J6 Heat

Temperature dependence of the volume of substances

Temperature dependence of the volume of solids

Substances change their volume with their temperature. This indicates that their state - their inner energy - changes as they take in or lose heat. Heat intake increases the kinetic energy of the smallest parts of the substance (molecules); they move then along expanded orbits around their equilibrium positions and thereby increase the space occupied by them, that is, the body. The cohesion of substances influences this in an essential manner. Gases expand most, because they lack cohesion almost completely, fluids much less, solids least of all; while all gases expand almost equally, because the cohesion in all of them is almost zero, this is not so for fluids and solids.

Starting from the energy principle, the theory can compute expansion (in space - one dimensional is set equal to one third of it) and obtain a formula. According to measurements, the expansion of cubical crystals as well as that of certain other crystals follows the theoretical formula (Eduard Grueneisen 1877-1949). However, in most cases, you use the more convenient, empirically established formulae, which often yield within limited temperature ranges more exactly the length as a function of the temperature. For most substances, you can only employ the empirical formulae (alloys, non-cubical crystals, glass, et al.).

Measurements of the absolute and relative expansions differ. During the former, the test body is given different temperatures and measured at each, during the latter, which is more simply performed, the body and a check body, the expansion of which is known within the temperature range under consideration and as small as possible (its is best to employ quartz glass), are given the same treatment and their relative changes of length are determined.

A straight metal bar is longer at 100º than at 0º. In order to find the law by which its length changes with the temperature, you measure its length l_t at different temperatures and compare it with its length l₀ at 0º. The fraction (l_t - l₀)/l₀ - that is, the ratio of the elongation due to the temperature increase and the length at 0º - is found empirically to be, in first approximation, proportional to t. Denoting the fraction (l₁ - l₀)/l₀ by a, one has (l_t - l₀)/l₀ = a·t or l_t = l₀·a·t. The length at the temperature tº is larger by l₀·t·a than at 0º. The coeffcient a is called the linear coefficient of expansion (linear, because is yields the linear, not the spatial expansion - the increase in volume). A platinum bar of length 1 m at 0º is 0.9 mm longer than at 100º, a copper bar of the same length 1.65.mm, a tin bar 2.30 mm. At 1º, the expansions are only 1/100 of these values, whence the temperature expansion coefficients a between 0º and 100º are:

This division by 100 yields only approximate values of a, because the proportionality between the temperature increases and expansion is not linear. As the temperature increases, a becomes gradually larger. - Among the solids, ice has an especially large coefficient of expansion: 0.000 027 (Dewar), quartz glass an especially small one: 0.000 001 5, Jena thermometer glass 59^III : 0.000 017 7.

The expansion of solids is always so small that it need only be taken into consideration during precision measurements or during engineering applications of bodies, which at 0º are so long that their changes of length during normal temperature variations are substantial. One metre is the distance of two marks of the platinum-iridium bar in Paris at the temperature of melting ice; it is 0.8 mm longer at 100ºC. Railway tracks, placed end to end, demand at adjacent ends a certain margin, since otherwise they would bend as they expand; the ends of an iron bridge must be allowed to move to a certain degree on their supports and are therefore placed on rollers.

A body, which is heated, expands in all directions, an isotropic solid body uniformly in all directions (lead, gold, silver, copper, very slowly cooled down glass, the crystals of the cubic system like diamond, rock salt, lead glance), an anisotropic solid differently in different directions. For isotropic substances, the cubic coefficient of expansion is three times as large as the linear coeffcient. Imagine that you have cut from an isotropic body a cube with edges of length l at 0ºC. It has then the volume l³. If its coefficient of expansion is a, the length of its edges at tº is l(1 + at) and the cube's volume is

v_t = [l(1 + at)]³ = l³·(1 + at)³ = l³·(1 + 3at + 3a²t²+ a³t³).

Compared with the other terms, the terms with a²t² and a³t³ vanish, whence v_t = l³·(1 + 3at). For the volume of the cube, 3a has the same meaning as a for the length of its edge. But this is only true for isotropic bodies, a rather small group. Certain crystals, especially Island spar and certain kinds of marble, cannot expand or contract in certain directions. (A proposal by Brewster) was to cut in this direction out of marble a cylinder and to use it as a presumably invariable pendulum.) Rubber and silver iodide contract as the temperature increases, quartz glass also at low temperatures.

The agreement of the expansion coefficient of certain types of glass and platinum, also of certain types of glass (Friedrich Otto Schott 1851-1935) and iron (also of selected alloys) make it possible, to melt metal wires air tight into glass. This is required whenever electric current is to flow through the wall of an air tight glass vessel, as in incandescent lamps, x-ray tubes, mercury vapour lamps, mercury rectifiers, etc. The joint, carefully produced, can undergo large temperature variations and considerable pressures without leaking. Uneven expansion ruptures the glass at the fusion points. Industry also employs the contraction, which accompanies decreasing temperatures, for example, by placing on wheels hot iron rims, which on cooling will adhere more strongly. You can even make an air tight joint ofa glass tube and an external steel sleeve.

Compensation pendulum

The temperature dependence of the length of the pendulum in pendulum clocks must be eliminated as otherwise they would not keep the time. A clock with a steel pendulum bar, which runs correctly at 15ºC would at 25ºC be slow by 5 seconds each day. You can compensate by means of mercury (George Graham 1675-1751 1715). The extension of the steel pendulum bar (Fig. 375) lowers the pendulum's vibration point; however, the mercury, as it fills a larger part of the glass container due to its expansion, lifts its centre of gravity and thereby also the pendulum's vibration point.

The frame pendulum of John Harrison 1693-1776 compensates well: A frame connects the pendulum bar to the clock work (Fig. 376). The lines S are steel rods, Z zinc rods (or brass rods); the rods of the frame are linked below by the rod EF, the point A is linked rigidly to GH. The steel rods, as they lengthen, tend to lower the disk L; this is impeded by the zinc rods, the upper ends of which can rise with the cross bar, thus also the point A. Since zinc extends almost twise as much as steel, the lengths of the steel and zinc rods can be adjusted so that the vibration point does not move.

The best compensation has the pendulum of Riefler 1847-1912, made out of nickel-steel: 35.7% Ni, 64.3 % steel (Invar, Guillaume) (Fig. 377). It consists of the nickel rod S, the disk L, the compensation body CC₁ and the correction nuts M and M'. The resting plane A, by which the disk l rests on the compensation tube C₁, passes exactly through the centre of L. In order to be able to change the compensation in a relatively wide range (corresponding to different pendulum bars), the body C is made out of two different metal tubes with widely different expansion coefficients. In seconds pendulums, the lower C is made out of nickel plated brass, the upper C₁ out of steel; together they are 10 cm long. The possibly remaining error of first class pendulums is for 1ºC ⁺/_- 0.005 sec daily, of second class pendulums ⁺/_- 0.02 sec.

Compensation balance wheel

In chronometers, the balance wheel takes the place of the pendulum and the elasticity of a spring that of the force of gravity. An uncompensated balance wheel oscillates slowlier at high temperatures than at low ones, becaus the spring is less elastic and the diameter of the balance wheel increases; this increases the distance of the individual mass points from the axis of rotation, that is, the momen of inertia of the balance wheel and thus causes an increase in force, if the velocity of the balance wheel is to remain constant; however, the elasticity of the spring decreases with the temperature, whence one has to ensure that the moment of inertia decreases with rising temperature. This is done by the balance wheel of Thomas Earnshaw 1717-1785: If you form from two parallel strips of metal with different constants of expansion, say, brass and steel, a band and connect them (by rivets or solder, heating will cause a change of shape of the band. One half (brass) expands more than the other (steel), whence the band curves so that the brass strip lies on the convex, the longer side; if the double band is curved prior to heating, as the temperature rises, the curvature becomes greater. Earnshaw exploited this fact.The ring of the balance wheel is made out of brass and steel - the brass outside - and cut at two opposite lying points (Fig. 378). As the temperature rises, the length of the diameter ab increases and the distances of the neighboring masses of the points a and b increase their distances from the axis of rotation, that is, increase the moment of inertia. However, due to the stronger expansion of the external side of the ring, the curvature of each of the two segments becomes larger, so that the ends of the segments together with their neighbouring mass get closer to the axis and the moment of inertia decreases.The spring of the balance wheel, made out of two metals with different coefficients of expansion, in order to make the spring independent of the temperature, yields an even better uniformity of the running of the clock. Recent even better compensations employ 45 & Ni-steel (Elinvar, Guillaume).

Temperature dependence of the volume of fluids

In the case of drop forming substances, only spatial expansion is of interest. As in the case of solids, it is generally proportional to increases in temperature, but becomes much greater due to the small cohesion of fluids. The fluids with the largest expansion are those, which only remain fluid under great pressure: Liquid sulphuric acid, fluid carbonic acid, etc. The coefficient of expansion of carbonic acid, which becomes fluid at 63 atm, is at 20ºC about 0.015 - considerably lager than that of air under the same conditions. The coefficient of expansion of fluids increases, except for mercury, strongly with the temperature, in general, the more the closer it comes to the boiling point. Letting the volume at 0º and atmospheric pressure be 10 000, then

This is the reason why on the scale of an alcohol thermometer the degree marks between 30º and 80º are further apart than between 0º and 30º; on the mercury thermometer, they are everywhere approximately equal. (Expansion of the glass must be allowed for!)

Water behaves completely differently from other fluids. It contracts between 0º and 4º, whence at 4º its specific volume is smallest and its density largest; from 4º onwards, it expands and has at 8º the same density as at 0º. A float, say a hydrometer, rises in water, as the temperature climbs from 0º to 4º, and then sinks as the temperature increases.

The change of density and its effects are demonstrated by simultaneously cooling a vertical column of water at a temperature above 4º from its centre upwards and downwards (Fig. 379) and measuring the temperatures at the bottom and top simultaneously (Hope 1766-1844). The upper thermometer does not show at first a substantial temperature change, the bottom one drops steadily to 4º; it then stops and the top one starts to drop and only stops at 0º. Reason: Cooling to 4º increases the specific weight of water, whence the cooled water sinks to the bottom. The lower layers are thereby strongly cooled. However, the bottom is only reached by water of 4º, whence the termometer stops at this temperature. Water with a temperature of less than 4º rises with the reduction of its specific weight and accelerates the lowering of the temperature of the water at the top. Now the water with a temperature of 0º has the least specific weight, whence it rises to the top; hence the thermometer at the top only stops at 0º.

The decrease of the specific weight of water with the temperature explains why standing bodies of water freeze on top, but can have relatively high temperatures at the bottom.The low heat conductivity of the ice cover and the water protect the bottom against further cooling (as well as the animals living in it). In flowing water, there can be places, where the velocity is not very large and under-cooled water exists due to vortices; further cooling at the bottom then becomes possible and ice is formed. The specific weight of ice is smaller than that of water, whence eventually the bottom ice rises and swims on the surface. - While it freezes, water expands more (by about 10%) and can then overcome great resistance, burst water conducts and bottles with water, walls in the spaces of which it freezes, etc.

The coefficient of expansion of a fluid can be measured with a calibrated thermometer-like glass instrument.You fill it completely, heat it to a given number of degrees, when the fluid due to its expansion escapes partly through the calibrated capillary; you then cool it again to the intial temperature. The fluid then does not fill the capillary entirely and the empty space indicates by how much the fluid, that filled it initially, has expanded during the rise in temperature. You must be sure to allow for the expansion of the glass vessel. During sudden strong heating, the fluid sinks at first due to the expansion of the vessel before starts to rise due to its own being heated up.

Measurement of expansion of mercury and water

The expansion of a vessel does not affect the mesaurements, if the method of Pierre Louis Dulong 1785-1838 and Alexis Thèrése Petit 1791-1820 (improved by Regnault 1847, Fig. 380a) is employed. Columns of the same fluid in a tube of th shape AA'BB' are heated to different temperatures - one is surrounded by melting ice, the other by oil, which may be heated - and are stopped from exchanging heat through the layer of air bb' (Fig. 380b). As long as they have the same temperature, the fluids have the same specific weight and stand equally high in a and a', b and b', If their temperatures differ, their specific weights are not the same and the fluid levels change. The theorem of fluids in communicating tubes and of different specific weights forms the foundation for the computation. Dulong and Petit (1818) have measured in this manner the averaged expansion coefficient of mercury between 0º and 100º at1/5000 per 1º temperature increase. The average expansion coefficient between 0º and º is: = {0.181 82 + 0.000 78·/100}10^-3 , valid between 0º and 100º (Thiesen, Scheel, Sell 1896). The same approach was applied to the measurement of the expansion of water between 0º and 40º by Thiesen,Scheel and Diesselhorst in 1900.

Temperature dependence of the volume of gases

When determining the changes of the volumes of gases due to a change in the temperature, you must take into consideration the pressure acting on the gas. That is unnecessary for solids and fluids, because - except in special cases as that of liquid sulphuric acid or liquid carbonic acid - the external pressure does not affect the volume noticeably. If an iron rod at 0º has the volume v₀ and at 100º the volume v₁₀₀, then it has under the same temperature conditions at each pressure almost the same volume. However, if air has at 0º the volume v₀ and at 100º the volume v₁₀₀, we cannot at all tell what volume it will have under the same temperature conditions next time, because the volume of a gas depends on the pressure acting on it.

The first idea will be to keep the pressure constant and then to examine the volume as the temperature increases. One can do so withuut difficulty in the case of solids, because the corresponding increase in the kinetic energy of the molecules is large enough to overcome the external pressure and shift the borders of the volume. This is different in the case of gases. The solid walls of a gas container rather become softer as the result of heating up and melt than give in to the increase in volume of the gas. Hence, the pressure, by which the gas acts on solid walls, increases in a vessel with solid walls and the walls respond. From the pressure, which can be measured, you can compute by means of the Boyle-law the volume, which the gas would occupy, it it were subject to the initial pressure - the pressure prior to the rise in temperature.

As the temperature of the gas increases, we can either measure its volume increase at constant pressure or its pressure increase at constant volume. Experience shows (Charles 1787, and then more exactly Gay-Lussac 1802*): At constant pressure, the volume of all all gases expands for every 1º approximately by a = 1/273 = 0.00366 of their volume at 0º - all! All have the same expansion coefficient! (That is only approximately correct, but almost so, so that we can accept it here as being completely correct.) If the volume of a gas is v₀ at temperature 0º and pressure p₀ (we mean here the atmospheric pressure), while the pressure remains constant, then at 1º, 2º, 3º ···tº the volume becomes v₀+v₀·a·1, v₀+v₀·a·2 ··· v₀+v₀·a·t, that is, at constant pressure, v = v₀+(1 +a·t), where v₀ is the volume of the gas at 0º. However, if the pressure has changed from p₀ to p, a change due to pressure also occurs. At the pressure p₀, the volume at t₀ is (1 +at), at the pressure p and temperature t, by Boyle's equation p₀·v₀(1 +at) = pv it becomes v = p₀·v₀(1 +at)/p.

* The first person to discover this law seems to have been Volta. It was published in the Annali di Chimica of Brugnatelli in 1793: "Della uniforme dilatazione dell'aria per ogni grado di calore, cominciando sotto la temperatura del ghiaccio, fin sopra delle del' ebollizione dell' acqua e di ciò, che sovente fa parer non equabile tal dilatazione, entrando ad accrescere a dismisura il volume dell' aria".

This equation allows you to compute the volume of a gas at any pressure p and at every temperature t, if you know its state at one definite temperature under one definite pressure. As normal pressure p₀ you have 1 atm and as normal temperature 0º.

The equation pv = p₀v₀(+at), the equation of state of the ideal gas, also shows how the pressure of the gas changes, if you assume that instead of the temperature its volume remains constant, that v is identical to v₀. The equation then becomes

p = p₀(1 + at) = p₀+ p₀·a·t,

That is, the pressure increase is proportional to the temperature increase; it grows by 1/273 of the pressure at 0º for every 1º increase. The coefficient now is called the tension coefficient.

Both the equations v = v₀(1 +at) and p = p₀(1 +at), in which we assume the expansion- and tension- coefficients to be the same, express the law of Gay-Lussac: vp = p₀·v₀(1 +at). The proportional increase of the volume with the absolute temperature can be represented by a straight line. Provided the gas obeys this law without a limitation, since a=1/273, this straight line would intersect the temperature axis at t = - 273.2 (Fig. 381), that is, the gas would not have a volume at the absolute zero. A similar result applies to the linear dependence of the pressure on the temperature. Corresponding values of p and v at constant temperature have already been represented by a curve. We can now represent the corresponding values of p, v, t, the points of which lie on a surface (Fig. 382. A given point on this surface describes a definite state of the gas in terms of its temperature, pressure and volume. Hence we call this surface the surface of state of the gas. It is characterized as follows: A plane, parallel to the pv-plane (all points of which have the same temperature t) intersects the surface of state in an equal-sided hyperbola (isotherm) SR, also NM; a plane, parallel to the pt-plane (all points of which have the same volume v) intersects it along a straight line cc, inclined to the negative temperature axis (isochore); a plane, parallel to the vt-plane (all points of which have the same pressure p) intersects it also along a straight line bb, inclined to the negative temperature axis (isobar).

You obtain the isotherms, corresponding to given temperature points, on the surface of state by placing through these points of the temperature axis planes parallel to the pv-plane. You see then the isotherms on parallel planes which lie behind each other. On the surface of state, they become flatter towards the back (relative to the origin of the coordinate system. If you project them forwards on to the pv-plane, they yield Fig. 383.

We list together corresponding pressures and volumes of a gas at the temperature tº, which at 0º and the pressure p₀ has the volume v₀, under the assumption that once the volume is kept constant, the second time the pressure, the third time pressure and volume have become p and v. - Both: the increase in volume at constant pressure and the increase in pressure at constant volume allows us to find the value of a.

Range of validity of the pv-law

The surface of state is idealized, because no gas obeys the Boyle-Mariotte law or the Gay-Lussac Law in the completeness, on which these laws are based. The laws suffer from deviations, which become larger as the pressure grows and the temperatures drop. In order to be able to survey the manner in which p·v changes with p, you make p·v into the ordinate, p into the abscissa of a coordinate system [(pv), p-diagram], keep the temperature constant and graph the isotherm for this temperature. You repeat this for several temperatures. Thus, you obtain a system of isotherms (Fig. 384). Every isotherm - above a definite temperature - passes through a minimum. In terms of differential calculus, this means: The tangent of the isotherm at this point is parallel to the p-axis, close to it effectively the same p·v relates to the same p independently of p, that is, p·v is there independent of p, that is, the law applies there strictly. This point is called the Boyle-point. The Boyle-points of all isotherms form the Boyle-curve, which is like a parabola. This curve intersects at any point the pv-axis. This point corresponds to the pressure p = 0, that is, to infinite dilution. The temperature of the isotherm through this point is called the Boyle-temperature. [It is important for the liquefaction of gases; it becomes possible by the Joule-Thomson method only when the gas has first been cooled to the Boyle-temperature.].

In summarizing, we can say: The pv-law does not apply exactly, but for many gases it is a very good approximation within wide ranges. It applies the more strictly, the smaller is the pressure, and fully strictly for the infintely diluted gas, that is, it applies to a limiting state. It is strictly valid to higher pressures than normally at the Boyle-temperature. Moreover, only the infinitely diluted gas has at all temperatures the same temperature coefficient and the same tension coefficient, namely a = 0.003 660 4 = 1/273.2 (Henning and Heuse).

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