Titration Calorimetry

Calorimetry literally means “heat measurement.” For consistency with other forms of energy and to avoid confusion, the modern unit for heat is the joule which equals 0.2390 calories. The unit for heat rate is the watt or J/s with prefixes m for milli (10⁻³), μ for micro (10⁻⁶), and n for nano (10⁻¹²). Heat can be measured in only three ways, referred to as “temperature change,” “heat conduction,” and “power compensation” (Hansen 2001). Table 1.1 lists characteristics of these methods.

Titration calorimetry is a relatively rapid way of obtaining thermodynamic data on reactions in solution. Enthalpy changes for solution of solids, for sorption of solutes on suspensions of sorbents, for reactions of gases with solutes, and for mixing of liquids and solutions can all be done in calorimeters equipped to handle gases, liquids, and solids, e.g., see Russell et al. (2006). However, this brief is largely limited to consideration of methods involving titration of a solution of one reactant into a solution of a second reactant. Titration calorimetry has three applications, analytical determinations of concentrations of reactants in solution, determination of enthalpy changes for reactions in solution, and under certain conditions, simultaneous determination of equilibrium constants and enthalpy changes for reactions in solution. This last application provides a full complement of thermodynamics for reactions in solution, i.e., the Gibbs energy change (Δ_rG = −RTlnK), the enthalpy change (Δ_rH), and the entropy change (Δ_rS = (Δ_rH − Δ_rG)/T); R is the gas constant, and T is the Kelvin temperature.

The equilibrium constant for a reaction is defined as

Calorimetry has been used to measure rates of reaction since the late 1700s when Antoine Lavoisier used an ice calorimeter to measure the rate of heat produced by a guinea pig (Lavoisier and LaPlace 1780). In proving that respiration was simply a slow combustion, Lavoisier also measured the rates of consumption of oxygen and production of CO₂. Lavoisier’s experiments demonstrate many of the advantages of calorimetry for kinetic measurements; rates can be measured directly and noninvasively in any media. Measurements of rates instead of measuring the amount of product accumulated over time (or of reactant lost) are faster, simpler, and more sensitive than most other methods, particularly for slow reactions. Aside from methods that count radioactive decay rates, heat conduction and power compensation calorimetry are the only methods that measure rates directly. Calorimetry has been shown to be capable of measuring rates of reactions with half-lives greater than 1000 years (Hansen 1996). Calorimetry makes no requirements of the system except that it fits within the reaction vessel and not react with any other materials in the reaction vessel. Systems can be gaseous, liquid, solid, or even a living organism, e.g., plant tissue, insects, microorganism cultures, and animals.

In many cases, the goal of ITC measurements is to infer quantitative values for reaction parameters such as equilibrium constants, enthalpy changes, and rate constants. To do this, a mathematical model of the reaction(s) going on in the calorimeter is first constructed. Then the predictions of the model are compared against the experimental observations. Because the model includes several unknown parameters, it is necessary to vary the parameters until the model predictions are “as close as possible” to the observed data.

Many analytical applications of titration calorimetry were developed beginning in the 1960s when thermistors became available. Thermistors, with time constants <1 s and sufficient sensitivity to resolve a few micro-degrees, provided a convenient way to make rapid measurements of temperature that made continuous titration and single injection methods of analysis feasible with very simple equipment. These methods, variously known as thermometric titration, enthalpy titration, enthalpic injection, calorimetric titration, etc., were the forerunners to what eventually became known as isothermal titration calorimetry (ITC). In the titration methods, endpoints and thence concentrations of reactants are indicated by a change in heat production. In the injection methods, the measured amount of heat is divided by the Δ_rH value to determine the amount of analyte. Several articles and reviews on analytical applications of titration calorimetry are given in the bibliography.

After studying Sects. 1 through 5, try your hand at answering the following questions to see if you remember and understand the major concepts. The answers are given in Sect. 8, but do not peek until you have given it an honest try.