New Formulas for Estimating the Temperature of Mixture of Hot Systems

Illustrations and questions based on the equation “heat gained equals heat lost” have always been restricted to mixing systems at two different temperatures. This article shows how to extend the above equation to systems with more than two different temperatures. Using the current formula the following procedures are generated. The method of mixing two systems at a time (pair mixing) is illustrated. The technique of taking any of the systems (preferably the one with the lowest temperature) as the only heat gainer is also illustrated. A new formula based on constancy of heat content is introduced. The various methods are tested and found to be perfect and should be incorporated into the teaching and learning in this aspect of heat study.


Introduction
Heat is a form o f energy co mmon ly associated with other forms of energy and processes especially those involving motion. In most cases there is always a temperature change when a body absorbs or dissipates heat [1] .
Heat has b een v arious ly des crib ed as en erg y that increases or decreases the temperature of a body absorbing or d issipat ing it. Any akoha [2] d escrib es heat as energy associated with a temperature d ifference. These show the inter relat ionship o f heat and temperature. Ho wever, not always does a body's temperature change on absorption or d iss ipation o f h eat [3] as is the case wh en a bod y is undergoing a phase change (e.g. at melting point, bo iling point or transition point). The energy absorbed by the body at this point does not lead to increase in the temperature of the body because the energy is used to free the atoms and mo lecu les fro m strong inter ato mic and intermo lecu lar forces lead ing to g ain in potent ial energy that result to change in phase [4,5] . For a body undergoing phase change at the transition temperature, specific heat capacity is zero but heat capacity is not zero, so specific heat capacity is a characteristic of a phase. The heat absorbed at this transition temperature is called the latent heat. With the above facts, h eat is mo re app rop riately defined as th e en ergy th at increases the tot al int ern al energy o f a body absorb ing it [1,3,5] . Temperature which has been described as a measure o f deg ree o f hotness o r co ld ness [2 ,3,4 ] is an int ens ive property, and properly defined, is a measure of the average kinetic energy of the atoms and molecules of a body [5] .
When two bodies at different temperatures are in contact and conduct heat, heat flows fro m the body at high temperature to the one at low temperature until the two bodies attain a constant temperature. Such constant or resulting temperature has always been calcu lated using the law of conservation of heat energy and for mixtures the law is expressed in the form of the equation, 'Heat gained = heat lost'. The law is of general applicat ion so also should be the equation but the illustrations and examples have always been restricted to mixing of bodies at only two different temperatures where the heat gainer or loser is obvious from the temperature values. No attempt has been made to show how to apply the law or the equation to a mixture resulting fro m mixing many bodies at different temperatures. One obvious reason could be that with more than two bodies, the bodies with temperatures in between the lowest and highest temperatures cannot be categorically classified as heat gainers or losers before mixing not until the temperature of the mixture is determined. A lso when the systems to be mixed are of different materials and at same temperature, the heat loser or gainer is not discernable. So there is need to seek other methods that are applicable to mixing any number of systems at same or different temperatures. As of now only the existing formu la is in use and only for systems at two different temperatures. It is the aim of this article to show how the existing fo rmula "Heat gained equals Heat lost" can be extended to mixing t wo or more different systems with dissimilar temperatures and to use the existing formu la to generate other formulas applicable to mixing t wo or more bodies at two or more different temperatures.

Generating Equations for the New Methods
In many text books [1,2,3,4,5] to mention a few, illustrative examples and exercises in the calorimet ric study of heat have always been restricted to mixing bodies at only two different temperatures applying the equation "heat gained = heat lost". This procedure can be applied to mixing bodies at more than two different temperatures via the pair mixing method that utilizes the method in vogue as follows.
Consider n bodies each of mass m 1 , m 2 , m 3 , …. m n , specific heat capacity (sp ht) s 1 , s 2 , s 3 , …s n and at temperature t 1 , t 2 , t 3 , … t n being mixed in the sequence (m 1 + m 2 ); (m 1 + m 2 ) + m 3 ; (m 1 + m 2 + m 3 ) + m 4 ; (m 1 + m 2 + m 3 + m 4 ) + m 5 ; and so on (i.e. pair mixing). Let t 1 < t 2 < t 3 …. < t n . By the equation "heat gained = heat lost", let t a be the temperature of th is first mixture. m 1 s 1 (t a -t 1 ) = m 2 s 2 (t 2 -t a ) or m 1 s 1 t a + m 2 s 2 t a = m 1 s 1 t 1 + m 2 s 2 t 2 or (m 1 s 1 + m 2 s 2 ) t a = m 1 s 1 t 1 + m 2 s 2 t 2 (1) This first mixture of heat capacity (m 1 s 1 + m 2 s 2 ) at temperature t a is mixed with m 3 to get a second mixture at t b . Hence (m 1 s 1 + m 2 s 2 )(t b -t a ) = m 3 s 3 (t 3 -t b ) or (m 1 s 1 + m 2 s 2 )t b + m 3 s 3 t b = (m 1 s 1 + m 2 s 2 )t a + m 3 s 3 t 3 (2) Fro m (1) (m 1 s 1 + m 2 s 2 )t a = m 1 s 1 t 1 + m 2 s 2 t 2 , hence m 1 s 1 t 1 + m 2 s 2 t 2 + m 3 s 3 t 3 = m 1 s 1 t b + m 2 s 2 t b + m 3 s 3 t b (3) In general if t f is the final temperature of the mixture m 1 s 1 t f + m 2 s 2 t f + m 3 s 3 t f + -----+ m n s n t f = m 1 s 1 t 1 + m 2 s 2 t 2 + m 3 s 3 t 3 + ------+ m n s n t n (4) Equation (4) will be used to generate various general methods applicable to mixing bodies at two or more different temperatures as follo ws. For simp licity let n = 3.

One Body Heat Gainer or Looser Method
For a mixture of three bodies, n = 3. So (4) reduces to (4a). m 1 s 1 t f + m 2 s 2 t f + m 3 s 3 t f = m 1 s 1 t 1 + m 2 s 2 t 2 + m 3 s 3 t 3 ..(4a) Assuming m 1 is the only heat gainer i.e. make m 1 s 1 (t f -t 1 ) the subject of the equation in (4a) to get (4b).
m 1 s 1 t f -m 1 s 1 t 1 = m 2 s 2 t 2 + m 3 s 3 t 3 -m 2 s 2 t f -m 3 s 3 t f or m 1 s 1 (t f -t 1 ) = m 2 s 2 (t 2 -t f ) + m 3 s 3 (t 3 -t f ) (4b) In the same equation, in the same manner, if m 1 is the only heat loser, then make m 1 s 1 (t 1 -t f ) the subject of equation (4a) to obtain equation (4c) m 1 s 1 (t 1 -t f ) = m 2 s 2 (t f -t 2 ) + m 3 s 3 (t f -t 3 ) (4c) Equations (4b) or (4c) is used in the "one body heat gainer or loser" method and affords a means of extending the current method to any number of bodies being mixed by assuming that any one body (preferably the body at the lowest temperature) gains heat and the rest loose heat and the general heat gained equals heat lost is applied. This method is illustrated by solution A(i) and B(i) in figure 1.

Heat Content Method
In equation (4), the left hand side (LHS) is the sum of the various final heat contents making up the total heat content of the resulting mixture and the right hand side (RHS) the sum of the init ial heat contents of the bodies. This introduces a new method based on heat content method with the equation 'sum o f final heat content = sum of init ial heat content' where heat content of any body is the product of its mass, sp ht capacity and its temperature i.e. heat content = mst or heat capacity times t, where heat capacity of a body is ms. Th is process involves calculating the sum of final heat content at t m (hereafter the mixture temperature) and equating it to the sum of init ial heat content as the equation stipulates. This is illustrated in solutions C(i) in figure 2.

Heat Content Method with Assumed Mixture
Temperature T a For a wider application of the heat content procedure, the resulting mixture temperature t m is replaced with an assumed mixture temperature t a which can be any temperature fro m the low to the h igh phase change temperature of the liquid body of the mixture. The replacement of t m with t a in equation (4) gives equation (4d) for n = 3. m 1 s 1 t a + m 2 s 2 t a + m 3 s 3 t a = m 1 s 1 t 1 + m 2 s 2 t 2 + m 3 s 3 t 3 . (4d) Since all the terms in equation (4d) are known, if the sum of the in itial heat contents minus the sum of the heat contents at t a is ze ro, t a is same as t m , if not the difference (positive or negative) is divided by the sum of the heat capacities of the systems at the assumed mixtu re temperature and the quotient (is in temperature units) added to the assumed mixture temperature to obtain the actual mixtu re temperature.
The detailed method follows the steps: (a) Choose an assumed mixture temperature t a as stated above. (b) Calculate the sum of (i) the heat capacities of the bodies at t a (ii) initial heat contents of the bodies (iii) the heat contents at t a . (c) Subtract the sum of the heat contents at t a fro m the sum of the initial heat contents of the systems. (d) Div ide the difference by the sum of the heat capacities and add the quotient (temperature) to t a to obtain the required mixtu re temperature as illustrated in solutions D(i) and E(i) to questions in figure 3.

Heat Content Method with Assumed Mixture Temperature T a Equals Upper or Lo wer Phase Temperature of The Liquid Content of the Mixture
It is possible that the final mixture temperature fro m step (d) above is h igher than the upper transition temperature (of vaporization) or belo w the lower transition temperature (of fusion) of the liquid content of the mixture, th is shows that part of the liquid has evaporated (higher temperature) or still in solid state (lo wer temperature). The amount of liquid evaporated or still in solid state needs be known and this involves a recalculation. To avoid this recalculat ion, t a will be taken as any of the phase change temperatures and this introduces the heat content method which utilizes the assumed mixtu re temperature t a as the same as the upper or the lower phase change temperature of the liquid content of the mixture. In the heat content method with assumed mixtu re temperature t a equals upper or lower phase temperature of the liquid content of the mixtu re; the procedure is followed as in 2.1.3 up to step d. If the operation in step d is negative ( for t a = 0℃ say for ice to water) the mixture temperature is 0℃ and the mass of ice not melted is the absolute value of the difference fro m step c above divided by the appropriate latent heat of fusion. If the operation in step d is positive (for t a = 100℃say for water to steam) the mixture temperature is 100℃and the mass of water evaporated is the positive value of the difference fro m step c above divided by the appropriate latent heat of evaporation. The whole procedure is illustrated in the solutions F(i) and G(i ) of samp le questions in figure 4. Tab le 1 summarizes the various new methods. Heat content with assumed temperature ta Sum of initial heat content = sum of heat content at ta. If LHS -RHS = 0, tm = ta. If LHS -RHS = ±ve, t m = ta ± (LHS -RHS)/sum ht capacity of mixt ure.
Heat content with assumed temperature (i) ta = upper transition temp of the liquid in the mixture (e.g. 100℃ for water).
(ii) ta = lower transition temp of the liquid in the mixture (e.g. 0℃ for water).
Formula is as above.
If tm is +ve, tm = ta and mass of liquid evaporated is (LHS -RHS)/sp latent ht capacity of evaporation of the liquid in the mixture. If tm is -ve, tm = ta and mass of ice not melted is absolute value of (LHS -RHS) ÷ sp latent ht capacity of fusion of the solid melting to liquid in the mixt ure.

Results and Discussion
Same and or different samp le questions (A to G) have been solved using the new methods introduced above as illustrations. Figure 1 shows the use of one body heat gainer or loser method, figure 2 heat content procedure, figure 3 heat content with assumed mixture temperature and figure 4 heat content method but with assumed appropriate transition temperature.

Generating Equati ons for the New Methods
Equation (4) wh ich is the base on which the new methods or formu las are derived is based on the method in vogue and therefore the new methods are correct. This process of mixing the bodies in steps of two (here called "Pair mixing procedure") mixes bodies at more than two different temperatures. It is lengthy but is a suitable check method for authenticating answers from any of the other methods hence its inclusion in some of the figures e.g. A(ii) and B(ii) of figure 1.

. Heat content Method
Heat content equation is a thermodynamic law. Just as the principle of heat gained equals heat lost, the total final heat content will be equal to the total in itial heat content which is also expressed mathemat ically in the expanded form as in equations (1) or (2) o r (3) or (4). The p rinciple of mixing that utilizes the equation heat gained equal heat lost is based on the assumption that the mixing is done adiabatically in wh ich case no heat or matter leaves or enters the system. This being the case the final heat content of the mixture is equal to the sum o f the in itial heat contents (that is the law of conservation of heat). Figure 2 below shows the application of the heat content method. The heat content method with assumed mixture temperature t a is not much different fro m the simp le heat content method only that it has the advantage of detecting liquid eg water evaporated or residual ice in the mixtu re when t a is either of the transition temperatures. The method of solution using an assumed mixture temperature is illustrated in figure 3 below.
The choice of an assumed mixtu re temperature t a in the region where specific heat capacity is not zero is because sum of heat capacities will be needed in the calculation of mixtu re temperature t m and the liquid body is the one to evaporate or be produced from its solid. A positive heat difference fro m sum of the init ial heat contents of the systems minus the sum of the heat contents of the bodies at the assumed temperature indicates a surplus heat for raising the temperature of the mixture above t a by an amount equal to the positive heat difference divided by the sum of heat capacity of the mixture and this quotient is added to t a to get the mixture temperature t m . A negative heat difference treated similarly gives the mixture temperature t m as illustrated in solutions D(i) and E(i ) in figure 3. In continuation of 3.1.3 above, where t a equals the upper transit ion temperatu re e.g . 100℃ fo r water or the lo wer transition temperature e.g. 0℃ for water, a positive heat difference (for t a = 100 o water) evaporates mass of water equal to the positive heat difference div ided by the specific latent heat o f ev aporat ion o f water and a n egat ive h eat difference indicates that heat is still needed to melt mo re ice of mass equal to the absolute value of the negative heat difference divided by the specific latent heat of fusion of ice, see F(i) and G(i) in figure 4. The use of any of the phase change temperatures (e.g. 0 ℃ or 100 ℃ for water, the mixtu re has water as the liquid with the least phase change temperatures) as t a eliminates a recourse to fresh calculation as is the case with any other method.

Conclusions
The three procedures viz the one body heat gainer or loser method, the heat content method (with its variants) and pair mixing, have been applied in solving common examples and the same answer obtained shows that the new methods are faultless and of general application. The versatility of the two new methods is shown by their use in solving sample questions. Solution B(i ) of figure 1 and D(i) of figure 3 prove that difference in temperature determines the direction of heat flow. It is reco mmended that the various formu las be incorporated in the teaching and learning of th is aspect of heat study.