Conversion of work into heat and of heat into work
The first main theorem, Impossibility of perpetuum mobile
The first main theorem of Thermodynamics comprises experienced facts relating to the magnitude of converted energy - conversion of mechanical work into heat or of conversion of heat into other forms of energy. It can be given different forms, for example, as follows: A given amount of mechanical work yields, converted into heat, always the same amount of mechanical work. A given magnitude of work is thus always equivalent to the same amount of heat and vice versa.
Once the units in which energies are measured have been defined, the relationships between heat and work or between heat and another form of energy can be formulated . The units for mechanical energy are erg and mkg*, also 107 erg - the Joule, the Watt second. The unit for heat is the calorie. The number of units of mechanical energy, which are equivalent to a calorie, is called the mechanical heat equivalent.
How large is the amount of heat, generated by 1 mkg* work? One of the earliest measuring techniques by Joule 1847/1879 employed generated heat to raise the temperature of quantities of water, measured in kg, by a measurable number of degrees. The work is obtained directly in mkg*.
Water is in a cylindrical vessel A (Fig. 356); in the water is a shaft a with vanes b, which are rotated by a hanging weight G as shown. In order that the vanes do not push the water ahead of them, but work through it with as much friction as possible, there are radial separating walls with openings through which the vanes pass with little margin. The weight G falls and turns the shaft. When it has fallen as far as the string admits, it has performed all of the work, which it can perform; it can work again, once it has been rewound to the top. In this way, arbitrarily much work can be transmitted to the vanes and the water.
The friction between the vanes and the water generates heat and hence raises the water's temperature, shown by a thermometer. (Account is taken of the heating up of the equipment and the outwards loss of heat during computation of the generated heat.) The falling weight in kg* times the height of fall in m yields the work spent on turning the vanes in mkg*. When 427 mkg* work has been performed, 1 kcal has been generated, that is, the work of 427 kg*, transferred to the water, caused the same as the input of 1 kcal heat would have caused, whence 427 mkg* are equivalent to 1 kcal.
This number, which determines the work in terms of mkg* equivalent to 1 kcal of heat, is called the mechanical equivalent of heat; its has been determined over and over again by many methods (Fig. 357). For example, Joule determined it also in 1843 from the work required to compress air in a container to a certain pressure and from the accompanying increase in temperature of the air; Mayer 1842 found it from the expansion of a heated gas: His was the first method used for calculations; Quintus Icilius 1824-1885 1857 employed the heat of an electric current. Whatever approach was used, people obtained always approximately the same value. In 1935, the most probable value was 426.9 mkg* (Jaeger and Steinwehr, "Electric Heating", 1915).
Since 1 kcal = 426.9 mkg* and 1 mkg* = 9.81·107 erg = 9.81 Joule, then 1 cal = 4.186·107 erg = 4.185 Joule and, moreover, 107 erg = 0.239. We found earlier for the gas constant the value R = 8.313·107 erg/degree per degree. If we replace the mechanical measure of work by the measure of heat, we find
R = 8.313·0.239cal/degree = 1.98 cal/degree.
Everybody knows that heat can generate work,even if it is only striking a match on a rough surface, during which the igniting part of the match reaches its inflammation temperature. During every mechanical work like turning, drilling, filing, the machined and machining parts heat up. Every moving part of a machine gets warm in its bearings and at other locations where friction occurs; heat, generated in such a manner, is useless work, whence you try to limit it by greasing moving contacting surfaces.
It is also easily verified that one can convert work into heat. Moreover, the disappearance of mechanical work as it generates heat is demonstrated, for example, by the fall of the weight in Joule's experiments. The weight, which has fallen to its lowest position, has no more of work stored. However, what about conversion of heat into work? The steam engine proves that heat can perform work. But does it disappear in the process? The engineer Hirn of Kolmar 1815-1890 1865 has shown that steam, which flows directly into the condenser of a steam engine, yields more heat than if it moves, to start with under otherwise equal conditions, the piston of the steam engine, that is, has performed work. Moreover, he has measured with a test person in a completely closed room the quantity of oxygen used and the heat generated. At perfect rest, this person generated 5.2 kcal per gram of used oxygen. If it pedalled a bicycle like contraption, that is, performed work which consumed one part of the oxidation heat, this number decreased to 2 kcal.
The equivalence of heat and work was first claimed by the Heilbronn doctor Mayer in 1842. He had observed in Java (Indonesia, during blood letting) that venous blood has there a light red colour, while, due to a considerable content of carbolic acid, it is dark red in middle latitudes. It was known that animal heat (like that of combustion) arises during the oxidation of nourishment and in the process carbonic acid is generated. Starting from this insight, Mayer gave his observation the explanation: In order to maintain its temperature at a constant level in a hot climate where less heat is lost, the human body demands less heat, whence it develops less carbonic acid. Mayer also suggested that the heat of the oxydization of nourishment is converted into mechanical energy*. Independently of Mayer, Helmholtz 1847 has proposed equivalence of heat and work in a quite general form and formulated it mathematically in his paper: "About the conservation of force". The equivalence of heat and work, which is present everywhere in Nature independently of human insights and particular forms of work, proves their internal relationship and justifies the concept, that both of them are only different forms of the same something - energy.
* This is the reason for evaluating the nourishing value of food in calories. The utilisation of the calories, supplied to an organism, depends on the digestion of the food and other metabolic processes. In general, carbohydrates (sugar, starch), fats, albuminous bodies are the three main nutritive substances of nourishment. The conversion heat (heat tone, cf. below) per 1 g of cane sugar is 4.0 kcal, of fat 9.3 kcal, of casein 5.9 kcal. All food can be a source of energy for the work of the muscles, but carbo-hydrates are preferred. The daily quantity of food (Karl Voit 1831-1908 1875) for an adult of weight 67 kg with medium working effort (carpenter, bricklayer) is: 118 g protein, 56 g fat, 500 carbohydrate = 3055 kcal. - The heart uses about 150 of these 3000 kcal, that is about 5% (Höber, "Treatise of the Physiology of the Human", 1920). In fact, it performs at 70 pulse beats per minute in 24 hours about 21 000 mkg*, the equivalent of about 50 kcal. In actual fact, the muscle of the heart needs about three times as much, since it, like a caloric machine, can only convert about 1/3 of the consumed energy into work. The performance of the heart (mkg*.sec) is about 1/300 PS.
Another form of the first main theorem says: The total content of energy U of a body is uniquely determined by the state of the body. It is the energy, which for example the body would yield in the form of heat, if one could reduce its temperature to absolute Zero and make the pressure zero. Depending on its state, a body has another energy constant, for example, 1 g of steam at 100º has a different one to 1g of water at 100º. However, the state of a body is not already characterized by its aggregate state (gaseous, fluid. solid); you must know, in addition, its temperature, pressure and density.
State of a body. Equation of state
The state of a body - of a homogeneous, isotropic body of given chemical constitution, at uniform temperature and uniform internal pressure - is determined by its mass m, its volume v and its temperature t. In this state, all its other properties depend on m, v and t. In particular, its pressure depends, apart from on the temperature T, only on m/v - its density - in other words: On v/m, the volume of its mass unit, its specific volume. The function p = f(v,t) is called the equation of state of the substance. It resolves all questions regarding its behaviour in the face of arbitrary changes of temperature, volume and pressure!
The equation of state for the ideal gas has the simplest form. The pv-Law of Boyle-Mariotte and the Law of Gay Lussac regarding the temperature dependence of pv is pv = p0v0(1 + at). It answers all the questions regarding the behaviour of the ideal gas during changes in volume, pressure and temperature. When gasses get close to their liquefaction, they differ more and more from the ideal state and their equation of state changes. The relevant equation is (p + a/v²)(v - b) = RT, the equation of state of van der Waals 1873, where p, v and T denote the pressure, volume and absolute temperature, R the gas constant, a and b are constants, which depend on the character of the gas; b is a volume correction, a allows for the mutual attraction of the gas particles. For large v, the equation becomes the equation of state of the ideal gas, for small v and corresponding t the equation of state of drop forming substances. Thus, this equation embraces the ideal state of a gas and the fluid state as special cases (also the transition from the one state into the other, that is, condensation, and also the critical point,above which condensation becomes impossible).
If you measure pressure, in fractions of the critical pressure, the volume in fractions of the critical volume, the temperature in fractions of the critical temperature, the equation of state assumes for all substances the same form. It does not involve anything signifying the peculiarity of a substance (a definite substance). Two arbitrarily selected substances yield identical diagrams of state; you must only compare pressure, volume and temperature in the stated manner - one says: in corresponding states. The law of agreeing states allows (according to an expression due to Zeeman), as far as thermic behaviour is concerned, to see in every substance a copy of another one, although in a changed scale. - Hitherto (1935), there does not exist an equation of state which embraces all three states of aggregate.
The first main theorem in the form U2 - U1=Q + A
The first main theorem can also be formulated mathematically, if one start from the energy content of a body in two different states 1. and 2. with the energies U1 and U2 (Planck). If U2 is the larger value, then U2 - U1 is the increase of energy in the transition from State 1. to State 2. Its energy can increase by a body accepting a quantity of heat Q and simultaneously a quantity A of another form of energy, which we will denote briefly by work; the energy can be mechanical, electrical, chemical, etc. One has then altogether U2 - U1=Q + A.
The state of a body then determines simultaneously its energy content. For example, if a given, measured volume of a gas undergoes an arbitrary change of state - changes in temperature and pressure - and returns in the end to its initial state, that is, to the initial temperature and pressure, its final energy content is the same as it was at the start. Such a process, which returns to the initial state - but can in between be arbitrary - is called a cyclic process. We can now give the first main theorem the form: If a body passes through a cyclic process, it has in the end the same energy as at the start, energy losses and gains cancel each other during such a process .
Hence follows the impossibility of the perpetuum mobile! In fact, the hope for the perpetuum mobile rests on the assumption that, without provision of energy (in contradiction to the first main theorem), one can continuously gain energy by letting a body or a system of bodies pass through certain cyclic changes and return to the initial state, that is, form a sequence of cyclic processes. However, as has been said already, a gain of work in a cyclic process is impossible. In a cyclic process, State 1 equals State 2, whence U2 = U1. However, then Q + A = 0, that is, in a cyclic process, the sum of the accepted heat and the consumed work vanishes.
Application of the first main theorem to an ideal gas
The change of the energy content of a body is especially clear if the body is a gas. If it is as a whole at rest, its energy is just its internal energy U. Hence, if we take it from a state U1 to another state U2 and measure the associated external effects (Q + A), the equation U2 - U1=Q + A indicates the magnitude of the change of energy and whether it has grown or dropped. The required experimental conditions are readily met in Joule's set-up of 1845 (Fig. 358). R and E are two closed metal containers with mechanically rigid walls, linked by the tube D; they can be connected or separated by a tap. They are fully immersed in water. R contains at first air, compressed to 20 atm and E a vacuum. The experiment is as follows: You measure accurately the temperature of the water, then open the tap between the vessels when air will flow from R to E until the pressure in both vessels is the same; after some time, the temperature is measured again. The result: The temperature of the water has not changed, whence the air, compressed to 20 atm has expanded (without performance of mechanical work) to twice its volume, without changing noticeably its temperature. Yet more exact measurements have yielded a minute cooling, but for the socalled permanent gases the action is minimal, and it is yet smaller as the gas at rising temperature or decreasing pressure approaches the state of an ideal gas. Hence, according to the equation U2 - U1=Q + A, the result of the experiment is U2 - U1= 0 (since there are neither thermal nor mechanical external effects), whence U2 = U1. The temperature remained constant, although the volume doubled. In other words: The inner energy of the ideal gas remains constant in spite of a large change of volume, that is, it only depends on its temperature and not on its volume.
However, the experimental set-up of Fig. 358 is not sufficiently sensitive. Any temperature increase of the air would have to be very large in order to become noticeable in the presence of large masses of metal and the large mass of water; according to a later calculation, under these conditions, it might be only 1/200º (Pollitzer). A much more sensitive set-up was developed by Lord Kelvin and employed in measurements with Joule (Fig. 359): A pump drives uniformly air through a long tube, the temperature of which is held constant. At a location, which is especially protected against heat conduction, the tube is plugged by compressed wool, silk or similar material. Through its pores flows the air with friction uniformly. In front of the plug, the pressure is higher than behind it. A very sensitive thermometer is located immediately behind it (in 1935, it was a thermo-element or platinum thermometer). Under lower pressure, the air flows slowly and uniformly (without noticeable kinetic energy). In the set-up of Fig. 358, a limited mass of air rushes into the vacuum, in that of Fig. 359, an unlimited flow of air moves slowly and steadily from the space at higher pressure into that at lower pressure; by artificial deceleration of the outward stream, the air transits immediately into the second state and indicates its temperature immediately. The result: Also here, the changes of temperature of the gases is so small that one can expect for ideal gases temperature constancy, that is, ultimately one can say: The inner energy of an ideal gas depends only on its temperature and not on its volume.
We have only introduced the Joule-Thomson experiment with a view to the
properties of ideal gases; it is of great technical importance
for real gases. Two processes are here superimposed: 1.
external, 2. internal work:
1. Driving the gas through the throttle demands work; on the other hand, the gas performs work behind the throttle by pushing other quantities of gas ahead of it. The difference of these performances of work is called the external work of the gas. It is measured by the difference of the products pv (Pressure times specific volume) behind and ahead of the throttle; in both products you substitute in the case of differential small pressure changes (dp) the same temperature.
2. Superimposed on the external work is the internal one of the gas overcoming molecular forces of attraction. It is always positive and especially large at low temperatures and then always larger than the external work. The effect depends in magnitude and sign on the temperature of the condensed gas: At room and lower temperatures all gases become colder, only hydrogen and helium become warmer; at sufficiently high temperatures, all gases become warmer, while, on the other hand, hydrogen also cools, if it has been cooled at the start to - 80ºC. This change in sign is called inversion, the corresponding temperature the inversion temperature of the Joule-Thomson-Effect. Only when the temperature of a gas lies below it, it can in this manner continue to become cooler. Carl von Linde 1842-1934 has used the Joule-Thomson-Effect in his air-liquefaction machine.
The first main theorem in the form dQ = dU + pdV
Another form of the first main theorem is due to Clausius: Introduce into the system an infinitesimal quantity of heat dQ and ask: What does it cause and what happens to it? It increases the inner energy U of the body by dU. But the increase dU is not equal to dQ, because one part of dQ is used to perform the work connected with the increase in volume of the body due to its gain in heat. The energy increase is reduced by this work - we assume that only volume-work is performed. How large is this volume-work? We assume that the only external force, which has to be taken into account, is the pressure acting on the surface, which - as happens especially frequently - acts at all points perpendicularly to it and equally strongly. In order to compute the work, required to overcome this force, we only need consider the change in volume of the body as a whole (not its change in individual directions). Denote the change in volume by dV, the pressure by p. The computation then yields - we can only quote here the result - for the volume work p·dV, whence we obtain for the change in inner energy dU = dQ - pdV or dQ = dU + dV. This equation is one of the most used mathematical forms of the first main theorem. We obtain it from U2 - U1=Q + A by assuming that U2 - U1 is so small, that we can call their difference dU, moreover calling the heat introduced from outside dQ and setting A = - pdV; it is negative, because the work is performed by the system (not as before transferred from outside to the system). It will not be possible to use this form of the first main theorem in the further presentation, since it demands knowledge of differential calculus, which is important as a basis of the computations of many theoretical predictions of Thermodynamics. It yields different equations involving observable quantities and thereby leads to experimental testing of the first main theorem.
Starting from dQ = dU + pdV and using the theorem that the inner energy of an ideal gas only depends on its temperature, you can obtain (by means of simple computations of calculus) important thermal properties of gases: If cp and cv are the specific heats at constant pressure and constant temperature, you have for every gas cp - cv = R/m, where R is the gas constant and m the molecular weight. Hence the difference of the specific heats of an ideal gas is constant, the difference of the molecular heats Cv = mcv and Cp = mcp is even the same for all gases: mcp - mcv = R.
Moreover the computation shows that also cp and cv depend only on the temperature and not on the volume, a fact confirmed by the first measurements of Regnault. However, according to measurements, cp is also independent of the temperature over a wide temperature range. But then this must also be valid in the same range for cv (since cp - cv = const). Planck therefore completed the definition of the ideal gas by the demand that his cp and his cv are completely independent of the temperature and volume.
The computation also yields an increase in temperature of a gas during adiabatic compression and a decrease during adiabatic expansion. (A change of volume without heat input or output is called adiabatic (Greek: a, dia, bainw = go; heat does not pass through), that is, the heat content remains constant; an isothermal change of volume takes place at constant temperature, that is, with input and output of heat.) Moreover (as accompaniment of pv = const during isothermal change of volume) pvg = const, where g = cp/cv during an adiabatic change of volume. A comparison of this equation with pv = const shows that the volume of a gas during adiabatic compression decreases more slowly (as a result of its increase in temperature) than during isothermal compression. The adiabatic curves (Gibbs calls them isentropic) in the pv-plane therefore drop more steeply to the v-axis than the isotherms (hyperbolas) (Fig. 360).
Adiabatic compaction and dilution of a gas are decisive during the propagation of sound in a gas. In this context, we are only interested in air. Compaction and dilution occur every second in a sound wave hundred, even thousand times. The accompanying equally fast warming and cooling of the air cannot adjust due to the rate of these changes, quite apart from the fact that air conducts heat very badly. Hence compaction and dilution in sound waves take place adiabatically. The formula v = (p/d)1/2 for the velocity of propagation of longitudinal waves in a gas contains the elasticity p of the gas. But a gas has two kinds of elasticity, depending on whether its compression is isothermal or adiabatic. The elasticity, measured adiabatically, is larger than the isothermally measured one, in fact, in the ratio of the specific heats cp/cv = k, whence in the formula for the velocity of sound, which involves the adiabatically measured ratio, v = (kp/d)1/2 (Laplace 1816). With this correction of the formula v = (p/d)1/2, due to Newton , the computed velocity of sound agrees with the measured one.
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