## Determining Enthalpies of Substances

James Richard Fromm

The modern concept of enthalpy developed through the nineteenth century from the older concept of heat as part of the discipline of thermodynamics. Although we defer the further development of chemical thermodynamics to later sections, heat and enthalpy will be considered now.

Quantitative measurements of the heat flows resulting from many different physical and chemical processes led to the idea that each pure substance had a characteristic heat content, or enthalpy. The heat content of a substance can be increased by adding heat from an external source. When a substance is heated, there is a heat flow from the environment into the substance and that heat flow can be measured. It is found that the amount of heat required to change the temperature of a pure substance is an extensive property of the substance and its value is characteristic of the substance. This property is what is now called heat capacity.

#### Heat Capacity

The idea that a different quantity of heat was needed to change the temperature of different substances over the same range became quantitatively obvious early in the eighteenth century. Experimentally, much more heat is required to increase the temperature of a kilogram of iron than a kilogram of wood. This amount of heat is, for any narrow temperature range, a constant characteristic of the substance being heated or cooled. This constant, called the heat capacity C of the substance, can be easily measured. It is related to the heat content or enthalpy and in the form most often used by chemists it is defined as Cp = dH/dT The heat capacity at constant pressure Cp is the change in enthalpy required per change in temperature, the pressure being held constant. In other words, heat capacity is the rate of change of heat content with temperature. As its absolute temperature increases, any material must contain more heat. While Cp is really a constant only over a narrow range of temperatures it is usually approximately a constant over a much wider range of temperatures, even the entire solid or liquid temperature range of a material. This approximation is often useful in simplifying calculations.

Since heat capacity is an extensive property of substances, it is more conveniently expressed in intensive form. The intensive forms of heat capacity are either specific heat, which is heat capacity divided by mass, or molar heat capacity, which is heat capacity divided by amount of substance. The heat capacity of liquid water is approximately constant at 4.184 kJ/kg K (1.000 kcal/kg K) over its entire range. The specific heat of solid water, or ice, is less at 1.84 kJ/kg K as is that of gaseous water, or steam, at 1.97 kJ/kg K.

Specific heat, which has the units of J/g K or in older works cal/g K, is a characteristic property of all substances. The word specific carries the general meaning of per unit of mass in chemistry, and that is its meaning here. When the substance is pure, it is more convenient to use the amount of substance present (in moles) rather than to use units of mass. The amount of heat required to change the temperature of a substance, when given in molar units, is known as the molar heat capacity of the substance and its units are then J/K mol. The symbol C is often used for the molar heat capacity rather than for heat capacity in general.

Heat capacity, like other properties of matter, varies slightly with the temperature and also with the pressure. Values given in standard tables such as those in this section and more extensively elsewhere refer to 25oC and 100 kPa (one bar). Older tables refer to 25oC and one atmosphere, 101325 Pa. The very small difference between these values is significant only in extremely precise work. The larger changes in heat capacities with temperature are also often neglected. The subscript p is sometimes added to the symbol for molar heat capacity, C, to indicate that the value refers to a constant-pressure situation. Heat capacity at constant volume, which is much less commonly used, is indicated by the subscript v. The molar heat capacities of pure substances at constant pressure are included as part of the tables of thermodynamic properties of pure substances, and the molar heat capacities of aqueous solutes are included as part of the tables of thermodynamic properties of aqueous solutes.

Example. Let us calculate how much heat is required to raise the temperature of 200 g of water 10K.

x kJ = 0.2 kg x 10 K x 4.18 kJ/kg K = 8.36 kJ.

Example. Let us calculate how much heat is required to raise 70 kg of water, about the mass of an average adult man, from 25oC to 37.5oC, the normal temperature of the human body.

The molar mass of water is 18.0152 g/mol so 70 kg of water must be 3886 mol of water. The molar heat capacity of water is 75.291 J/K mol, so the amount of heat required is (3886 mol)(12.5 K)(75.291 J/K mol) = 3657 kJ.

##### Heats of Fusion of Substances When a solid substance is converted into a liquid, a certain amount of heat is required. This amount of heat is an extensive property and its value is characteristic of the substance. Thus the ice calorimeter used by Lavoisier measured heat by measuring the number of grams of ice melted. Such a process normally occurs at a specific fixed temperature known as the melting point. The melting point of water is 273.16 K, or 0oC. The amount of heat necessary to melt a substance at its melting point is called its heat of fusion. The heat of fusion of water is 333.51 kJ/kg and the molar heat of fusion of water is 6.009 kJ/mol.

When heat is steadily added to a substance, as when a pan of ice is heated on a stove, the heat is used to increase the temperature of the substance until a temperature is reached above which the substance can no longer exist. If the substance heated is a solid, under normal conditions that temperature is the melting point at which the substance becomes a liquid (if the substance goes from a solid to a gas rather than to a liquid, the temperature is called the sublimation point rather than the melting point). The substance then undergoes a change of state, as from solid to liquid. This change of state occurs at a constant temperature such as the melting point. During this change of state, the heat input is used to supply the heat of fusion required. Once all of the material has changed state no further heat can be used to supply heat of fusion and the temperature again increases, the heat capacity of the substance now being that of the substance in the new state, which may be significantly different from that of the previous state as in the case of water. Heats of fusion are symbolized by DHfus. (The symbol D used here to represent a difference is more often the upper-case Greek letter delta.)

Experimental measurements of the specific heats and heat capacities of substances made it clear the the heat properties of solids were quite different from those of the liquids into which those solids melted. What became even more obvious, however, was that a lot of heat was involved in converting a substance from one form to another. For pure substances, the heats which are involved in these phase changes are found to be characteristic of the substance and of the type of change. They are called the heat of fusion for melting processes, heat of vaporization for boiling processes, and heat of sublimation when the solid-gas phase change occurs directly. They have the modern units of J/mol; the values found in standard tables are those measured at one atmosphere or 100 kPa pressure and at the normal boiling, melting, or sublimation point of the substance at that pressure.

The heats of vaporization, of fusion, and of sublimation are now known as the enthalpy of vaporization, enthalpy of fusion, and enthalpy of sublimation. When values are given in terms of moles of substance rather than mass, they are called molar enthalpies of vaporization, molar enthalpies of fusion, and molar enthalpies of sublimation. The name enthalpy is modern and the term is part of the modern concepts of thermodynamics. The heat properties of matter were known, and in many cases accurately measured, over a century before the modern concepts of thermodynamics were formulated.

#### Heats of Vaporization of Substances

Conversion of a substance from a liquid to a gas, like conversion from solid to liquid, requires an increase in the enthalpy or heat content of a substance. This heat is called the heat of vaporization of the substance, usually symbolized as DHvap. This amount of heat and the temperature at which the conversion normally occurs, the boiling point, are characteristic of the particular substance. Water has a normal boiling point of 373.16 K (100.00oC), a heat of vaporization of 2483.20 kJ/kg, and a molar heat of vaporization of 38.371 kJ/mol. Boiling points and heats of vaporization of other substances are given in the Table above.

When a liquid is heated its temperature increases to the boiling point. Thereafter it remains constant in temperature until completely vaporized, since the addition of further heat goes to supply the heat of vaporization. The heat can thereafter go to increasing the temperature of the vapor, whose heat capacity is probably quite different from that of the liquid.

Example. Let us compute the total heat required (or change in heat content) to convert one kilogram of ice at -40oC to water vapor at +150oC. (Remember that a Celsius degree and a kelvin (K) are the same size.)

Since several things happen as one does this and since one cannot integrate across discontinuity in what happens, the total change in enthalpy is the sum of several enthalpy values for ice, water, steam and conversions between them. In integral form:

DH = -40(integral)0Cp(c)dT + DHfus + 0(integral)100Cp(l)dT + DHvap + 100(integral)150Cp(g)dT

which yields, assuming that the heat capacities are not temperature-dependent,

DH = (1.84 kJ/kg K)(40 K) + 333.51 kJ/kg + (4.184 kJ/kg)(100 K) + 2483.20 kJ/kg + (1.97 kJ/kg K)(50 K) = 3407 kJ/kg

The heat capacities for water in its different physical states are all different and each must be used over that temperature range in which that state exists. This is true for all substances; the heat capacities of different physical states of the same substance are not the same.