k depends on the ass of water). Eq. (6) becoes Table II. Data obtained in water vaporization experients. is the ass of water t i its initial teperature 1 the elapsed tie for water to start boiling and 2 the elapsed tie for water to evaporate copletely. Run (g) t i ( C) 1 (s) (s) Black Q k (t t R ). (8) In the heating process we observed that the teperature varies linearly with tie t t R t R. The power lost in the 1 heating process is then heating Q k 100 t. R (9) 1 During boiling the teperature of the recip- Table III. Ties and teperatures for (a) cooling 100 g of water in an environent at t R 19.7 C and (b) cooling 300 g of water at t R 21.0 C. In both cases cooling occurred in a copper vessel with 155 g. (a) (b) (s) t ( C) (s) t ( C)

5 Table IV. Latent heat of fusion of ice whose standard value is Jg -1 and latent heat of water evaporation whose standard value is Jg We note that Eq. (1) only applies when t f 0 C which is not the case for runs 5 and 6 of the fusion experient. Run (Jg -1 ) (Jg -1 ) (Jg -1 ) (Jg -1 ) [Eq. (1)] [Eq. (3)] [Eq. (5)] [Eq. (15)] Black (343) (351) ient reains at t 100 C. Hence the power lost during boiling is heating Q k (100 t R ). (10) Integrating we find Q heating k (100 t R ) 1 2 Q boiling k (100 t R ) 2. (11) Using the data fro Table III in the integral for of Newton s law of cooling [integral of Eq. (7)] k 1 1n t( ) t R (12) t( 0) tr one obtains the (ass dependent) cooling paraeter: k s -1 and k s -1 for 100 g and 300 g respectively. For the (ass independent) quantity k one obtains 100 k W C -1 and 300 k W C -1. Inserting the average value 1.75 W C -1 back into Eq. (11) we obtain the following heat losses during heating and boiling: Q heating 1.75(100 t R ) Q boiling 1.75(100 t R ) (13) Taking into account these heat losses in Eq. (4) the energy balance equations are P 1 (100 t i ) P (14) Eliinating P in these equations we obtain 2 c (100 t ) W i 1. (15) 1 W Taking 4.18 Jg -1 C -1 and inserting data given in Table II in this equation one obtains the results displayed in the last colun of Table IV. They iproved with respect to the results arising fro Eq. (5) (the average deviation is less than 10% now). However one should notice the large uncertainty estiates. In these experients to deterine the latent heat of vaporization uncertainties are ainly associated with difficulties in easuring exact ties when water starts to boil and when water evaporates copletely. The heat absorbed by the etallic vessel and the water losses during heating and stea condensation in the recipient walls are other sources of uncertainty. 9 Coents In suary we have carried out Black s experients on the latent heats of water which ay be done nowadays for teaching therodynaics. The first one is usually perfored in introductory physics laboratories either at high school or college level in order to obtain the latent heat of fusion of ice but we pointed out the need for using the water equivalent of the calorieter when the final teperature is different fro 0 C. On the other hand the deterination of the latent heat of vaporization using Black s experient requires a odel for heat losses. We used Newton s law of cooling but nevertheless it turned out to be difficult to obtain the accepted value with a precision uch better than 10%. The uncertainties are siply too large to ake this ethod of great practical value. The ost coon ethod to deterine the latent heat of water evaporation is the so- 30

6 called Berthelot s ethod. 311 We would like to stress that astering the scientific ethod also requires understanding why soeties soe experients do not work well. Failed experients suitably presented and discussed ay be of pedagogical value. References 1. S.C. Brown The caloric theory of heat A. J. Phys (Sept. 1950). 2. L.W. Taylor Physics: The Pioneer Science. Volue I. Mechanics Heat Sound (Dover Publications New York) Ch. 20 p This reproduces the original Lectures on Eleents of Cheistry given by Black at the University of Edinburgh published fro his anuscripts by John Robinson (Longan & Rees London 1803). 3. H.S. Allen and H. Moore A Textbook of Practical Physics (MacMillan London 1965) p P.H. Bligh and R. Haywood Latent heat Its eaning and easureent Eur. J. Phys (1986). 5. See Ref. 2. Note that Black s theroeters were graduated in Fahrenheit degrees but teperatures have been converted here to Celsius. 6. See Ref. 2. Italics added to call attention to the fact that both ties have to be easured in regular heating conditions. 7. S.Y. Mak and C.K.W. Chun The easureent of the specific latent heat of fusion of ice: two iproved ethods Phys. Educ (May 2000). 8. H.U. Fuchs The Dynaics of Heat (Springer New York 1996) p C.D. Galles Revival of Black s experient A. J. Phys (Nov. 1979). 10. J.W. Dewdney Newton s law of cooling as a laboratory introduction to exponential decay functions A. J. Phys (Dec. 1959). 11. H. Lindean and A. Lavie Instruent for the easureent of the heat of vaporization of water A. J. Phys (Oct. 1961). 31