J8 Heat

Specific Heat of substances. Calorimetry

Specific heat (Black 1760)

The first question is now: How does the temperature of a substance increase when it depend son its heat intake? How much heat is required in order to raise its temperature by a given number of degrees?

We employ as a heat unit, by which we measure quantities of heat, that amount which raises the temperature of 1 gram distilled water from 14.5ºC by 1ºC. It is called a calorie (cal) - gram calorie or small calorie; 1 000 cal are called a kilogram calorie (kcal). The next question is: Do you need the same amount (1 cal) in order to raise the temperature of one gram of another substance by 1ºC? No! For mercury, you use about 1/30 calorie (exactly 0.33) cal, for copper 0.091 cal, for iron 0.111 cal, for alcohol 0.58 cal, etc. or in other words: The quantity of heat, which is just enough to raise the temperature of 1 g of water by 1ºC, would suffice to do so for 30.3 g of mercury or 11.0 g of copper or 32.3 g of lead or 0.02 g of iron or 1.72 g of alcohol, etc. We see that the amount of heat, which a substance requires for a given increase in its temperature, does not depend on the quantity of its mass, but on its chemical character; you call the amount of heat, required to raise 1 gram of a substance by 1ºC, its specific heat.

Specific heats of various substances, strictly, their mean values for the temperatures between 0ºC and 100ºC, are:

Of all liquid and solid substances, water (with 1) has the largest specific heat, that of ice is 0.505, of water steam 0.462. Thus, water as steam and as liquid has almost the same specific heat, and indeed it is half as large as in its liquid state.

Certain of the relations between the specific heats of various substances become clearer, when you relate, instead of to the same mass (1 g), the quantity. required to cause a temperature increase of 1ºC, to equally many atoms or to equally many molecules. You relate it to the gram atom or the gram molecule and call it the atomic heat and molecular heat; it is equal to the product of the atomic or molecular weight and the specific heat, related to 1 gram. However, this definition of specific heat does not take into account the change of volume, associated with the intake of heat. If the external pressure remains unchanged, then one part of the heat intake is used to increase the volume of the body and only the remainder to raise the temperature, which is the true specific heat. Also that first part would be measured as specific heat, if as a consequence of a sufficiently high rise in the external pressure the volume were to remain constant. You call the difference of the specific heat and the volume change heat the specific heat at constant volume - C_v - and the really measured value the specific heat at constant pressure - C_p. For solids and fluids, the difference is without consequence due to their small change in volume, but for gases it is so large ( in the atomic and molecular heat about 2 cal), because the external work is large due to the large change of volume.

The amount of heat, by which c_p is larger than c_v, in mechanical measure, equals the work performed by the gas as it expands and overcomes the pressure. We will perform this calculation for 1 mol: In order to heat 1 mol of molecular weight M, that is, Mg gas, by 1 º, once at constant pressure, another time at constant volume, you require the first time Mc_p = C_p, the second time Mc_p = C_v cal. whence the work of expansion is C_p - C_v cal. We can compute this work in a different manner: let the volume of the mol before heating be v and its temperature 0ºC, that is, T = 273º. If we heat the mol at constant pressure by 1º, it expands by va (where a = 1/273), that is, by v/T. During the expansion, it overcomes the constant pressure p, whence it performs the work pv/T and therefore C_p - C_v = pv/T. However, pv = 1.986·T cal, that is C_p - C_v = 1.986 cal.(cf1, cf2.)

Law of atomic heat (Dulong and Petit)

The atomic heat of the solid elements lie at room temperature, as has been discovered first by Dulong, Alexis Thèrése Petit 1791-1820 1818, in the mean at 6.4 cal/º. The law is not fulfilled strictly. Metalloids with small atomic weights have already at room temperature much smaller atomic heat; there are more exceptions at deeper temperature. Furthest downwards deviate boron, beryllium, carbon and silicon with 2.7 - 3.7 - 3.8; but their atomic heat approaches at rising temperature the value 6.4. For carbon at 980ºC, it is about 5.5, that is, about 7 times as large at at - 50ºC. The law is satisfied best by metals. If you compute the work of the volume change and subtract it from C_p, all C_v would be about 6.0. This means: You must add to each atom for a 1º increase in temperature without change of volume , irrespectively of what substance, always the same amount of heat. For chemical compounds holds the law of Franz Ernst Neumann: The mol heat of a compound equals the sum of the atomic heats of the individual elements. Naturally, it only is valid as far as the law of Dulong and Petit.

Classical theory of atomic heat (Gibbs, Boltzmann)

Intake of heat by a body means: Intake of molecular energy of motion, which distributes itself to the molecules and atoms. However, the heat content of a body does not only comprise kinetic energy. This is even not so for gases (except for those the molecules of which contain only one atom); these can rotate in the molecules as well as oscillate, whence a part of the energy of vibration is always potential energy. In order to raise the vibration energy during a rise in temperature, each substance requires more energy in the form of heat than would correspond to the mere increase in the mean kinetic energy. And more so the solids! Mutual attractive forces hold its atomic structure together. Each atom is thereby held in spite of its thermal motion in the vicinity of a mean location, about which it can only oscillate. Hence a solid body with a uniform temperature distribution has a certain amount of mechanical energy - kinetic and potential - and on the average a certain amount of each of them comes to each atom. However, the laws of Dulong-Petit and Neumann say: This amount of energy, which is available on an average to each atom, grows for all atoms at a given temperature by the same amount during an increase by 1ºC in arbitrary solid elements and compounds. This has led to the conjecture: Not only the change of the atomic energy is the same at a given temperature in all solid bodies and for all atoms, but the atomic energy itself is the same. Now, when heat transits from one well tempered body to another such body, then, according to this concept, in the thermal equilibrium the total heat energy will be distributed proportionally to the number of atoms to both bodies. Boltzmann has justified this result by the Theorem of Equal Partitioning: The total energy of motion of two arbitrary interacting mechanical systems distributes itself in the mean according to the number of their degrees of freedom.

Starting from pV = Nmv²/3, you can show*: The kinetic energy per degree of freedom in a gas of rigid molecules (monatomic) at temperature T is RT/2N, where R is the gas constant and N the number of molecules in the mol. In a solid body, it is twice as large, that is, RT/N (kinetic and potential energy, both assumed to be equal).

* We write: pV = 2N/3·mv²/2 = 2N/3·mean kinetic energy or also 2N/3·mean kinetic energy of 3 degrees of freedom, which by Boltzmann's law is a universal function of the temperature f(T) per degree of freedom. Then mv²/2 = 3f(T) and pV=2Nf(T), that is, both together yield f(T) = RT.2N = kT/2.

With R/N = k, we can write (1/2)·kT and kT. You ascribe a spherical atom, which can move freely in three dimensions, three degrees of freedom, whence it has at the absolute temperature T the energy mean (3/2)·kT. If a gas contains N monatomic spherical atoms, it contains therefore at the temperature T the energy (3/2)·NkT and demands the increase in energy (3/2)·N·k·DT, in order to raise its temperature by DT degrees. A molecule, which consists of two rigidly linked atoms, has five degrees of freedom and at the temperature T in the mean the energy (5/2)·kT. If a mol of a two atom gas, like hydrogen or oxygen, has N molecules of five degrees of freedom each, then the energy which must be spent on heating it by 1º (without it performing external work) is (5/2)·kN, which is the mol heat C_v. However, since kN = R (Gas constant), you have for C_v at five degrees of freedom per molecule (5/2)·R and at n degrees of freedom (n/2)·R. However, the specific heat of a body can only then change, if the number of degrees of freedom changes, but that means: The specific heat can only increase or decrease in jumps of integral multiples of (1/2)·R or must remain constant. In other words, it cannot vary continuously with the temperature. However, observations contradict this conclusion.

In spite of this contradiction, the classical theory has proved itself brilliantly during computation of the specific heats of simple gases, equally during that of monatomic solids, for which besides the kinetic energy also the potential energy has an influence. The mean value of the two energies is assumed to be equal. The total energy is then equal to twice the kinetic energy. Since for monatomic substances with freely moving atoms you have to consider three degrees of freedom, the total energy of the atoms of a solid is 2·(3/2)·RT, that is, the mol heat C_v is 3R = 5.958 cal/degree in close agreement with the rule of Dulong and Petit. In general, the measured mol heats of the solids are somewhat larger, because the theoretical value relates to c_v, the measured one to c_p.

The monatomic gases are of special interest. A knowledge of c_pand c_v gives insight into the structure of gas molecules: The molecules move linearly, collide with each other and the wall, while their velocities and directions change continually. The molecule consists of atoms, which can oscillate within the molecule about the common centre of gravity as well as rotate inside the molecule. Hence the molecules can have beside translation and oscillation energies also rotation energy. We call the energy of translation its external energy K, the oscillatory and rotatory energy its internal energy, their sum the total energy H. How much of the total energy belongs to the external energy? Clausius found K/H = (3/2)(c_p- c_v)c_v. Setting c_p/c_v = k, one has K/H = (3/2)(k - 1). For hydrogen, oxygen, nitrogen, oxide of nitrogen and carbon monoxide, each of which have two atoms in a molecule, k = 1.40, that is, K/H = (3/2)·0.40 = 0.6, whence 60% of their total energy is translation energy. You can only imagine that a molecule lacks internal energy in the case of a monatomic gas. The K = H, that is, 1 = (3/2)(k - 1) or k = 0.66. Apart from antimony and bismuth, this unusual value occurs with the vapours of metals ( as was first discovered by Kundt, Warburg for mercury vapour) and the inert gases argon, helium, neon, etc. (Rayleigh).

The work performed during a change of volume, that is, C_p- C_v = R, is only very approximately confirmed by experiment, indirectly by measurements with argon (Pier), which is monatomic. For monatomic gases, C_p/C_v = 1.66. If really C_p- C_v = R, then C_v ·(1.66-1) = R or C_v·(2/3) = Rr/, that is C_v = 3R/2, and that is really confirmed by measurements. The value also agrees with the demand of Boltzmann's law. For the mol heat C_v of the monatomic gases follows from the equal distribution law for N molecules

C_v = (3/2)·(R/N)·T·N = 3R/2 = 3 cal/ºC,

as argon (2.979) really has from room temperature to 2 350º. This shows at the same time that its atoms really behave like mass points with three degrees of freedom.

As regards biatomic gases, we must distinguish at middle temperatures two groups. The first, which includes Cl₂. Br₂ and J₂ , have C_v = 3R = 5.95 cal/ºC, that is twice the value for monatomic gases, a result that had not been fully explained in 1935. A second group, which includes H₂, O₂ and N₂, has for C_v only the value (5/2)R = 4.96 cal/ºC. The two atoms of the molecule are here possibly linked by extremely strong directional forces, so that they do not at all oscillate with respect to each other at middle temperatures and the oscillations contribute only at very high temperatures to thermal motion and then affect C_v. Indeed, C_v increases for H₂ and N₂, that is, H₂ rises from 5 cal at room temperature to 5.72 at 2 500º and N₂ to 5.92 at 2 550º. The part of the heat, which for polatomic gases originates from the degrees of freedom of rotation, must vanish at sufficiently low temperatures and every polatomic gas must therefore have at lows temperature the mol heat of a monatomic gas, that is C_v=(3/2)·R = 2.979. This theoretical prediction has really been confirmed by measurements of the specific heat of hydrogen (Eucken, 1912):

Hence, according to its specific heat, hydrogen is already monatomic at - 233 ºC ; its specific heat is clearly smaller already at 0 ºC as is demanded by the classical theory of biatomic gases (C_v = (5/2)R = 4.965).

Quantum theory of specific heat (Einstein 1907, Nernst-Lindemann, Debye, Born, von Karmán)

The law of equal distribution and the simplifying assumption that the potential energy equals the kinetic energy yield a foundation for the laws of Dulong-Petit and Neumann. However, only to an explanation of their general character! They leave unexplainable the facts that C_v is much too small for several substances with small atomic weight (especially carbonic acid), that the values of C_v for the other substances are not exactly equal as the theory demands, and, above all, that C_v changes with the temperature (for example, diamond has effectively at -253ºC the specific heat zero) - indeed, that at very low temperatures all bodies behave like carbon, that is that C_v is much too small for them.

All these difficulties disappear on application of the Quantum Theory of Planck to the heat oscillation of solid bodies. The basic concept of this theory is: Before Planck, no one doubted that changes in Nature take place continuously; it was considered a matter of fact, that, for example, a body, the energy content of which drops, passes steadily through all values of the energy between its initial and final values - just like a sphere, which rolls down an inclined plane, passes through all heights between the initial and final height. However, during his examination of radiation, Planck was forced to the assumption, that the (sub-microscopic) carriers of the vibrations can only change their energy content in jumps of always the same magnitude, just as - transferred to the (sub-microscopic) carrier from the oscillation - a sphere jumps down a staircase from step to step or the falling weight of a pendulum clock, which spends its potential energy in jumps, the size of which depends on the subdivisions of the control wheel, or Corti's organ, which lets us sense the glissando of a storm from the lowest to the highest tone steadily, although as a consequence of the gradations of the length of its fibres it is only excited in steps (quantum like or quanted).

Accordingly, Planck starts from the concept: A quantity of energy consists of very small, but finite elements, energy quanta, similarly to how a mass is built out of atoms. If a body accepts energy, it takes in at least one or two or another integer multiple of a quantum. For example, imagine that a sphere is to be lifted up a stair case step by step. If the available energy is too small for lifting it for a whole step, the energy used does not at all manifest itself. Only lifts by entire steps can be registered. If this sphere rolls down the stairs, it loses its energy step by step, each jump corresponding to the height difference between two steps. Also here, a quantity of energy , which is smaller than corresponds to an entire step, does not manifest itself.

This thought experiment also intermediates between the quantum and classical theory: Imagine that the steps become lower and lower and, in order to meet a given total height difference, their number increases inversely to their height; we then arrive at the arbitrarily far continued subdivision of the energy, with which Physics only computed prior to Planck. The step height e - in order to continue the concept - differs from stairs to stairs depending on the frequency n of the vibrating system*. According to Planck, it must be set proportional to it. With the proportionality factor h - Planck's constant - we have h = e ·t, that is, energy x time, an effect - hence Planck's constant is also called elementary effect quantum.

*In this way, we indicate simultaneously that the Quantum Theory for the time being can only be applied to periodic motions like the oscillation of the molecules of a solid body or the rotations of molecules.

Thus, not the energy elements e are are the constant part in the atomic structure of energy, but e·t, the products of the energy element e and the time t during which the energy is converted. Only when n is constant, can one speak of constant elements of energy, as in the atomic oscillations of monochromatic (homogeneous) light , but also in the atomic oscillations in a solid body. In order to gain an appreciation of energy quanta, you must know the constant h. It is know to about 3 decimals: h = 6.55·10^-27 erg sec. In green light (l 5000,n = 6·10¹⁴ ), the energy element takes only 4·10^-12 erg. X-rays have the largest known frequency - 1000 times larger than that of violet light - correspondingly also the largest known energy elements. The lower the frequency, the larger the wave length, the smaller the quanta. During very slow oscillations, they approach the energy of practically arbitrary divisibility, as has been assumed in classical physics.

According to the Quantum Theory, the distribution of energy per degree of freedom differs from the equal distribution law; it depends on the frequency n. In equilibrium. one degree of freedom does not take on the average energy k·T, but only a fraction of it - and indeed one which is the smaller the larger the frequency n and the lower the temperature T, at which equilibrium exists: For large T or small n, the fraction approaches unity, for small T or largen, it becomes vanishingly small. The fraction, by which you must multiply k·T, is (hn/kT)/(e^hn/kT- 1) (we will denote it below by s).

The idea, which took Planck to his radiation law, was applied by Einstein to the heat oscillations of solids. He assumed: Oscillating atoms of solids take in only k·T·s per degree of freedom (in the expectation to be able in this manner to explain the abnormally small specific heats of the elements with small atomic weights; during equal elastic forces, the lighter atom must oscillate faster, that is, lag behind the equal distribution value kT.) According to Quantum Theory, the place of kT is then taken by (hn)/(e^hn/kT- 1). However, if you expand e^hn/kT in powers of hn/kT and neglect already the second order term, you find hn/hnkT, that is, also here is kT the energy per degree of freedom of the solid body. If n is very small or T very large, you may neglect the higher powers of hn/kT in the expansion. For the sake of simplicity, Einstein introduced one single frequency n for each individual element. The heat content of the 3·N-degrees of freedom is then 3N(hn)/(e^hn/kT- 1) and the mol heat C'_n = 3R·(hn/kT)²·(/kT)/(e^hn/kT- 1)² (C_n according to the classical, C'_n according to the quantum theory.)

Hence the atomic heat of a solid substances is not a constant, as is claimed by the Dulong-Petit law, but depends on n/T, whence for a given n, that is, for a given substance, it depends on its temperature T. For T = 0, the atomic heat itself vanishes; its value rises gradually with rising temperature and approaches for high temperature asymptotically the classical value 3R, whence it is a limit law, which is only valid for small values of hn/kT, that is, for slow atomic oscillations or high temperatures. The curve E in Fig. 392 shows this shape of C'_n . The ordinate is C'_n /3R, the abscissa kT/hn, that is, T/q, where q = hn/k. q = N/R·hn, where N is the Loschmidt number and R the gas constant, has the dimension of a temperature, changes only with the characteristic period n, that is, it only differs from element to element. The shape of C'_n is the same for all elements the same, if it is presented as a function of T/q, that is, T is measured in fractions of q. It is called the characteristic temperature, responsible for the energetic behaviour of a body. Fig. 392 informs: At very low temperatures (low compared with the characteristic temperature), the atomic heat sinks far below the value of Dulong-Petit, at high temperatures it approaches it steadily. For example, if the temperature is half the characteristic one (T/q = 0.5), then the atomic heat is 0.7 of the value of Dulong-Petit, that is about 4 cal/ºC

We arrive at the result: The Classical Theory demands C_n = 3R. The Quantum Theory yields Einstein's value C'_n . Both theories yield the same value, if n/T is very small, that is, for small periods of high temperatures. In contrast, if n/T is very large, the results of the two theories differ essentially from each other, for in the limiting case, if n/T is infinite, one has C'_n = 0 (Fig. 392), it must decrease with a falling temperature, if n remains constant.

The fact that carbon has at ordinary temperatures abnormally low atomic heat is explained according to the above in that its n and therefore also its q is especially large. At a sufficient rise in temperature, its atomic heat should approach the Dulong-Petit value 3R = 6 cal/ºC. And indeed, that is what it does. The atomic heat of diamond is at 222º absolute only 0.76/ºC, at intermediate temperatures about 1.7 cal/ºC, however, at 1 258ºC, it is 5.5 cal/ºC (H.F.Weber).

We can now understand, why the degrees of freedom of those axes of rotation contribute, about which rotation can take place without the position of the molecule changing in place.The moment of inertia is then very small and hence the corresponding period, which is proportional to the square root of the moment of inertia, is very small, that is, the corresponding period n very large. In the case that the temperature is not extremely high, also T/n is is small and therefore, according to Quantum Theory, the part of the specific heat, which belongs to that rotation, is very small, whence, in the first approximation, this contribution can be neglected.

A special support for the quantum theory of the specific heat is the mol heat of the diamond (crystallized carbon), the proverbial exception to the Dulong-Petit rule. The mol heat of diamond is at room temperature 1.4, at - 183ºC only 0.03 and at -230ºC effectively 0 (Nernst-Lindemann). For all other solid substances, this decrease of the specific heat manifests itself only at much lower temperatures. The characteristic temperature of diamond lies especially high at 1860ºC; for example, it is for aluminium at 400ºC, for silver at 210ºC, for lead at 90ºC.

The fraction h/k is (6.55·10^-27)/1,37·10^-16) = 4.78·10^-11, so that n_diamond = 3.9·10¹³ and n_lead = 1.9·10¹². (The period of the shortest wave of light is about 9·10¹⁴, of the longest heat wave 5·10¹².

The characteristic period n depends essentially on the forces between the atoms; it grows, when these grow, that is, when the mutual bond becomes firmer. This leads to a relationship between n and the melting temperature of a substance. Apparently, atoms escape the easier from the mutual range of attraction, that is, transit more easily from the solid into the liquid state, the less solidly they are tied to each other. This concept (Lindemann 1910) yielded n = 2.8·10¹²(T_s/a·V^2/3)^1/2 as an approximate value, where T_s is the (absolute) melting temperature of the substance, a its atomic weight and V its atomic volume. According to this, diamond (also boron and silicium) has by its specific heat a special position among the elements, because it has with a relatively small atomic weight and high density an especially high melting point.

However, it is insufficient to consider solely one period. The force, which causes an atom to oscillate, arises from the neighbouring atoms, which also oscillate. The oscillations of an atom therefore arise by a superposition of a multitude of oscillations. This multitude - the spectrum - of oscillations is taken into account in Debye's theory. (It arrives at a formula, which agrees almost with one, derived by Nernst and Lindemann on the basis of totally different considerations.) Debye considers the solid body as a continuum in the sense of the theory of elasticity and treats heat motion as a superposition of all possible elastic eigen vibrations. However, with a view to atomic structure, he extends the elastic spectrum upwards only so far that one gram atom receives 3N eigen vibrations. In this way, he also arrives at a characteristic temperature q_m. It is especially important that n_m and q_m can be computed from the elastic and density properties. For Al, Cu, Ag and Pb, the elastic constants, computed from the atomic heat, and the ordinary elastic constants agree very well with each other; the concept that heat oscillations and elastic oscillations are identical is thereby justified. For very low temperatures, the atomic heat is directly proportional to the third power of T/q_m; in agreement with experiment, it decreases at low temperatures more slowly than according to Einstein's theory.

The space lattice theory of specific heat of Born and v. Kármán develops - in contrast to Debye's continuum theory - a strict atomistic theory of solid bodies and computes from the space lattice model, on the one hand, the elastic behaviour, on the other hand - as superposition of the eigen oscillations of this point lattice - the thermal behaviour. The final formula is similar to that of Nernst-Lindemann. Special significance attaches for several substances (rock salt, sylvite, fluorite) to the clearly emanating relationship between optical and thermal behaviour. Also, this general lattice theory arrives at the proportionality of the atomic heat to the third power of the absolute temperature at lowest temperatures.

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