Basic concepts: Temperature, Heat
Sensation of heat and thermal state
We sense immediately the thermal state of a body through our skin* and call accordingly a body hot, warm, tepid, cool, etc. The fact that a thermal state makes us sense it only through feelings of warmth or coldness causes us to allot decisive importance to this physiological manifestation of heat. However, the reactions of our skin are useless for the assessment of the thermal state of a body, because they depend on the instantaneous state of our own body. The same object is found to be cold or warm by a hand touching it, depending on whether it became colder or warmer: If it was colder, it finds the body to be warm, because it became warmer, if it was warmer, it finds the body to be cold, because it was cooled down. Moreover, in certain states of the hand, the same temperature may appear to have different degrees of warmth: A body made out of metal at room temperature seems to be colder than a wooden one under otherwise equal conditions (a knife is colder than its wooden handle!), because metal removes heat faster from your hand than wood. The sensation of heat also becomes stronger with an increase of the area of your skin touching the body; moreover, under the same conditions, it is much stronger on your neck than on your hands and in your mouth.
* The skin altogether is not heat sensitive, but only certain places in it (heat points, coldness points), where the corresponding sensitive nerves are located. The effect exercised there is taken by the nerves from the final organ via the spinal chord to the cerebral cortex. We sense the arrival of this stimulus as heat or cold.
We see that the physiological effects of heat exerted by a given body cannot enable us to describe its state of heat uniquely, for example, in order to indicate when and whether it was in the same state which it had before. However, we must have a unique, physical description of the thermal state of a body. We will obtain it as follows: Experience tells us that most substances increase (decrease) their volumes as they accept (give away) heat - gases do so most, solids least; moreover, given two bodies at different temperatures, the warmer one gives away heat (cools down), the colder one takes in heat (warms up), and eventually thermal equilibrium is established between them. Experiments show that thermal equilibrium also occurs when arbitrarily many, not equally warm bodies touch each other long enough. If we join three bodies A, B, C into a ring, there will occur eventually thermal equilibrium at the locations AB and BC. Hence there must also be equilibrium at AC; otherwise the thermal equilibrium in the ring would not be a general state - which would contradict the assumption.
This combination of change of volume by heat and thermal equilibrium enable us to compare the thermal states of any two bodies A and B without bringing them into contact with each other. We only need establish the contact of A with a third body C, observe whether and how its volume changes until thermal equilibrium occurs and repeat the same action with the bodies B and C. (The body C assumes here the place of a thermometer or better thermoscope - this means: The place of a body, the change of volume of which becomes visible during creation of thermal equilibrium.) We allot to each thermoscopic volume of the body C, when it is in thermal equilibrium with another body, a definite number - like to the larger or smaller volume of mercury on the temperature scale - and call this number its respective temperature**. We also allot the same number to the body in thermal equilibrium with it and call it its temperature.
** There exist in Nature thermal states, but the concept of temperature only exists through our arbitrary definition. A temperature signifies the thermal state by a number. This temperature number has just the property of an inventory index, with the aid of which we can recognize again the same thermal state and, if required, find and reinstate it (Mach ).
We now have a physical criterion, which informs us about the thermal state of a body. If, to start with, we establish contact between the bodies A and C until C does not any more change its volume, then we see that thermal equilibrium exists between them and we find out which number we must allot to this state as its temperature. Correspondingly, we obtain such a number for the body B. If we arrive in this way at the same number for A and B and establish contact between them, neither of them gains or loses heat - they are in thermal equilibrium. By determination of the temperature number - the number which is allotted to a given volume of C - we can now, without establishing contact between A and B, discover whether their thermal states are the same or different and, if they differ, which of them would lose heat when they are brought into contact. The Rule: Bodies, the temperatures of which are equal to that of a third body, have the same temperature and the principle by which we allot a definite thermoscopic number - a definite number, namely the temperature (degree) - form the basis of measurements of temperature (thermometry).
Definition of temperature
Every property of matter, which can be related quantitatively to temperature, is suitable for the definition of temperature, for example, the length of a body which, in general, increases with the temperature. However, not all substances expand equally at the same temperature increase*, whence you must select arbitrarily the substance to be used for the definition. By what principle does one obtain numerical values for the temperature? First of all, we must fix the starting point - the zero - from which we can start to count. In every day life, the temperature of melting ice at a pressure of 1 atm is employed. (The pressure affects the melting temperature of ice insignificantly, so that one normally does not specify its instantaneous magnitude). Next, you must specify its unit - one degree - , that is, you must state how many degrees are there to be between zero and a second, arbitrarily fixed temperature (fixed point). Celsius 1742 selected as second fixed point the temperature of water, boiling at 1 atm, more accurately, the temperature of its rising vapour. (The boiling point depends noticeably on the pressure, whence one must also measure the air pressure and convert the observed temperature to the normal pressure. During this conversion, you must lower or raise the boiling temperature by 0.037º for every 1 mm of mercury above or below 1atm.) Celsius was also responsible for the subdivision into 100 degrees.
We still have to decide - arbitrarily - according to what rule the temperature between the two fixed points and beyond is to be measured. It will be simplest to adopt a linear relationship between the corresponding property of the body (here the length of the thermometric body) and the temperature. If we denote its length at 0º by l0 and at 100º by l1, then we have for a linear relationship
l1 = a + bt. (1)
For t = 0º and t = 100º, we have l0 = a and l1 = a + b·100, whence a = l0 and
b = (l1 - l0)/100 (2)
and (1) yields
|lt = l0 + t·(l1 - l0)/100||(3)||or||t = 110·(lt - l0)/(l1 - l0)||(4)|
Equation (4) is the definition of the temperature t.
There exist besides Celsius' temperature subdivision between the freezing and boiling points of water into 100 degrees yet the 80 degrees subdivision of Réaumur and the subdivision of Gabriel Daniel Fahrenheit 1686-1736 (Fig. 355). Celsius' scale is used for scientific work, the Fahrenheit scale only in England and the USA, the Réaumur scale - in 1935 - only by room and bath-room thermometers. The signs + and - do not signify a contrast between degrees of heat and cold, but only the position of a given temperature with respect to another temperature - the temperature of melting ice. They neither indicate addition nor subtraction. The temperature of 30º is also by no means twice as high as that of 15º. The zero point is merely an arbitrarily selected temperature, from which degrees are counted and which we refer to as zero - a relative zero.
For a long time, the temperature for scientific purposes was determined by substituting in Equation (4) for l the length of the column in the mercury thermometer. However, since the glass tube expands with rising temperature, the measurement depends on the difference in the thermal expansion of glass and mercury and even on the type of glass used. Moreover, measurement of temperatures below the coagulation point of mercury and near its evaporation point demand another substance for thermometry. Gases are employed for accurate measurements; the normal thermometer then is a gas thermometer of constant volume, in which the pressure increases at rising temperature according to the Law of Jacques César Charles 1746-1823. By Equation (4), you must set
t = 110·(pt - p0)/(p1 - p0). (5)
In this case, the expansion of the vessel is of little importance and all gases yield by this equation nearly equal results - practically completely equal results, if the gases are very diluted, that is, almost ideal gases.
According to the equation pV = NkT, the pressure p for a gas thermometer of constant volume is proportional to the temperature T. Denoting the factor of proportionality by A, one writes pt = AT and obtains from (5) the relationship
t = 100·(T -T0)/(T1 - T0). (6)
Since the difference T1 - T0 between the temperatures of the boiling and freezing points of water equals 100, you arrive at
t = T -T0,
where T0 is the temperature of the freezing point on the T-scale. You find the numerical value of T0 by again setting p0 = AT0 and p1 = AT1 from the relation T0/(T1 - T0).= p0/(p1 - p0), whence T0=100·p0/(p1 - p0). Measurements by Henning and Heuse 1921 have yielded the pressure p0 at the freezing point and p1 at boiling point of water for gases under small pressure:
T0 = 273.20º.
T is called the absolute temperature, its zero, at which the kinetic energy of the molecules vanishes**, the absolute zero, the temperature T0 = 273.20º, which coincides with the zero of the Celsius scale, the absolute temperature of the freezing point. The connection between the two scales is T = 273.20 + t.
**Zero point energy. How big is the energy at the absolute zero point? According to the classical molecular theory of heat, you have there a perfect state of rest, whence the energy vanishes. The ideal gas, the molecules of which only displace (not also rotate) and the molar energy of which is 3/2·RT, has therefore for T = 0 zero energy. In Quantum Theory, it is not a matter of course that the energy vanishes at T = 0; this question had not been resolved in 1935!
Heat, a form of energy. Brownian motion
Until the start of the Nineteenth Century, the general opinion was: Heat is a substance (phlogiston) - warming up of a body means taking in heat, cooling it off means getting rid of it - its content determines the temperature of the body, and since a body in spite of its content of heat substance does not change its weight, this substance has no weight. This view became untenable through the discovery of Rumford 1798 that arbitrary amounts of heat can be generated by work. While drilling a gun barrel, he employed the heat, which arose through the friction of the drill at the wall of the barrel, to warm up water and evaporate it. He showed that he could evaporate any amount of water by continuing the work (with a blunt drill!). Hence he concluded: Heat cannot possibly be a substance, because the barrel and the drill have finite dimensions, whence they cannot contain and provide unlimited amounts of heat substance; the heat substance, which was entering the water, must therefore have been created, but matter cannot be created. However, motion was being transferred to the drill and the barrel continuously and heat was being generated as long as the transfer of motion lasted, whence the heat could only be due to the motion, that is, it must itself be motion - motion of the molecules of the body, generated by friction. (More accurately: Energy is transferred to the drill and the barrel, whence heat must be one form of energy - energy which every mass, that is also a molecule possesses due to its motion.) Rumford's views were supported by many experiments, especially by Davy who has demonstrated that pieces of ice, when rubbed against each other, melt without an external supply of heat.
The motion, which apparently is destroyed, is not lost, it has only been passed on to molecules and atoms. It is not directly detectable, but the motions are detected in the persistent, undestructable, microscopic trembling of microscopic particles, suspended in a fluid, discovered by Robert Brown 1773-1858 1827 on pollen in water. For example, gamboge (gum-resin) yields in water a lively yellow emulsion. You can find in it molecules (1000 and more) conglomerated into microscopic, spherical formations. These tremble microscopically, one may say: Eternally and on their own. They tremble the faster, the smaller they are, the more liquid is the fluid and the higher the temperature, and independently of external, mechanical effects (vessel perfectly at rest), the thickness of the fluid layer (between two cover glasses), the strength of illumination and immediately (in the same manner tremble microscopic droplets and the gas bubbles in cavities of certain minerals, filled with fluid). External sources of energy seem to be excluded; one believes that Brownian motion is identical to theoretically predictable molecular motions and the internal heat content is the actual agent ( Smoluchowski , Einstein). The molecules of water impact all the time against the solid particles, which move when pushed especially strongly in some direction . During a given time interval, while a particle receives in the mean as much impulse, for example, from the right to the left as from the left to the right, at some instant , however, more from the left or from the right. The smaller are the particles, the more probable it is that one direction of impulse predominates and they move strongly. If an area is so small that only a few molecules can hit it simultaneously, especially strong or weak impacts are less easily balanced than when the area is so large, that many molecules can hit it simultaneously. The result of this irregularity is the Brownian Motion. You cannot see an individual particle, because it changes its direction of motion a billion times per second. You see its mean path! During trembling, it follows a zigzag path of atomic partial paths. Only because their sum attains in the course of time a noticeable magnitude, you can observe displacements.
The molecules of water also interchange impacts and thereby execute totally disordered motion. As a result of collisions with other molecules, a single molecule describes certainly a zigzag path of atomic distances, which it follows at changing velocities; a given location of a fluid is passed in changing directions by molecules at changing velocities - no direction is preferred, it is a state of ideal disorder. Felix Ehrenhaft 1879-1952 1907 has discovered a type of molecular motion in gases, which is analogous to the Brownian Motion in fluids. The concept of molecular thermal motion of molecules has turned out to be very fertile for the Theory of Gases; theoretical predictions and measurements agree so well (for very diluted gases) that one cannot doubt even without a knowledge of Brownian Motion the correctness of the kinetic concept of heat. The molecules of solid substances cannot - in contrast to fluids and gases - move quite freely; they are tied to a certain position of rest, about which they can only oscillate.
Connection between heat and temperature
If a body accepts energy in the form of heat, it distributes itself in the course of time between all molecules. The kinetic energy of a molecule, in the mean, grows larger, the larger the accepted amount of heat and the fewer molecules are involved. The measure for the averaged kinetic energy is the body's temperature. Hence, if a given amount of heat enters a body with many (few) molecules, every individual molecule receives comparatively little (much) energy, and the temperature remains comparatively low (increases comparatively fast). You say: The body possesses a large or small heat capacity.
Heat capacity is the link between the temperature and heat content of a body; if it is larger, the temperature of the body rises less while it accepts a given amount of heat. However, for example during evaporation of water, the temperature of the water does not rise in spite of heat input. It would be an error to conclude from this: The water possesses then an infinitely large heat capacity. Moreover, the heat serves here and during similar processes to change the aggregate state. Since it does not manifest itself here by a rise of temperature of the body, it is called latent.
If two bodies with different temperatures touch each other, the energies of motion between the molecules balance, heat flows from the more highly tempered body to the more lowly tempered body. The direction of flow depends only on the temperature of the bodies, which they had prior to their coming into contact, not on the amount of heat.
You can demonstrate the relationship between heat and temperature by a vessel filled with water. The amount of water corresponds to the amount of heat, the level of the water in the vessel to the temperature. The same amount of water is in a narrow vessel higher than in a wider one just as the same amount of heat raises in a body of less heat capacity the temperature higher than in a body of larger heat capacity. If you link vessels in which water has different levels, as in Fig. 183, water flows from the vessels with higher levels to those with lower levels until all have the same level; the same occurs in the heat exchange between differently tempered bodies until all of them have the same temperature. - However, this comparison fails in one respect. Heat cannot be compared with an indestructible mass like water; it is a form of energy, which arises out of another form of energy and can change into a different form of energy.
Unit of heat
The calorie has been defined as the unit of heat; it is that amount of heat (cal) required to raise the temperature of 1 g of pure water from 14.5 º to 15.5º. Since, by experience, the amount of heat required to raise the temperature of a given amount of water by 1º is almost the same for every temperature, it is almost always sufficient to measure a quantity of water k by finding out by how many degrees Dt one can warm up m gram of water. Then k = m·Dt cal. Industry employs most of the time kcal - 1000 times the amount of heat of 1 cal. It is the amount of heat which raises the temperature of 1 kg water by 1º, more accurately, from 14.5º to 15.5º.
Note: 1 g of water demands 1 cal for raising its temperature by 1º; other substances use different amounts of heat, for example, aluminium 0.214 cal, mercury 0.033 cal. These amounts of heat are called the specific heats of the respective substances. For a strict definition of specific heat, we must first of all define the heat capacity of a body. It is the number of units of heat, which is required to raise a body's temperature by 1º. Hence we have the definition of the specific heat of a body: It is the ratio of the amount of heat, required to raise the body's temperature by 1º, to the amount of heat, which raises the same weight of water by 1º. Hence specific heat is the ratio of two quantities of the same kind; it is independent of the units of measure and the temperature scale.
Moreover note: A gas has a different specific heat, depending on whether it expands while taking in heat or whether its expansion is stopped. The first is called specific heat at constant pressure and denoted by cp, the second, specific heat at constant volume cv. You have that cp > cv, because gas works during expansion, that is, it uses at equal temperature more heat than when it does not expand.
Molecular theory of heat. Thermodynamics
By the middle of the Nineteenth Century, the theory of heat substance was eliminated by proofs that a given amount of work always generates the same amount of heat and, conversely, that this amount of heat can perform equally large work; namely, after the equivalence of heat and work had been established. This concept that heat is a form of energy is called the mechanical theory of heat.
The molecular theory of heat leads us to the micro-cosmos of molecular motion and teaches us to interpret heat processes as mechanical processes. However, motions of molecules can only be recognized by their actions. We do not recognize the entirety of disordered motions of infinitely many molecules as motion, but as heat; the totality of the impacts of gas particles against a wall is not recognized as impact and reaction of individual particles, but as pressure on the wall. The space, within which the molecules move, the volume of the body, is important for the behaviour of the totality of its molecules, but has no role for an individual particle. Temperature is a concept for a macroscopic community and cannot be applied to a single molecule; it is only determined by the mean value of the kinetic energy of all molecules and cannot be related to the continuously changing instantaneous value of the kinetic energy of an individual molecule.
The theory of heat as a form of energy, that is, Thermodynamics, starts out from macroscopic phenomena. It is headed by the three main theorems of Thermodynamics, which arise from experience and describe most concisely all phenomena; all thermodynamic processes can be derived from them. The molecular theory of heat attempts to prove these theorems from the laws of Mechanics. For the first main theorem - the law of conservation of energy, applied to the theory of heat - the proof is readily executed. The proof of the second main theorem - it determines the direction of a thermodynamic conversion (work into heat or heat into work) - requires an extension of Ordinary Mechanics to Statistical Mechanics. Classical Mechanics fails in the face of the third main theorem - concerning the thermodynamic properties of matter with a very small content of kinetic energy - and must be replaced by Quantum Mechanics. The three main theorems rule over the entire theory of heat; their meaning can be explained by means of heat phenomena, which are relatively simply described.
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