The Thermodynamics of Home-Made Ice Cream Donald L. ~ i b b o nand ' Keith Kennedy Calgon Corporation, Box 1346, Pittsburgh, PA 15230 Nathan kading2 Stanford University, Palo Alto, CA Mardsen Quieroz Department of Mechanical Engineering, Brigham Young University, Provo, UT Introduction: Why Ice Cream? One of a free societys most serious problems is an ignorant citizenry. Ignorance leads to poor decision-making when the citizens are called on to make choices, whether those choices are made a t the ballot box or at the local store. Intentional secrecy, the "What you don't know won't hurt you-I know what's best for you" approach, received a lot of publicity during the Iran-Contra affair. In matters scientific, however, it is commonly not intentional secrecy that is the problem. Citizens remain largely ignorant of the technological workings that form the basis of their society because of a widely held perception that "science is too hard" to understand. But much of fundamental science, well taught, is within the grasp of most people. I t can be done, as recent articles on Jerl Walker's success in teaching physics a t Cleveland State University so eloquently testify. But teaching, to be effective, has to be tailored to the particular group of students being taught. If principles are taught abstractly, most students of any age will simply not make the effortto understand. But principles taught in a *amework of common experience often become quite obvious. It's the context, not the principle, that makes learning difficult. Quite obviously, if citizens understand how things work, what the principles are on which the devices of daily life are based. then all sorts of benefits fall out. Unrealistic hopes for miraculous "technological fures" fade, while innovation becomes more likelv. Rates of obsolescencewould eo down, andequipment, both personal and public, would l&t lower, because it would be used more effectively. Thus it is cle&li in society's best interest to do a better jib of teaching science to its citizens. How best to go about implementing that fundamental idea? One good ap~roachis to brim science closer to home, in fact, bring i t i i h t into the kitchin! We propose here the princide that eatine the results of the experiment is an impoAant aid to d e r s t a n d i n g the results! At least the experiments are rarely complete failures that way. Ice Cream as the Basis for a Chemistry Course We suggest that making ice cream can be the basis of an entire course in chemistry. Consider the principles on which the process is based. First of all, ice-cream making is fundamentally a matter of controlling heat transfer. What's warm has to be made cold. The mechanism that makes i t all possible in a small manually operated system is the process of depression of freezing point in the system ice-water-salt. To begin to understand what we just said in a fundamental way, one must first consider all sorts of concepts:
Presented at the 1987Annual Meeting of the American Association for the Advancement of Science,Chicago, IL. 'Author to whom correspondence should be addressed. 'Undergraduate student.
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temperature itself the nature of heat and heat versus temperature the nroeess of meltine phase diagrams the effects of dissolved solids on liquid freezing points. Back of all these processes and definitions, the students should be encouraged to ask, ''Why? Why? Why does this happen? As the ice cream "freezes", many other process are taking place. Of course, the most obvious one is nucleation of ice crystals in the ice cream mix. Successfulice cream making requires maximum number and minimum size of ice crystals, so that the ice cream is "smooth". These two criteria require maximum rate of heat transfer. Why? What is the role of stirring the mix? It not only raises the rate of heat transfer. but also adds air to the mix. alterine the entire originalliquid-to-metal-to-liquidheat lransfer situation. What is the final result of the orocess? The startine mix does not become a solid.block of iee. What is it? It turns out that the ice cream never fullv freezes: i t remains about loq liquid at its hardest. What part is liquid? What is the difference between a \iscous liquid and a d i d ? Does that difference matter? Why and where? This listine is onlv a tinv fraction of the issues that can of making ice cream in the be discussedYduring"the classroom. And the interest certain to be aroused during these discussions can be focused on a more formal "setting" of those principles. Lab sessions can become the most interesting part of the course. The end result of such a course is students who are both motivated and knowledgeable about the world they see on tidaily basis.
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Experimental Setting: Getting There Is More Than Half The Fun The major experimental equipment has to be a n ice cream freezer and some method of measuring temperature. We had access to a digital readout with inputs for several thermocouples, but long thermometers would have done the job. My ice cream freezer is an old hand-cranked one. From vast experience, I know that it takes about 11 min to freeze a batch, if the ice-salt ratio is about right. We drilled several holes in the side of the bucket and wiredlepoxied in a thermocouple to monitor the brine temperature. One of the most severe challenges, one not surmounted in these experiments, is how to determine the temperature of the mix inside the central tub while it is being stirred by the paddles; some form of microtelemetry seems to be the answer. However, we performed three major experiments and talked about dozens of others. Heat Capacity We wanted to determine the heat capacity of the mix, to see how much heat had to be removed from the mix to freeze it. So we did an elementary calorimetry experiment. We added ice to a salt-water mixture, stirred until it was
T e m p . (c;5] 10
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Pulse Rate a s a 160 Brine Temperature
Function of Time
PULSE RATE (Bmts/Smnd)
loo
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Mix Temperature
Elapsed T i m e ( m i n u t e s )
Figure 1. Temperture equilibrium diagram for making marshmallowchocolate chip ice cream.
80C 0 2 4 Iar n d t ,
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quite cold (about -13 'C in one experiment), removed the remaining ice to avoid complication of melting, measured the temperature of a batch of ice cream mix, then put the mix into the brine in a steel container. We wrapped the brine bucket in towels to insulate it, and began stirring both the brine and mix, all the while measuring the temperature versus time in both containers. The plot of one batch of data is shown as Figure 1.Let's look at that graph and see what we can learn. First, we had reached equilibrium between the two materials in slightly over 11min. The change in temperature of the brine was from -10 'C to +2 'C AtB=-10-(+2)=1 'C (1) The change in temperature of the mix was from +20 'C to +2 'C AtM=ZO-2=lS .C
(2)
We know the volume of the brine (VB)and the volume of the mix (VM).We can weigh both the brine and the mix to get their densities in gramsfliter DB = weight of brine
v~ DM = weight of mix
v~ Heat capacity is measured in calories per degree per gram and will he designated HB and HM.For the overall heat transfer, we have (Total heat last)rl = (Total heat gained,, HBxAtBxVBxDg=HMxAtMxVMxDM
(5)
He and HM are both unknown. One way to quantify the experiment would be to run it brine-against-water, where H*., has the well-known value of 1 calPClg. That would give us HB and then we wuld calculate HM. Now, knowing HM,we can calculate how much heat has to be removed to make ice cream. First we'd have to make some. Determining the ingredients' starting and final temperatures would allow us to calculate AtM.Then, remember that question above about the ice cream never being fully solid? Only part of it actually freezes. We wuld make an assumption about its actual ice content (for purposes of discussion, let's assume 70%), and assume that the heat of crystallization of that part was approximately that of water. The ice cream making urocess consists of molinn the mix down to the freezing of water, then freezing whatever part of it will actually become solid, then continuing tocool
Figure 2. Pulse rate of person cranking the freezer plotted against time. the whole batch to its fmal temperature. Therefore, for 1L of ice cream mix,starting at +20 'C and ending at -10 'C, we have the following: (Total heat rern~ved)~ = HM x 20'C x 1L x DM
To get through all of these calculations, the only thing we would need, in addition to what we already know, is the heat of crystallization of water. Viscosity We also wanted to monitor the rate of change of viscosity during the freezing process, presuming that this was related to the rate of nucleation and growth of ice crystals in the mix. It was proposed that the pulse rate of the cranker might be a direct function of viswsity, so we checked that out. It is an experimental fad that the mix gets harder to crank as it gets colder and harder. The cranker's pulse is a way to calibrate or quantify that fad. The data from this experiment are shown as Figure 2. Dozens of questions come out of this: the nature of precision, how to fit curves to data; what do we expect the relationship to be (straightline, ever-increasing, etc.) and why; how to design experiments to check on only one thing, ete. A second way to check on the viscosity of the mix might be with an inductive ammeter on the power cord of a motorized freezer. These meters measure the current flow in a single wire of a two-conductor pair by simply clamping around the wire. As the resistance to flow (viscosity) of the mix goes up, it takes more power to turn the motor. Power, measured in watts, is equal to volts (a fixed value) multiplied by amps (the current flow). As the viscosity goes up, the power required goes up. We shouldn't do this for long on small motors without overload protection (such as a Waring Blendor) or we'll bum them out. However, ice cream freezers are equipped to handle running to a stall. The scale on our ammeter was very small, so this brought up some questions about sensitivity, data gathering, and principles of electricity and magnetism. Figure 3 shows the results of our experiment. Eventually the motor stalled, at maximum current flow, as the viswsity became too high for its power to overcome.
Volume 69 Number 8 August 1992
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P o w e r C o n s u m -p t i o n Data Baseline Vanilla 2.0q f
I OX NoCI 100% HZ0
I 61 .B 38.1
1 100% NaCI OX HZ0
COMPOSITION Time (minuter)
Figure 3. Current required by mixing motor plotted against time. Freezing-Point Reduction The most interesting question of all is: What makes the ice cream cold enough to freeze? Fundamentally, it's the reaction between salt (sodium chloride, NaC1) and ice, though water speeds up the process. What happens when we put the salt and the ice together? The temperature goes down.. .but why? It is amusing to hear the arguments that develop among professional scientists who have long since left their thermo books behind when this question is raised! Our version of the answer is shown below. Before we go into an explanation of why the temperature eoes down. let's look at an ex~erimentthat demonstrates this interesting phenomenon. It's a simpler version of the ice cream making process. We took a Waring Blendor, rigged a thermocouple down the inside of the jar, filled it with crushed ice and a double handful1 of salt, filled the Dore spaces with water to get the heat transfer going raster, Hnd turned on the ~lenhor.In about 5 s, the temperature had droDDed to -19 'C! ks well see in a moment. this is almost the minimum possible temperature for this system. Not bad for an uncalibrated thermocouple! This experiment is so easy to run that it's a fun one to do for yourself. Our f r s t step in fmding an explanation for what caused the temperature to drop, is to make a few d e f ~ t i o n sFirst, . the Blendor jar and the ice, salt, and water, constitute what is known as a "svstem". Systems can be either closed or open, depending on whether we allow anything - e n ei-ev. to flow in or out of the svstem dur-. matter..anvthine. ing the experiment. In our experiment the system was closed for the few seconds that it ran. No matter went in or out; for all practical purposes, all the heatlenergy for the experiment was contained in the jar and the ingredients. If it had run much longer, the system would have become "open" as a measurable amount of heat flowed in from the surroundings. Our second ster, is to understand equilibrium. If a system is in equilibhum, it has no tendency for net change. That doesn't mean it's the same in all Darts..iust that it has no tendency to modify itself. For example, if we took a hollow steel sphere, filled it halffnll of water, reduced its temperature to 0 'C, and insulated it extremely well, so that heat could not flow in or out, inside the sphere we would find three phases of material, all with the same compsition: liquid water, solid water (or ice), and water vapor. They would coexist at equilibrium. Some water molecules would be escaping from the surface of the water and from the ice surface, into the vapor phase. Likewise, some ice
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Figure 4. The sait-water phase diagram (notto scale). would be melting and some liquid freezing. Over the long run, at equilibrium, nothing would change. The third piece of background that we need before we go on is the ability to interpret a phase diagram. A phase diagram shows the temperatures at which various compositions of phases coexist at equilibrium. Figure 4 is just such a diagram for salt and water, in all their various compounds and mixtures, in the liquid and solid states. On the leR side is pure water, 100%HzO, and as we proceed to the right, the diagram represents progressively more salt-rich mixtures. Note the line sloping down to the right from 0 ' C ; this shows the compositions of the brines that would coexist at equilibrium with ice a t the various temperatures. For example, at -10 T; a brine with about 15% NaCl would be in eauilibrium with ice. There would be no tendency for the m&ture to change: it would remain salty water with ice floating in it. The ice would neither melt nor form more ice. If we added more salt to that brine, swn it would contain 23.3% NaCl, and at this point ice coexists with aliquid at the lowest possible temperature, called the eutedic point in this system. At a point colder than this, the whole mass crystallizes into twointimately intermixed or eutectic phases. That means that the wldest ice-brine mix will have 2 3 . 3 9 ~NaCl in it. And that i.; what we are shooting for when we make our ice cream. The key to this freezing-pint depression question has to do with the tendency to change, mentioned above. This is what drives the svstem to lower and lower temperatures. All systems tend "toward equilibrium, at the lowest possible enerw state. If we had some salt dissolved in water at 0 OC, thytendency for the water molecules to leave that liquid surface, to vaporize, would not be the same as over pure water. That is, the vapor pressures would be different. The vavor pressure over liauid water and ice at 0 'C is the same. 1n-fact, that's one very good way of defining eauilibrium. But if we add salt to the system, we upset the e
