## From Heat to Enthalpy: Calorimetry

James Richard Fromm

The term calorimetry is the modern name for the studies of the quantity or amount of heat. Early calorimetry is exemplified by the work of Brook Taylor, who in 1723 mixed different proportions of hot and cold water and noted the resulting temperatures. These and other results gave rise to the so-called law of mixtures: the temperature of a mixture of substances is the weighted mean of the temperatures of the two substances at two different temperatures which made up the mixture. The law of mixtures was well-known by the end of the first half of the eighteenth century.

Examples of the Law of Mixtures using Water.
1 liter at 20oC, + 1 liter at 30oC = 2 liters at 25oC
2 liters at 20oC + 1 liter at 30oC = 3 liters at 23 1/3oC
2 liters at 30oC + 1 liter at 20oC = 3 liters at 26 2/3oC

Calorimetric measurements led to the discovery that every substance requires a characteristic amount of heat to change its temperature over a temperature interval. The amount of heat required is approximately, but not exactly, uniform over any reasonable range of temperatures. The amount of heat required per unit of mass came to be known as the specific heat of the substance. The specific heat of a substance is an intensive property characteristic of the substance.

Water, the most conveniently available pure substance, was used as a standard for specific heat and its use became entrenched in two units of heat. In Europe the most common heat unit, the calorie, became established as the amount of heat necessary to raise the temperature of one gram of water one degree Celsius. In England, and later in North America, the most common heat unit became the British Thermal Unit, or BTU, which is the amount of heat necessary to raise the temperature of one pound of water one degree Fahrenheit. Both of these units, after successive minor redefinitions, have now been replaced by the SI unit of energy, the joule. One thermochemical calorie is now by definition exactly 4.184 joules and one BTU is now by definition exactly 1055.06 joules.

In the second half of the eighteenth century, two significant developments related to heat took place. The first, the technology and science of work in relation to heat, we shall defer to a later chapter. The second, a systematic understanding of the quantitative measurement of heat, originated from the systematic chemical studies of Antoine Lavoisier and his colleagues in France.

Antoine L. Lavoisier (a chemist) and Pierre L. Laplace (a mathematician and physicist) set forth a statement of the two rival theories of heat in a joint paper published in 1783. These were those claimed by physicists and chemists in the previous half-century. To the physicists, heat was the vis viva (living motion) of the atoms which comprise all bodies, while to the chemists, heat was a subtle material fluid, often called caloric. This paper pointed out that there are several consequences which are common to both of these theories. In either case, the total amount of heat is constant in simple mixing. In either case, any changes in the amount of heat, real or apparent, which appears when any system of bodies changes state (ice melts, water is heated, water boils) are reproduced in inverse order when the system returns to its first state. This latter consequence can be, and was, taken to mean that if some certain amount of heat is required to warm up and vaporize enough water for one working cycle of a steam engine, the same amount of heat is delivered to the condenser at the end of the cycle.

They then proceeded to develop the law of mixtures and to discuss in some detail the use of their ice calorimeter. They were able to use the idea that some of the heat in a body is "free" (and hence will affect a thermometer) while some of it is "combined" and doesn't affect the thermometer. They defined the three physical states of matter (solid, liquid, gas) in terms of their content of this "combined" heat. These same ideas were used by Lavoisier in his textbook Elements of Chemistry. He himself was convinced that heat, or the caloric fluid, was a subtle physically existing fluid.

#### Calorimeters

The ice calorimeter is simply a large insulated container of ice and water with a basket which can be used to remove the ice for weighing. The amount of heat evolved in whatever reaction takes place within the calorimeter is equal to the mass of ice melted multiplied by the heat of fusion of ice, 333.5l kJ/kg.

The water calorimeter is often used in undergraduate student laboratories. A styrofoam (expanded polystyrene) coffee cup and a styrofoam lid constitute the calorimeter. Calorimeters of this type are used in chemical research, though their insulation is of much higher quality and their thermometer is more accurate. In a water calorimeter, the mass of the water in the calorimeter is measured. The temperature is measured accurately once before the reaction begins and once after the reaction has taken place and the interior of the calorimeter has come to a uniform temperature. The difference in temperature multiplied by the mass and by the specific heat of water is equal to the total amount of heat evolved in the reaction. In accurate work the specific heats of the calorimeter itself, the reactants, the products, and the reaction container must also be considered.

Water calorimeters are used to measure heats of reactions which take place in sealed reaction containers as well as heats of solution of solid salts in water and heats of reaction which occur in solutions. For solution reactions, no reaction container is necessary if the specific heat of the solution has been measured beforehand.

Example. An ice calorimeter initially contains 1.352 kg of ice. A chemical reaction is allowed to occur within the calorimeter between 1.271 g of carbon and an excess of oxygen inside a reaction vessel. After a uniform temperature of 0oC is again attained, the calorimeter contains only 1.207 kg of ice. Let us calculate the heat of reaction, which in this case is a heat of combustion, per gram of carbon and per mole of carbon.

The mass of ice melted is (1.352 - 1.207) kg, which is 145 g /(18.015 g/mol) = 8.049 mol ice. The molar heat of fusion of ice is (333.51 J/g)(18.015 g/mol) = 6.008 kJ/mol so the heat produced was 48.36 kJ. The heat of reaction is then 48.36 kJ/1.271 g C = 38.05 kJ/g C. The molar heat of this reaction is (38.05 kJ/g C)(12.011 g C/mol C) which is 457.0 kJ/mol C. This is considerably more than the standard value of 393.509 kJ/mol, probably due to the melting of some ice by the heat of the room.

Example. A water calorimeter at 20.1oC contains 643.2 g of water. Addition of 2.744 g of KOH to the calorimeter produces a temperature rise to 23.4oC. Let us compute the molar heat of solution of KOH from these data.

The temperature rise, or difference between the initial and final temperatures, is +3.3oC or +3.3 K. Ignoring the solute, the calorimeter contains 634.2 g of water which is 634.2 g/(18.015 g/mol) = 35.204 mol water. The molar heat capacity of water is 75.291 J/mol K so the total heat of dissolution is 2.651 kJ. The heat was produced by dissolution of (2.744 g/56.109 g KOH/mol KOH) = 0.0489 mol of KOH, so the molar heat of solution of KOH is (2.651 kJ)/(0.0489 mol) = 54.2 kJ/mol.

In this example the heat capacity of those parts of the calorimeter which are heated is neglected, as is the heat capacity of the solute and any difference between the heat capacity of the resulting solution and the heat capacity of water. All of these factors would have to be considered in precise solution calorimetry.

Lavoisier's quantitative use of the ice calorimeter is worthy of considerable note. In his book Elements of Chemistry, he reported that one lb (pound) of hydrogen will, when burned, melt 295 lb 9 oz 3 1/2 gros of ice. For hydrogen, the chemical reaction involved would be

2H2(g) + O2(g) 2H2O + heat

The weight of ice melted was 295.0000 lb + 0.5625 lb + 0.0273 lb, or 295.5895 French royal lb, or 144.693 kg of ice melted. The heat given off is then 144.693 kg x 333.51 J/g = 48.256 kJ of heat. This quantity of heat was produced by the reaction of one lb hydrogen or 489.5058 g of hydrogen, which is 244.7529 moles H2. Thus the heat of combustion of hydrogen is 48.256 kJ/244.759 moles hydrogen = 197.2 kJ/mole hydrogen(0oC); the best modern value would be 285.830 kJ/mole H2(25oC) or 287.712 kJ (0oC). It is probable that this error, which is large, is due to the difficulty of weighing the hydrogen used accurately.

A much better result is found with the combustion of charcoal (carbon). Lavoisier reports that one pound of charcoal will, when burned, produce enough heat to melt 96 lb 8 ounces of ice. Hence 1 lb carbon melts 96.5 lb ice, or 489.5058 g carbon melts 47.2373 kg ice. Then, using the same calculation as above, 40.755 moles carbon produce 47.2373 kg x 333.51 J/g, or 1 mole carbon produces 385.97 kJ/mole (0oC) of heat on combustion. The best modern value is 393.509 kJ/mol (25oC).

Lavoisier also reported that combustion of one pound of phosphorus will melt 100 pounds of ice. This does not agree well with modern values if the reaction of combustion is taken as:

4P + 5O2 2P2O5 + heat.

One reason might be the hydrolysis of P2O5, which reacts rapidly with water. It has been employed as a laboratory drying agent.

In 1800, the concept of energy had not yet been formulated. Lavoisier lists both heat and light as elements in 1789, and although at that time it was realized that they were no doubt related the inclusive concept of energy had not been formulated. The question as to the nature of heat remained unresolved - the crucial experiment of Rumford belongs properly to the nineteenth century, though published in 1798. The practical use of heat, however, is now a study of great importance, and the replacement of drudgery with machine labor is a continuing process today. These machines still derive power in the same way as did the early steam engines, from heat, and the science of thermodynamics has many consequences for mankind.