Photoacoustic Calorimetry

“Any chemical species, which under ambient conditions (i.e., a temperature around 25°C, and a pressure close to 1 atm) will, for a combination of kinetic and thermodynamic reasons, decay on a timescale ranging from microseconds, or even nanoseconds, to a few minutes” can be classified as a short-lived compound. According to this definition, suggested by Almond, it is clear that the experimental methods described in previous chapters can only be used to study the thermochemistry of long-lived substances. The technique that we address here, known as photoacoustic calorimetry (PAC) or laser-induced optoacoustic calorimetry (LIOAC), is suitable for investigating the energetics of molecules with lifetimes smaller than about 1μs. It relies on the photoacoustic effect, which was discovered by Bell more than 100 years ago. With the assistance of Tainter, he was able to “devise a method of producing sounds by the action of an intermittent beam of light” and conclude that the method “can be adapted to solids, liquids, and gases”. Figure 13.1 shows a photophone, “an apparatus for the production of sound by light,” used by Bell to investigate the photoacoustic effect. The controversy around the origin of this phenomenon was settled by Bell himself and by Lord Rayleigh; their views were rather close to our present understanding: When a light pulse is absorbed by a substance, a given amount of heat is deposited, producing a local thermal expansion; this thermal expansion propagates through the medium, generating sound waves. The basic theory of the photoacoustic effect was described by Tam and Patel and some of its applications were presented in a review by Braslavsky and Heibel. The first use of PAC to determine enthalpies of chemical reactions was reported by the groups of Peters and Braslavsky. The same groups have also played an important role in developing the methodologies to extract those thermodynamic data from the experimentally measured quantities. In the ensuing discussion, we closely follow a publication where the use of the photoacoustic calorimety technique as a thermochemical tool was examined. Consider the elementary design of a photoacoustic calorimeter, shown in figure 13.3. The cell contains the sample, which is, for instance, a dilute solution of a photoreactive species.

Differential Scanning Calorimetry (DSC)

Physical and chemical changes may often be induced by raising or lowering the temperature of a substance. Typical examples are phase transitions, such as fusion, or chemical reactions, such as the solid state polymerization of sodium chloroacetate, which has an onset at 471 K: ClCH2COONa (cr) ⇋ NaCl (cr) + 1/n − (CH2COO)n − (pol) Differential scanning calorimetry (DSC) was designed to obtain the enthalpy or the internal energy of those processes and also to measure temperature-dependent properties of substances, such as the heat capacity. This is done by monitoring the change of the difference between the heat flow rate or power to a sample (S) and to a reference material (R), ΔΦ = ΦS − ΦR = (dQ/dt)S − (dQ/dt)R, as a function of time or temperature, while both S and R are subjected to a controlled temperature program. The temperature is usually increased or decreased linearly at a predetermined rate, but the apparatus can also be used isothermally. In some cases DSC experiments may provide kinetic data. According to Wunderlich, differential scanning calorimeters evolved from the differential thermal analysis (DTA) instruments built by Kurnakov at the beginning of the twentieth century. In these early DTA apparatus, the temperature difference between a sample and a reference, simultaneously heated by a single heat source, was measured as a function of time. No calorimetric data could be derived, and the instruments were used, for example, to determine the temperatures of phase transitions and to identify metals, oxides, minerals, soils, and foods. The attempts to obtain calorimetric data from DTA instruments eventually led to the development of DSC. The term differential scanning calorimetry and the acronym DSC were coined in 1963 when the first commercial instrument of this type became available. This apparatus was easy to operate, enabled fast experiments, and required only small samples (typically 5–10 mg). Its importance for materials characterization was immediately demonstrated and the DSC technique soon experienced a boom. New user-friendly commercial instruments were developed, and new applications were explored. It is, however, somewhat ironic that the method ows its still growing popularity to analytical rather than calorimetric uses.

Titration Calorimetry

Titration calorimetry is a method in which one reactant inside a calorimetric vessel is titrated with another delivered from a burette at a controlled rate. This technique has been adapted to a variety of calorimeters, notably of the isoperibol and heat flow types. The output of a titration calorimetric experiment is usually a plot of the temperature change or the heat flow associated with the reaction or physical interaction under study as a function of time or the amount of titrant added. A primary use of titration calorimetry is the determination of enthalpies of reaction in solution. The obtained results may of course lead to enthalpies of formation of compounds in the standard state by using appropriate thermodynamic cycles and auxiliary data, as described in chapter 8 for reaction-solution calorimetry. Moreover, when reactions are not quantitative, both the equilibrium constant and the enthalpy of reaction can often be determined from a single titration run. This also yields the corresponding ΔrGo and ΔrSo through equations 2.54 and 2.55. Extensive use has been made of titration calorimetry as an analytical tool. These applications, which are outside the scope of this book, have been covered in various reviews. The historical development of titration calorimetry has been addressed by Grime. The technique is credited to have been born in 1913, when Bell and Cowell used an apparatus consisting of a 200 cm3 Dewar vessel, a platinum stirrer, a thermometer graduated to tenths of degrees, and a volumetric burette to determine the end point of the titration of citric acid with ammonia from a plot of the observed temperature change against the volume of ammonia added. The capabilities of titration calorimetry have enormously evolved since then, and the accuracy limits of modern titration calorimeters are comparable to those obtained in conventional isoperibol or heat-flow instruments. The titration procedures described in the literature can be classified as continuous or incremental, depending on the mode of titrant addition. In the first case the titrant is continuously introduced in the reaction vessel at a programmed (not necessarily constant) rate during a run.

Bond Energies

Although standard enthalpies of formation provide information about the net stability of molecules and their transformations, they do not always indicate stability of individual bonds. This analysis normally involves parameters, loosely called “bond energies,” that reflect the amount of energy required to cleave chemical bonds. Bond energies are essential for understanding the nature of chemical bonds. They can be used to assess the results from quantum chemistry calculations (or from other, less sophisticated theoretical models) and thus support or oppose the descriptions of those bonds. Moreover, bond energy values also enable us to estimate the driving forces of chemical reactions by considering the strengths of all the bonds that are cleaved and formed. In fact, there are many reactions for which the standard enthalpies of formation of all reactants and products are not available (and cannot be easily estimated) but whose energetics can be predicted from the appropriate bond energies. In the previous chapters, we attempted to review all the important parameters in molecular energetics, but to avoid unnecessary distraction, we deliberately omitted bond energies from the discussion. The literature is plagued with a variety of concepts that fall into that designation but are not always synonymous. We can find names like bond strengths, bond enthalpies, bond energies, bond dissociation enthalpies, bond dissociation energies, bond disruption enthalpies, bond enthalpy terms, intrinsic bond energies, and symbols like D, D̄, 〈D〉, E, BDE, and so on. The meaning of these concepts it not always obvious and, unfortunately, some are occasionally misused. Now we look into each one of them. Consider a molecule AB, where A and B can be atoms or groups of atoms.

Isoperibol Reaction-Solution Calorimetry

The determination of enthalpies of reaction in solution, using isoperibol reaction-solution calorimetry, is often the easiest and most accurate method of determining enthalpies of formation of compounds that cannot be studied by combustion calorimetry. The technique was pioneered by Thomsen who, between 1882 and 1886, performed thermochemical measurements involving the solution of various substances in liquids (e.g., diluted acids). Many types of isoperibol reaction-solution calorimeters have been developed since then. The designs vary according to the nature of the reactions of interest. One of the most widely used consists of a vessel, such as the one shown in figure 8.1, immersed in a thermostatic water bath. The sample is sealed inside a thin-walled glass ampule A, fixed to an ampule breaking system B in the calorimeter head C. The calorimeter head also supports the temperature sensor D, the stirrer E, and an electrical resistance F, used for calibration of the apparatus. The Dewar vessel G, containing the solution to be reacted with the sample, is adjusted to C. The assembled calorimetric vessel is transferred to the thermostatic bath, and from then on, the experimental procedure closely follows that already described in section 7.1 for isoperibol static-bomb combustion calorimetry. The reaction is initiated at the end of the fore period by pushing down the plunger H and breaking the ampule against a pin situated at the bottom of the ampule breaking system B. As a result of the calorimetric experiment, a temperature-time curve such as the one in figure 7.2 is obtained. Note that figure 7.2 is typical of an exothermic process. In the case of an endothermic process, a decrease of the temperature of the calorimetric system is observed during the reaction period. The experiments are usually carried out at atmospheric pressure and the initial goal is the determination of the enthalpy change associated with the calorimetric process under isothermal conditions, ΔHICP, usually at the reference temperature of 298.15 K. This involves the determination of the corresponding adiabatic temperature change, ΔTad, from the temperature-time curve just mentioned, by using one of the methods discussed in section 7.1; the determination of the energy equivalent of the calorimeter in a separate experiment.

Gas-Phase Ion Energetics

The experimental methods designed to investigate the energetics of gas-phase ions have been another important source of thermochemical data, particularly throughout the past two or three decades. In this chapter, we discuss the main quantities that are measured experimentally and lead to reaction enthalpy values. The adiabatic ionization energy of any molecule AB (mono-, di-, or polyatomic), represented by Ei (AB), is the minimum energy required to remove an electron from the isolated molecule at 0 K: AB(g) → AB+(g) + e−(g) (4.1) The proviso T = 0 signifies that AB is in its electronic, vibrational, and rotational ground states and has no translational energy. The word isolated indicates the perfect gas model. The “minimum energy” condition ensures that AB+ is also in its electronic, vibrational, and rotational ground states and the translational energies of AB+ and e− are both zero; it also indicates that the products in reaction 4.1 do not interact, that is, they also conform with the perfect gas model.

Thermochemistry and Molecular Energetics

Thermochemistry has been defined in one of the most popular physical chemistry textbooks as “the study of the heat produced or required by chemical reactions”. The use of heat, instead of the more general word energy, immediately suggests a close association between thermochemistry and calorimetry—the oldest experimental technique for investigating the thermodynamics of chemical reactions. This view is, in fact, shared by many of our students and some of their teachers, together with the belief that thermochemistry, founded in the eighteenth century by Black, Lavoisier, and Laplace, has seen few major developments since the days of Berthelot and Thomsen, over 100 years ago. The notion that calorimetric studies are almost the sole source of thermochemical information prevails beyond the classroom. Also, the idea of thermochemistry as a science of the past is even conveyed by distinguished scientists and lecturers. Figure 1.1, taken from a delightful account by Herschbach, depicts thermochemistry as a mountain that was necessary to climb to conquer the structure and dynamics summits and reach the ultimate goal—the understanding of chemical synthesis. The picture is enlightening, but the timeline suggests that the climax of the thermochemical era dates back to the early decades of the last century, coinciding with the publication of Lewis and Randall’s Thermodynamics and the Free Energy of Chemical Substances. However, the golden years of calorimetry started in the 1930s, thanks to the work on organic compounds and key molecules, such as water and carbon dioxide, by Rossini and his colleagues at the National Bureau of Standards, and continued in the 1960s and 1970s. Thermochemical studies of organometallic compounds were pioneered by Skinner and his coworkers at Manchester University. Figure 1.1 can also be regarded from a different perspective, which is more correct and probably in keeping with Herschbach’s thoughts: Thermochemistry is not only the first mountain to climb but also the solid ground from which the remaining heights can be reached. The heaven, or the perfect understanding of chemical synthesis, rests on a detailed knowledge of thermochemistry, structure, dynamics, and their relationships.

Electrochemical Measurements

Electrochemical measurements have been playing an increasingly important role in the thermodynamic study of reactions in solution, not only because they provide data that are difficult (or even impossible) to obtain by other methods but also because these data can often be compared with the values determined for the analogous gas-phase reactions, thus yielding information on solvation energetics. Figure 16.1 was adapted from a scheme proposed by Griller et al. It summarizes the thermochemical information on the R–X bond that can be probed by electrochemical methods. The vertical arrows represent homolytic cleavages, and the horizontal arrows depict reduction or oxidation potentials. The authors have appropriately called the scheme in figure 16.1 a “mnemonic,” rather than a “thermochemical cycle,” because not all arrow combinations define thermochemical cycles. This can be made more clear by inspecting figure 16.2, where true thermochemical cycles are defined. For example, the enthalpy of reaction 7 is not the sum of the enthalpies of reactions 1 and 4 (as might be suggested by figure 16.1) but their sum minus the enthalpy of reaction 12. In fact, true thermochemical cycles in figure 16.1 can only be defined by considering parallelograms confined either to the upper or the lower part of the mnemonic. For instance, the enthalpy of reaction 7 is given by the enthalpy of reaction 4 plus the enthalpy of reaction 9 minus the enthalpy of reaction 3, but it is not equal to the enthalpy of reaction 6 minus the enthalpy of reaction 11 plus the enthalpy of reaction 10. Also, the enthalpy of reaction 1 (the homolytic dissociation of the R–X bond in the neutral molecule RX) can be given by the sum of the enthalpies or reaction 5 and 11 minus the enthalpy of reaction 3 or, for example, by the sum of the enthalpies of reactions 7 and 12 minus the enthalpy of reaction 4. The attractive feature of the mnemonic in figure 16.1 (or the thermochemical cycles in figure 16.2) is that it depicts the seven possible R–X cleavage reactions of RX, RX−, and RX+, as well as their relationships.

Kinetics in Solution

The main equations used to extract thermochemical data from rate constants of reactions in solution were presented in section 3.2. Here, we illustrate the application of those equations with several examples quoted from the literature. First, however, recall that the rate constant for any elementary reaction in solution, defined in terms of concentrations, is related to the activation parameters through equations 15.1 or 15.2. Equation 15.1 yields the enthalpy and the entropy of activation respectively from the slope and the intercept of a ln(k/T) versus 1/T plot (an Eyring plot). Equation 15.2 leads to the Arrhenius activation energy and the frequency factor, respectively, from the slope and the intercept of a ln k versus 1/T plot (an Arrhenius plot). All the parameters refer to the mean temperature of the plot, and Δ‡Ho is related to Ea by equation 15.3. Finally, recall that if the activation parameters are available for the forward (subscript 1) and the reverse (subscript −1) reaction, the enthalpy of this reaction is calculated by equation 15.4. In the preceding chapter on equilibrium in solution, it was pointed out that any analytical method suitable for determining equilibrium compositions of a reaction mixture at several temperatures can be used to obtain the enthalpy and entropy of that reaction. A similar statement can be made here: Any analytical method suitable for monitoring concentration changes with time at several temperatures can be used to derive the activation parameters of a reaction. Therefore, the analytical techniques used in equilibrium experiments are also applied in nonequilibrium (kinetics) studies. However, in this case, the choice of the analytical method will have an additional and important restriction, for it must consider the reaction rate. An instrumental technique suitable for determining the concentration of a given species under equilibrium conditions may be inappropriate for determining a fast concentration change of the same species.

Heat Flow Calorimetry

Heat flow calorimeters, also known as “heat flux,” “heat conduction,” or “heat leakage” calorimeters, are instruments where the heat output or input associated with a given phenomenon is transferred between a reaction vessel and a heat sink. This heat transfer can be monitored with high thermal conductivity thermopiles containing large numbers of identical thermocouple junctions regularly arranged around the reaction vessel (the cell) and connecting its outside wall to the heat sink (the thermostat). The determination of the heat flow relies on the so-called Seebeck effect. An electric potential, known as thermoelectric force and represented by E, is observed when two wires of different metals are joined at both ends and these junctions are subjected to different temperatures, T1 and T2. Several thermocouples can be associated, forming a thermopile. For small temperature differences, the thermoelectric force generated by the thermopile is proportional to T1 − T2 and to the number of thermocouples of the pile (n): E = nε ′ (T1 − T2) (9.1) where ε′ is the thermoelectric power of a single thermocouple (ε′ = dE/dT).