Some General Remarks Concerning Heat Capacity

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1 CHEM 331 Physical Chemistry Fall 2017 Some General Remarks Concerning Heat Capacity The 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 that 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 m 2 and temperature T 2 is introduced into an insulated vessel containing a mass m 1 of water at temperature T 1. 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 T 1 and T 2, and the temperatures are related by an equation of the form c 2 m 2 (T 2 - T) = c 1 m 1 (T -T 1 ) Here, c 2 is the specific heat of the metal and c 2 m 2 = C 2 is the heat capacity of the mass of metal used. The corresponding quantities for water are c 1, and c 1 m 1 = C 1. 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

2 Q = C dt or C = Physical Chemistry, 4 th Ed. Walter J. Moore So, in general, measured under a given constraint x, the Heat Capacity of a substance is defined as: C x = We have already specified two such constraints, constant volume and constant pressure, giving us the heat capacities C v and C p, 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. C v = = C p = = Since C v and C p 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 - P op dv = Q - P dv and du = dt + dv = C v dt + dv Equating these expressions, we have: Q - P dv = C v dt + dv Dividing by dt and restricting ourselves to constant P:

3 - P = C v + Recognizing the lead term is C p gives us: C p - P = C v + Rearranging a bit gives us the desired result: C p - C v = In general we find that C p > C v. 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 also 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, C p - C v > 0. Work to Surroundings Heat Inc Chaotic Motion Heat "Pull" Molecules Apart (Temperature) Inc Chaotic Motion (Temperature)

4 In our above expression for C p - C v, 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: C p - C v = = [P + 0] = R (You can only use C p - C v = R for an Ideal Gas.) Typically, for liquids and solids we find: C p C v This is illustrated by the heat capacity data for Water. C p (J/K mol) C v (J/K mol)

5 The Molecular Basis for the Heat Capacity We now do something we are typically loathe to do in a course on thermodynamics; admit that our systems are made up of molecules and atoms. However, in the case of our discussion of heat capacity, this admission is useful as it will allow us to understand why certain substances have the heat capacity values they do and draw attention to some useful generalization. However, we do not want to get carried away here and will only cite only a few generalizations. These limited generalizations will however be useful in our further discussion of thermodynamic potentials. Ideal Monatomic Gases Our first case involves the heat capacity of monatomic gases near Room Temperature. Here the results of the Kinetic Molecular Theory of Gases are quite useful. In brief, the Kinetic Molecular Theory postulates that: A gas contains a very large number of essentially volume-less "particles". The gas "particles" obey Newton's First Law of Motion. The gas "particles" do not interact with each other except for the brier period during which they collide with one another, and these collisions are completely elastic. A major conclusion of the Kinetic Molecular Theory is that the gas will obey the Ideal Gas equation of state. Another major conclusion concerns the relationship between the internal energy of the gas of its temperature: U = NRT where N is the number of moles of the gas present. In terms of molar quantities, this means: Notice that the Kinetic Molecular Theory has allowed us to determine that which we said could not be determined, namely the internal energy U. The Kinetic Molecular Theory uses this basic model of the gas and certain statistical methods in order to come to this result. The details of the derivation is well beyond the scope of this course. For our discussion here, this key result allows us to determine the constant volume heat capacity of a monatomic gas: Typically, it is that is actually measured, so:

6 since the gas is Ideal. Numerically, this reduces to" = (8.314 J/K mol) (1 cal / J) = 4.96 cal/k mol This result compares favorably with experiment. Gas (cal/k mol) He 4.98 Ar 5.00 Now to push our luck. In the above model, only translational motion need be considered; the gas "particles" are internally structure-less. We note that each particle has three degrees of translational motion available to it: So, of the RT of internal energy adduced by the Kinetic Molecular Theory, we can ascribe RT to each mode of translational motion. This observation is pushed further by what is known as the Equipartition Principle: In equilibrium there is associated with each degree of freedom an average energy of k B T per molecule, where k B is the Boltzmann Constant. On a per mole basis, this means that for each degree of freedom possessed by the gas "particle", we have: Ideal Diatomic Gas So, let's apply the Equipartition Principle to a diatomic gas "particle".

7 It is reasonable to assume that a diatomic gas molecule will be able to rotate with two degrees of freedom. Thus, its heat capacity can be calculated according to: (trans) (rotat) = 6.95 cal/k mol This result also compares favorably with experimental data: Gas (cal/k mol) N O CO 6.95 HCl 7.08 But this gives rise to a major problem. The diatomic molecule is assumed to not only possess rotational degrees of freedom, but should also possess a vibrational degree of freedom. According to the Equipartition Principle this should add another R to the heat capacity. (As the molecule's atoms oscillate back-and-forth, they juggle energy between vibrational Kinetic Energy and vibrational Potential Energy. So, the vibrational degree of freedom should contribute: (vib KE) (vib PE)

8 to the heat capacity.) However, this additional R contribution is not observed experimentally. Where did the contribution to the heat capacity from the vibrational degree of freedom go. This was a major problem in pre-quantum physics. What quantum physics tells us is that the vibrational energy is not accessed continuously. If we treat the "spring" between the atoms as Hookean, then the classical energy profile is parabolic. However, in the quantum world, the energy is actually quantized, as pictured below. Because the spacing between energy levels is relatively large, collisions between molecules at Room Temperature are not sufficiently energetic as to excite molecules vibrationally. This means that at Room Temperature, the vibrational degree of freedom is not assessable. Hence the additional R contribution to the heat capacity will not be observed. (It should be noted that the translational and rotational energies are also quantized. However, the spacing between the energy levels in both cases is extremely small and so these degrees of freedom are assessable at Room Temperature.) At much higher temperatures, however, the vibrational energy levels become assessable and this degree of freedom will begin to manifest itself. The temperature profile for the vibrational contribution to is provided below.

9 A Textbook of Physical Chemsitry Arthur W. Adamson vib is the Characteristic Temperature for the vibrational degree of freedom and marks the transition from no virational contribution to to that of a full contribution. It can be measured spectrocopically. Characteristic temperatures for select diatomic molecules are provided below. Note that most characteristic temperatures are well above Room Temperature. Gas vib (K) N O CO 3080 NO 2700 HCl 4150 HBr 3680 Cl Br I A short note concerning the number ofdegrees of freedom for a molecule. In general, a molecule containing n atoms will have 3 degrees of translational freedom per atom. However, typically, we consider the translational motion of the molecule to be concerted and view things from the point of view of its center of mass. Therefore the molecule will have 3 degrees of translational motion associated with its center of mass. As for its rotational motion about the center of mass, if, as above, the molecule is linear, there will be 2. If the molecule is non-linear, then 3. This leaves us with the following for numbers of vibrational degrees of freedom: # degrees = 3n - 5 (if linear)

10 # degrees = 3n -6 (if non-linear) Finally, a simple rule of thumb is that the heat capacity will have some temperature dependence. So, it is typical to take the constant pressure heat capacity as having the following functional form: = A + BT + CT 2 where A, B and C are empirically determined constants for a particular sample. A Textbook of Physical Chemsitry Arthur W. Adamson Solids A rudimentary ball-and-spring model of a simple solid is as pictured.

11 Physics Paul A. Tipler In this model, each atom possess 3 translational degrees of freedom and 3 virational degrees of freedom. This means: (trans) (vibrat) Or, 5.96 cal/k mol Again, the vibrational degrees of freedom are accessed only at higher temperatures. Nonetheless, a value of 6.2 cal/k mol was determined experimentally by Dulong and Petit, and is codified by the Law of Dulong and Peitit. Pierre Louis Dulong Alexis Petit The relationship between [heat capacity] and atomic weight, known as the law of Petit and Dulong, was reported by its discoverers to the French Academy in the spring of 1819 and was published soon thereafter. These two investigators noticed that in the case of a number of solid elements the product of specific heat [read heat capacity on a per gram basis] and atomic weight was a constant.

12 [cal/g o C x g/mol = cal/mol o C. Hence the heat capacity, cal/mol o C, is a constant.] The law actually proved to be a fair approximation despite the uncertainty regarding atomic weights and the rather arbitrary assumptions made by the authors. Its promulgation represents an act of faith rather than being a sound scientific generalization, but this is not the only time this has occurred in science. The Development of Modern Chemistry Aaron J. Ihde The following graph shows that for Pb, Ag and Cu, the Dulong-Petit value is obtained at around 200K. It was Peter Debye that showed temperatures: has the following functional form at extremely low This result is known as Debye's Law and is found to hold true for a number of solids below about 15K.

13 Peter Debye