CHEM 331 Physical Chemistry Fall 2014 Some General Remarks Concerning Heat Capacity The Heat Capacity is one of those quantities whose role in thermodynamics cannot be underestimated. Measured using simple calorimetric measurements, it finds itself useful in calculations involving any number of thermodynamic "potentials". But, more on this later. Here we derive a few simple results involving the heat capacity which we will find useful as we proceed with our discussion of thermodynamics. The beautiful work of Joseph Black on calorimetry, the measurement of heat changes, was published in 1803, four years after his death. In his Lectures on the Elements of Chemistry, he pointed out the distinction between the intensive factor, temperature, and the extensive factor, quantity of heat. Black showed that equilibrium required an equality of temperature and did not imply that there was an equal "quantity of heat" in different bodies. He then proceeded to investigate the capacity for heat or the amount of heat needed to increase the temperature of different bodies by a given number of degrees. It was formerly a supposition that the quantities of heat required to increase the heat of different bodies by the same number of degrees were directly in proportion to the quantity of matter in each … But very soon after I began to think on this subject (Anno 1760) I perceived that this opinion is a mistake, and that the quantities of heat which different kinds of matter must receive to reduce them to an equilibrium with one another, or to raise their temperatures by an equal number of degrees, are not in proportion to the quantity of matter of each, but in proportions widely different from this, and for which no general principle or reason can yet be assigned. In explaining his experiments, Black assumed that heat behaved as a substance, which could flow from one body to another but whose total amount must always remain constant. This idea of heat as a substance was generally accepted at that time. Lavoisier even listed caloric in his "Table of the Chemical Elements." In the particular kind of experiment often done in calorimetry, heat does, in fact, behave much like a weightless fluid, but this behavior is the consequence of certain special conditions. Consider a typical experiment: A piece of metal of mass m2 and temperature T2 is introduced into an insulated vessel containing a mass m1of water at temperature T1. We impose the following conditions: (1) the system is isolated from its surroundings; (2) any change in the container itself can be neglected; (3) there is no change such as vaporization, melting, or solution in substance, and no ne temperature T somewhere between T1 and T2, and the temperatures are related by an equation of the form c2 m2 (T2 - T) = c1m1 (T -T1) Here, c2 is the specific heat of the metal and c2 m2 = C2 is the heat capacity of the mass of metal used. The corresponding quantities for water are c1, and c1 m1 = C1. The specific heat is the heat capacity per unit mass. More careful measurements showed that the specific heat was itself a function of temperature. … [Thus,] the heat capacity, being a function of temperature, should be defined precisely on in terms of differential heat flow Q and temperature change. Thus, in the limit, [it] becomes Q = C dT or C = Physical Chemistry, 4th Ed. Walter J. Moore So, in general, measured under a given constraint x, the Heat Capacity of a substance is defined as: Cx = We have already specified two such constraints, constant volume and constant pressure, giving us the heat capacities Cv and Cp, respectively. These two heat capacities are the most useful experimentally and are related to the the internal energy U and to the enthalpy H; again, respectively. Cv = = Cp = = Since Cv and Cp can both be measured for a given substance, it is not unreasonable that they should be related. It is this relationship which we now seek to establish. We start with our expressions for dU: dU = Q + W = Q - Pop dV = Q - P dV and dU = dT + = Cv dT + dV dV Equating these expressions, we have: Q - P dV = Cv dT + dV Dividing by dT and restricting ourselves to constant P: - P = Cv + Recognizing the lead term is Cp gives us: Cp - P = Cv + Rearranging a bit gives us the desired result: Cp - Cv = In general we find that Cp > Cv. This is not unreasonable. When heat enters a system at constant volume, it increases the "chaotic" motion of the system's constituents. This leads to an increase in the temperature. (What we mean by "chaotic" motion is the translational, rotational and vibration motion of the molecules. The first of these contributes to the kinetic energy of the molecules and thereby increases the system's temperature.) When heat enters a system at constant pressure, the system's volume will increase, doing work in the surroundings. The molecules of the system will be "pulled" apart requiring energy. And, there will be an increase in the "chaotic" motion of the system's constituents. Only this last will lead to an increase in temperature, for which there is less heat energy available. For a given amount of heat, there are more modes to absorb the energy without a temperature increase, if the heating is done at constant pressure rather than at constant volume. Hence, Cp - Cv > 0. Work to Surroundings Heat Inc Chaotic Motion (Temperature) Heat "Pull" Molecules Apart Inc Chaotic Motion (Temperature) In our above expression for Cp - Cv, the term: P can be thought of as the work against the external pressure. The term: is the work done against the internal pressure. Recall, the internal pressure of a system is: T = For an Ideal Gas, we have: = 0 and V = So, = This gives us the important result: Cp - Cv = = [P + 0] (You can only use Cp - Cv = R for an Ideal Gas.) Typically, for liquids and solids we find: Cp ≅ Cv This is illustrated by the heat capacity data for Water. Cp (J/K mol) Cv (J/K mol) 75.3 74.8 = R