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Elements of Chemical Thermodynamics
By Leonard K. Nash Dover Publications, Inc.
Copyright © 1970 Leonard K. Nash
All rights reserved.
ISBN: 978-0-486-15184-7
CHAPTER 1
Introduction
Thermodynamics is a notably abstract science with innumerable concrete applications. It unites extreme generality with extreme reliability to a degree unsurpassed by any other science. The magnificent generality of thermodynamics arises from the abstractness of its fundamental concepts: stripped insofar as possible of everything referring to specific things and events, such concepts are fitted for the interpretation of relations of the most diverse varieties. The extraordinary reliability of thermodynamics arises because, while calling on only a minimal array of axiomatic postulates, it cunningly contrives to discuss material phenomena without making any assumption whatever about the constitution (atomic or otherwise) of matter.
Thermodynamics treats of systems—parts of the world that definite boundaries separate conceptually (and often, with a good degree of approximation, physically) from the rest of the world. The condition or state of such a system is regarded as thermodynamically defined as soon as values have been established for a small set of measurable parameters. These parameters are so chosen that any particular state of the system will be fully and accurately reproduced whenever the defining parameters take on the set of values descriptive of that state. Temperature, pressure, volume, and expressions of concentration (e.g., mole fractions) are the parameters most used by chemists, and temperature in particular is distinctive of thermodynamic analyses generally. But not every state of every system can be characterized by a single well-defined temperature (and pressure), and from this circumstance arises two major restrictions on the applicability of classical thermodynamics. First, given the (Max-wellian) distribution of molecular velocities, a single molecule, or even a small group of molecules, does not have a definite temperature. Only to macroscopic systems will a temperature be assignable, and only to such systems will thermodynamics be applicable. Second, even a macroscopic system will manifest local inhomogeneities of temperature (and pressure) while it is undergoing rapid change. An entire macroscopic system will be characterizable by a unique temperature (and pressure) only when that system stands in an unchanging state of equilibrium—or in a quasi-static state only infinitesimally different from a true equilibrium state. And it is with such states alone that classical thermodynamics concerns itself.
Focused on the equilibrium states of macroscopic systems, classical thermodynamics tells us nothing about the paths by which different states may be connected, and nothing about the rates at which the paths may be traversed and the states attained. Thermodynamics is then, if you please, rather more the science of the possible in principle than the science of the attainable in practice. What thermodynamics finds impossible we cannot hope to achieve, and we are spared the investment of effort in a vain endeavor. But to achieve in practice what thermodynamics finds possible in principle may still require an immense endeavor. For example, thermodynamic calculations show that, under pressures of 30,000-100,000 atmospheres, diamond should be formed from graphite at temperatures of 1000-3000°K. However attempts to achieve this conversion were for long totally unsuccessful. Having confidence in thermodynamics, one attributes these failures to slowness of the reaction, and one perseveres in the endeavor to achieve in practice what thermodynamics indicates as possible in principle. And in 1954 the synthesis of diamond was at last achieved—through the discovery of an effective catalyst, and the construction of equipment competent to maintain the required conditions for hours rather than seconds.
TEMPERATURE
The theoretical term "temperature" first acquires empirical relevance only when we learn how temperature is to be determined. Let it now be specified that temperature is to be measured with a constant-volume gas thermometer, shown schematically in Fig. 1. When the gas in the sealed bulb has come to equilibrium at the temperature (T) prevailing in the enclosure, by appropriate manipulation of the leveling bulb, we bring the mercury meniscus always to the indicated fiducial mark. Having thus assured constancy of the gas volume (V), we obtain the pressure (P) by measurement of the height h. To establish what we mean by temperature, we have then only to write, for the measurements made at any two different temperatures:
T2/T1 [equivalent] [P2/P1]v,
where the subscript v underlines the constant-volume requirement. The temperature ratios determined from this formula prove unhappily variable with the identity and pressure of the gas present in the bulb. But whatever the gas used, this variability disappears when we repeatedly determine the pressure ratio with progressively diminished amounts of gas in the bulb, and then extrapolate to the limit P -> 0. That is, with any pure gaseous species we obtain the same temperature ratio from the limiting pressure ratio:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
To pass from the temperature ratios so defined to numerical values for individual temperatures, we need only adopt a convention that assigns some particular numerical magnitude to the temperature of some one standard reference point. By international agreement, we assign 273.16 as the temperature at the "triple point" at which ice, water, and water vapor coexist in equilibrium. Letting the subscript tp signify the triple point, and dropping all other subscripts, we find that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
This equation expresses the so-called "ideal-gas temperature scale" and, anticipating a conclusion reached on p. 67, we shall henceforth distinguish readings on this scale as °K, e.g., 273.16°K.
In convenience, other thermometric devices far surpass the constant-volume gas thermometer. And in the extremes of very high and very low temperatures, the use of these other devices becomes a matter not merely of convenience but of necessity. Ultimately, however, all such alternative devices are calibrated against (or at least indirectly referred to) the gas thermometer—which thus establishes the empirical meaning of the concept "temperature."
HEAT AND WORK
In much the same period that the equivalence of heat and work was first established, the limited efficiency of steam engines posed a problem from which developed the science of thermodynamics. Today we no longer conceive heat in the original "thermo-dynamic" sense, as a caloric fluid that "flows." Yet that conception is commemorated in a familiar unit—the calorie—which represents approximately the quantity of heat required to warm 1 gm of water 1°K. Note however that the fundamental unit of both heat and work is the joule (= 1 newton-meter = 1 watt-second = 1 volt-coulomb), and the calorie is today a derived unit defined as 4.184 joules. Because on the molar scale most chemical changes involve transfers of a great many calories, the unit of heat most used by chemists is neither the calorie nor the joule but the kilocalorie (= 1000 calories = 4184 joules).
Heat and work are different modes of energy transfer between a system and its surroundings. To indicate the direction of transfer, we use a sign convention that derives from the original thermodynamic concern with systems constituted by heat engines delivering a work output when supplied with a heat input. Thus work delivered by the system is called positive, and work supplied to the system is then appropriately negative— diminishing the net production of work. Heat supplied to the system is called positive, and heat given up by the system is then appropriately negative—since every heat loss means a smaller production of work.
Defining heat as energy transferred by thermal conduction and radiation, we may think to have distinguished it clearly from work, defined as energy transferred by other mechanisms. But such purely verbal definitions are wholly insufficient. We do better to proceed as we did with "temperature," by seeking for heat and work some simple operational definitions which, though incomplete, amply identify the empirical bearing of the abstract concepts.
For "heat" our definition takes as its cornerstone the so-called ice calorimeter shown schematically in Fig. 2. This conceptually simple but remarkably accurate device is primarily intended only for heat measurements at 0°C (273°K), but any of the manifold other ways of measuring heat can in principle be referred to the paradigmatic case constituted by the ice calorimeter.
Imagine chamber M completely filled with a mixture of ice and water, and the stopper pressed home until the water meniscus stands within the graduated region of the capillary tube C. The same ice-water mixture is used to fill the outer bath B: by thus enforcing temperature equality, we ensure that there will be no heat transfer between M and the surrounding bath B. But if the temperature reigning in the inner chamber R deviates even minutely from 0°C, heat will be transferred through the rigid but thermally conductive wall between R and M. If, for example, we bring about in R an exothermic chemical reaction, the consequent transfer of heat will melt some part of the ice in the mixture that fills M. What will be the effect? The resultant mixture will of course differ not at all in temperature from the 0°C of the original mixture. However, due to the fact that at 0°C ice is substantially less dense than water (recall: ice floats on water), any melting of ice will be accompanied by a decrease in the volume of the mixture. Thus the transfer of heat from R to M will be reflected in a measurable subsidence of the water meniscus in C.
Now even as "pressure" is expressible in terms of cm Hg, just so a quantity of "heat" could easily be expressed in terms of the cm3 of volume change produced by its transfer to or from an ice calorimeter. But actually we much prefer the joule and the calorie as units for heat, and a single additional determination suffices to convert all ice-calorimeter measurements to this basis. We have only to set up the ice calorimeter afresh, placing in R a small electrical resistor through which we pass a known current for a known time. We are then in a position to calculate the number of joules (or calories) of electrically generated heat transferred from R to M, and to read off from C the volume change produced by that transfer. And so we establish once and for all the conversion factor that permits us to pass from an observed volume change to a statement that some particular number of joules represents the "heat" transferred to or from the calorimeter. The great simplicity of the basic calculation should be evident from the following three examples.
* Example 1
Into an ice calorimeter was inserted a 10.00 ohm resistance through which a current of 1.000 ampere was passed for 4.00 minutes. The consequent decrease in volume was measured as 0.653 ml. Determine the conversion factor:
α = Number of joules of heart transferred/Number of ml volume change observed.
Solution. The formula for the electrically generated heat (in joules) is g2 g2 Rt, where g is the amperage, R the ohmage, and t the number of seconds during which the current passes. We have then:
Heat = (1.000)2(10.00)(240) = 2400 joules.
Therefore,
α = 2400 joules/0.653 ml = 3.68 × 103 joules/ml.
* Example 2
Instead of making an experiment, as above, one may prefer to calculate the conversion factor from highly accurate available data. Given 0.91671 gm/ml and 0.99987 gm/ml as the densities at 0°C of ice and water respectively, and 79.72 cal/gm as the heat of fusion of ice, calculate the conversion factor:
α' = Number of calories of heat/Number of ml volume change.
Solution. What will be the volume change when 1 gm of ice is melted? This we can most easily find by converting the given densities to the corresponding specific volumes, i.e., the volumes occupied by one gram of each material.
Spec. vol. ice = 1/density ice = 1/0.91671 gm/ml = 1.09086 ml/gm Spec. vol. water = 1/dens. water = 1/0.99987 gm/ml = 1.00013 ml/gm [??] Volume change per gm of ice melted = 0.09073 ml/gm
But this is the volume change that results whenever the ice calorimeter receives the 79.72 cal required to melt one gm of ice. Therefore,
α' = 79.72 cal/gm/0.09073 ml/gm = 878.7 cal/ml.
Is the figure obtained in the first example consistent with this more accurate result?
* Example 3
Just 0.01 mole of a compound was dissolved at 0°C in 10 ml of water in chamber R of an ice calorimeter. The consequent change in the volume reading shown on tube C was an increase of 0.0284 ml. Determine the direction and magnitude of the heat transfer when 1 mole of the compound is dissolved at 0°C in 1 liter of water.
Solution. Using the factor calculated in the last example, we see that the heat transfer in the actual calorimetric experiment was
0.0284 ml X 878.7 cal/ml = 25.0 cal.
Observe that the volume increased: this means that in the calorimeter ice was formed at the expense of the less voluminous water. Thus the direction of heat transfer was from the calorimeter to the solution in R. If the solution represents our "system," the heat transferred in this endothermic process will carry a positive sign. Hence the heat of solution of 1 mole of compound in 1 liter of water will be
+25.0 × 100 = +2.50 × 103 cal/mole. *
Heat is thus seen to be readily measurable, and the indicated style of measurement is itself a definition of what we mean by "heat." Turning then to "work," we can be much briefer. The only species of work of major concern to us is the familiar mechanical work of which we write:
Work (joules) = Force (newtons) x Distance. (meters)
Toward the end of this text we touch briefly upon electrical work, for which we write:
Work (joules) = Electromotive force (volts) × Charge trnasported. (coulombs)
Now both kinds of work can be used to lift weights—with 100% efficiency in principle, i.e., using frictionless pulleys, ideal electric motors, etc. We can measure (and to that extent define) "work" in terms of this effect.
We shall say that work has been done by a system, on its surroundings, if weights are observed to rise in the surroundings as a result of interaction with the system. And we shall say that work has been done on a system, by its surroundings, if weights are observed to descend in the surroundings as a result of interaction with the system. From the observed rise or fall, the work transfer can at once be calculated. Suppose that, in a field where the gravitational acceleration is g, a particular mass M is observed initially at height hi and finally at height h;. Then:
Work (joules) = Vertical force × Vertical distance = Mass x Gravitational acceleration × (hf - hi) = M (kgm) × g (meters/sec2) × Δh (meters)
Observe that in [DELTA]h(= hf – hi) the operator Δ is used as it always will be—demanding that we form the difference [final value of indicated parameter minus the initial value thereof]. The sign convention for work given on p. 4 is then taken care of automatically. For when the weight rises Δh will be positive and, indeed, to make the weight rise, work must have been done by the system. And when the weight descends Δh will be negative and, indeed, through the descent of the weight, work will have been done on the system. Thus a value for work (w) correct in both sign and magnitude is given by the simple relation:
w = Mg Δh. (a)
Ample illustration of work calculations will be found in the next section, which introduces one of the key concepts of thermodynamics.
REVERSIBILITY
The amount of work extractable from even the simplest process of change depends markedly on just how that process is conducted. This point we highlight by subjecting the set-up sketched in Fig. 3 to a style of analysis first suggested by Eberhardt.
Let our system be a long spring, suspended from a hook at its upper end and bearing a light pan at its lower end. Let the surroundings be all the rest of the world, and most particularly a group of weights totaling 1600 gm. As state I of the system we have the spring stretched to the length at which it supports 1600 gm in the pan, as shown. As state II of the system we have the spring relaxed to the length at which it supports the empty pan. Assuming the spring ideal in its conformity to Hooke's law—and many real springs closely approach the ideal—the elongation of the spring will be directly proportional to the load it sustains. In that case, if the elongation produced by 1600 gm is the 81.6 cm shown at the extreme left, the elongation produced by 800 gm will be 40.8 cm; that produced by 400 gm will be 20.4 cm; and so on. We then construct a set of nine shelves at the heights shown at the extreme right, measured from the level of the pan when, with a 1600-gm load, the system stands in state I. And now the question: how much work can be obtained from this system when it passes from state I to state II?
Possibility (a). Temporarily securing the pan, we slide all 1600 gm horizontally onto the bottom shelf. On then releasing the pan, we see the system pass through a series of progressively less violent oscillations until at last it settles down in state II. Now all the weights remain at precisely their original level, so that no work whatever has been accomplished. Thus we write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where the arrow signifies the forward change from state I to state II, and the subscript indicates the first of a number of possible ways of conducting that change.
(Continues...)
Excerpted from Elements of Chemical Thermodynamics by Leonard K. Nash. Copyright © 1970 Leonard K. Nash. Excerpted by permission of Dover Publications, Inc..
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