Chemical Energetics: an introduction to chemical thermodynamics and the First Law

# Thermochemistry and calorimetry

On this page:
Thermochemical equations and standard states
Enthalpy of formation
Hess' Law and thermochemical calculations
Calorimetry
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The heat that flows across the boundaries of a system undergoing a change is a fundamental property that characterizes the process. It is easily measured, and if the process is a chemical reaction carried out at constant pressure, it can also be predicted from the difference between the enthalpies of the products and reactants. The quantitative study and measurement of heat and enthalpy changes is known as thermochemistry.

Thermochemical equations and standard states

In order to define the thermochemical properties of a process, it is first necessary to write a thermochemical equation that defines the actual change taking place, both in terms of the formulas of the substances involved and their physical states (temperature, pressure, and whether solid, liquid, or gaseous.

To take a very simple example, here is the complete thermochemical equation for the vaporization of water at its normal boiling point:

H2O(l, 373 K, 1 atm) → H2O(g, 373 K, 1 atm) ΔH = 40.7 kJ mol–1

The quantity 40.7 is known as the enthalpy of vaporization (“heat of vaporization”) of liquid water.

The following points should be kept in mind when writing thermochemical equations:

• Thermochemical equations for reactions taking place in solution must also specify the concentrations of the dissolved species. For example, the enthalpy of neutralization of a strong acid by a strong base is

H+(aq, 1M, 298 K, 1 atm) + OH–(aq, 1M, 298 K, 1 atm) →
H2O(l, 373 K, 1 atm) ΔH =56.9 kJ mol–1

in which the abbreviation aq refers to the hydrated ions as they exist in aqueous solution. Since most thermochemical equations are written for the standard conditions of 298 K and 1 atm pressure, we can leave these quantities out if these conditions apply both before and after the reaction. If, under these same conditions, the substance is in its preferred (most stable) physical state, then the substance is said to be in its standard state. Thus the standard state of water at 1 atm is the solid below 0°C, and the gas above 100°C. A thermochemical quantity such as ΔH that refers to reactants and products in their standard states is denoted by ΔH°.

• In the case of dissolved substances, the standard state of a solute is that in which the “effective concentration”, known as the activity, is unity. For non-ionic solutes the activity and molarity are usually about the same for concentrations up to about 1M, but for an ionic solute this approximation is generally valid only for solutions more dilute than 0.001-0.01M, depending on electric charge and size of the particular ion.

• Any thermodynamic quantity such as ΔH that is associated with a thermochemical equation always refers to the number of moles of substances explicitly shown in the equation. Thus for the synthesis of water we can write

2 H2(g) + O2(g) → 2 H2O(l) ΔH = –572 kJ

or

H2(g) + ½ O2(g) → H2O(l) ΔH = –286 kJ

Enthalpy of formation

The enthalpy change for a chemical reaction is the difference

ΔH = Hproducts – Hreactants

If the reaction in question represents the formation of one mole of the compound from its elements in their standard states, as in

H2(g) + ½ O2(g) → H2O(l) ΔH = –286 kJ

then we can arbitrarily set the enthalpy of the elements to zero and write

Hf ° = ΣHf °products – ΣHf °reactants = –286 kJ – 0 = –268 kJ mol–1

which defines the standard enthalpy of formation of water at 298K.

The standard enthalpy of formation of a compound is defined as the heat associated with the formation of one mole of the compound from its elements in their standard states.

In general, the standard enthalpy change for a reaction is given by the expression

Δ ΣHf °products – ΣHf °reactants (1) must know this!

in which the ΣHf ° terms indicate the sums of the standard enthalpies of formations of all products and reactants. The above definition is one of the most important in chemistry because it allows us to predict the enthalpy change of any reaction without knowing any more than the standard enthalpies of formation of the products and reactants, which are widely available in tables.

The following examples illustrate some important aspects of the standard enthalpy of formation of substances.

• The thermochemical equation defining Hf ° is always written in terms of one mole of the substance in question:

½ N2(g) + 3/2 H2(g) → NH3(g) ΔH° = –46.1 kJ (per mole of NH3)

• A number of elements, of which sulfur and carbon are common examples, can exist in more then one solid crystalline form. The standard heat of formation of a compound is always taken in reference to the forms of the elements that are most stable at 25°C and 1 atm pressure. In the case of carbon, this is the graphite, rather than the diamond form:

C(graphite) + O2(g) → CO2(g) ΔH° = –393.5 kJ mol–1

C(diamond) + O2(g) → CO2(g) ΔH° = –395.8 kJ mol–1

• The physical state of the product of the formation reaction must be indicated explicitly if it is not the most stable one at 25°C and 1 atm pressure:

H2(g) + ½ O2(g) → H2O(l) ΔH° = –285.8 kJ mol–1

H2(g) + ½ O2(g) → H2O(g) ΔH° = –241.8 kJ mol–1

Notice that the difference between these two ΔH° values is just the heat of vaporization of water.

• Although the formation of most molecules from their elements is an exothermic process, the formation of some compounds is mildly endothermic:

½ N2(g) + O2(g) → NO2(g) ΔH° = +33.2 kJ mol–1

A positive heat of formation is frequently associated with instability— the tendency of a molecule to decompose into its elements, although it is not in itself a sufficient cause. In many cases, however, the rate of this decomposition is essentially zero, so it is still possible for the substance to exist. In this connection, it is worth noting that all molecules will become unstable at higher temperatures.

• The thermochemical reactions that define the heats of formation of most compounds cannot actually take place; for example, the direct synthesis of methane from its elements

C(graphite) + 2 H2(g) → CH4(g)

cannot be observed directly owing to the large number of other possible reactions between these two elements. However, the standard enthalpy change for such a reaction be found indirectly from other data, as explained in the next section.

• The standard enthalpy of formation of gaseous atoms from the element is known as the heat of atomization. Heats of atomization are always positive, and are important in the calculation of bond energies.

Fe(s) → Fe(g) ΔH° = 417 kJ mol–1

• The standard heat of formation of a dissolved ion such as Cl– (that is, formation of the ion from the element) cannot be measured because it is impossible to have a solution containing a single kind of ion. For this reason, ionic enthalpies are expressed on a separate scale on which Hf° of the hydrogen ion at unit activity (1 M effective concentration) is defined as zero. Thus for Ca2+(aq), Hf° = –248 kJ mol–1; this means that the reaction

Ca(s) → Ca2+(aq) + 2e–(aq)

½ H2(g) → H+(aq) + e–(aq)

(Of course, neither of these reactions can take place by itself, so ionic enthalpies must be measured indirectly.)

Hess’ law and thermochemical calculations

You probably know that two or more chemical equations can be combined algebraically to give a new equation. Even before the science of thermodynamics developed in the late nineteenth century, it was observed that the heats associated with chemical reactions can be combined in the same way to yield the heat of another reaction. For example, the standard enthalpy changes for the oxidation of graphite and diamond can be combined to obtain ΔH° for the transformation between these two forms of solid carbon, a reaction that cannot be studied experimentally.

C(graphite) + O2(g) → CO2(g) ΔH° = –393.51 kJ mol–1

C(diamond) + O2(g) → CO2(g) ΔH° = –395.40 kJ mol–1

Subtraction of the second reaction from the first (i.e., writing the second equation in reverse and adding it to the first one) yields

C(graphite) → C(diamond) ΔH° = 1.89 kJ mol–1

This principle, known as Hess’ law of independent heat summation is a direct consequence of the enthalpy being a state function. Hess’ law is one of the most powerful tools of chemistry, for it allows the change in the enthalpy (and in other thermodynamic functions) of huge numbers of chemical reactions to be predicted from a relatively small base of experimental data.

Germain Henri Hess (1802-1850) was a Swiss-born professor of chemistry at St. Petersburg, Russia. He formulated his famous law, which he discovered empirically, in 1840. Very little appears to be known about his other work in chemistry.

Because most substances cannot be prepared directly from their elements, heats of formation of compounds are seldom determined by direct measurement. Instead, Hess’ law is employed to calculate enthalpies of formation from more accessible data. The most important of these are the standard enthalpies of combustion. Most elements and compounds combine with oxygen, and many of these oxidations are highly exothermic, making the measurement of their heats relatively easy.

For example, by combining the heats of combustion of carbon, hydrogen, and methane, we obtain the standard enthalpy of formation of methane, which as we noted above, cannot be determined directly:

The standard heat of atomization refers to the transformation of an element into gaseous atoms:

C(graphite)→ C(g) ΔH° = 716.7 kJ

... which is always, of course, an endothermic process. Heats of atomization are most commonly used for calculating bond energies.

Introduction to Hess's Law at the ChemTeam site

Calorimetry

How are enthalpy changes determined experimentally? First, you must understand that the only thermal quantity that can be observed directly is the heat q that flows into or out of a reaction vessel, and that q is numerically equal to ΔH° only under the special condition of constant pressure. Moreover, q is equal to the standard enthalpy change only when the reactants and products are both at the same temperature, normally 25°C.

The measurement of q is generally known as calorimetry. A very simple calorimetric determination of the standard enthalpy of the reaction

H+(aq) + OH–(aq) → H2O(l)

could be carried out by combining equal volumes of 0.1M solutions of HCl and of NaOH initially at 25°C. Since this reaction is exothermic, a quantity of heat q will be released into the solution. What we actually measure is the resulting temperature rise; if we multiply ΔT by the specific heat capacity of the solution (which will be close to that of pure water, 4.184 J/g-K), we obtain the number of joules of heat released into each gram of the solution, and q can then be calculated from the mass of the solution. Since the entire process is carried out at constant pressure, we have ΔH° = q.

> For reactions that cannot be carried out in dilute aqueous solution, the reaction vessel is commonly placed within a larger insulated container of water. During the reaction, heat passes between the inner and outer containers until their temperatures become identical. Again, the temperature change of the water is observed, but in this case the value of q cannot be found just from the mass and the specific heat capacity of the water, for we now have to allow for the absorption of some of the heat by the walls of the inner vessel. Instead, the calorimeter is “calibrated” by measuring the temperature change that results from the introduction of a known quantity of heat. The resulting calorimeter constant, expressed in J K–1, can be regarded as the “heat capacity of the calorimeter”. The known source of heat is usually produced by passing an electric current through a resistor within the calorimeter.

> ΔH = ΔU + Δ(PV) = qV + ΔngRT (8)

Since the process takes place at constant volume, the reaction vessel must be constructed to withstand the high pressure resulting from the combustion process, which amounts to a confined explosion. The vessel is usually called a “bomb”, and the technique is known as bomb calorimetry. In order to ensure complete combustion, the bomb is initially charged with pure oxygen above atmospheric pressure. The reaction is initiated by discharging a capacitor through a thin wire which ignites the mixture.

Since the process takes place at constant volume, the reaction vessel must be constructed to withstand the high pressure resulting from the combustion process, which amounts to a confined explosion. The vessel is usually called a “bomb”, and the technique is known as bomb calorimetry. In order to ensure complete combustion, the bomb is initially charged with pure oxygen above atmospheric pressure. The reaction is initiated by discharging a capacitor through a thin wire which ignites the mixture.

Problem Example

A sample of biphenyl (C6H5)2 weighing 0.526 g was ignited in a bomb calorimeter initially at 25°C, producing a temperature rise of 1.91 K. In a separate calibration experiment, a sample of benzoic acid C6H5COOH weighing 0.825 g was ignited under identical conditions and produced a temperature rise of 1.94 K. For benzoic acid, the heat of combustion at constant pressure is known to be 3226 kJ mol–1 (that is, ΔU° = –3226 kJ mol–1.) Use this information to determine the standard enthalpy of combustion of biphenyl.

Solution.The calorimeter constant is given by

The heat released by the combustion of the biphenyl at constant pressure is then ΔU:

(The negative sign indicates that heat is released in this process.) From the reaction equation

(C6H5)2(s) + 19/2 O2(g) → 12 CO2(g) + 5 H2O(l)

we have Δng = 12 - (19/2) = –5/2. Converting to ΔH, we substitute into

ΔH = qV + ΔngRT

ΔH° = ΔU° – (5/2)(8.314 J mol–1 K–1) = –6440 J mol–1

This is the amount of heat that is lost by the system if reaction takes place at constant pressure and the temperature is restored to its initial value.

Although calorimetry is simple in principle, its practice is a highly exacting art, especially when applied to processes that take place slowly or involve very small heat changes, such as the germination of seeds.

| Calorimeters can be as simple as a foam plastic coffee cup, which are often used in student laboratories.

Research-grade calorimeters are more likely to occupy entire table tops... | ... or even entire rooms!

The ice calorimeter is an important tool for measuring the heat capacities of liquids and solids, as well as the heats of certain reactions. This simple yet ingenious apparatus is essentially a device for measuring the change in volume due to melting of ice. To measure a heat capacity, a warm sample is placed in the inner compartment, which is surrounded by a mixture of ice and water. The heat withdrawn from the sample as it cools causes some of the ice to melt. Since ice is less dense than water, the volume of water in the insulated chamber decreases. This causes an equivalent volume of mercury to be sucked into the inner reservoir from the outside container. The loss in weight of this container gives the decrease in volume of the water, and thus the mass of ice melted. This, combined with the heat of fusion of ice, gives the quantity of heat lost by the sample as it cools to 0°C. |

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