Thermal Properties Of Matter

Type of physical science: Classical physics

Field of study: Thermodynamics

When heat is added to a substance, it will respond by changes in temperature and volume. Substances will also conduct heat through themselves. An understanding of the thermal properties of matter has affected the development of areas of physics, including thermodynamics, statistical mechanics, quantum theory, and solid-state physics, and has both scientific and commercial applications.

Overview

When two objects at different temperatures are placed in contact, there will be a net flow of energy from the object at higher temperature to the object at lower temperature. This flow of energy will continue until thermal equilibrium is reached and the two objects acquire the same temperature. The energy that moves between two objects as a result of their difference in temperature is defined as heat, or thermal energy. From a microscopic point of view, this thermal energy can be pictured as the energy resulting from random motion of the molecules making up a substance. The changes in a substance that occur when heat is added or removed, or when the temperature of a substance is changed, and the rate of flow of heat through a substance are examples of the thermal properties of matter.

When heat is added to a system, the system will respond by an increase in temperature.

The heat capacity of the system (C) is defined as the ratio of the amount of heat added to the system (q) divided by the change in the temperature of the system (T), that is, C = q/(T). One drawback in this definition is that heat capacity is proportional to the mass of the system. It is convenient to define a new term, the specific heat (c), as the heat capacity for a fixed quantity of substance. Two different types of specific heats are usually used.

The specific heat per gram (c_g) is equal to the heat capacity of 1 gram of the substance. Alternatively, the specific heat can be defined for a sample containing a specific number of molecules of a substance. The number of molecules is usually given in units of moles, where 1 mole of molecules is a number equal to the number of atoms in 12 grams of an isotopically pure sample of carbon, approximately 6 x 10²³. The specific heat per mole (C_m), sometimes called the molar heat capacity, is equal to the heat capacity of a sample containing 1 mole of molecules. For a sample of substance with a mass of m grams, and containing n moles of molecules, the specific heat per gram of substance is c_g= q/m(T), while the specific heat per mole of substance is C_m= q/n (T).

The specific heat of a substance depends both on the properties of the substance and on the conditions used to add heat to the system. Heat can be added to a system under conditions where the volume of the system is held constant or under conditions where the external pressure applied to the system is held constant. When heat is added to a substance at constant volume, all the heat goes into raising the temperature of the substance. When heat is added to a substance at constant pressure, however, some of the heat will be converted into work associated with the change in volume of the substance. A general relationship for the difference between the specific heat measured at constant pressure and at constant volume can be found from thermodynamics.

For substances that expand when heated, as is usually the case, the specific heat measured for conditions of constant pressure will be larger than that measured under conditions of constant volume. This results from the fact that some of the heat is used to expand the substance against the applied pressure. For substances where the volume decreases as temperature increases, the specific heat measured at constant volume will be larger than the value found at constant pressure, since the work performed on the substance resulting from the decrease in volume will be converted into an equivalent quantity of heat.

Heat added to a gas can be used to increase the average translational kinetic energy of the gas particles. For gases composed of molecules, some of the thermal energy will also be used to increase the total energy of the molecule contained in vibrational motion of the atoms making up the molecule and in rotation of the molecule. Since the number of molecular vibrations increases as the number of atoms making up the molecule increases, the specific heat per mole of gas molecules in general increases as the size of the molecules increases. The equipartition theorem, a result from statistical mechanics, predicts that there will be a contribution to the specific heat of a gas from translational, rotational, and vibrational motion of the molecules, which can be calculated from the theorem. The values observed for specific heats for gases are in general agreement with the predictions made from the equipartition theorem at high temperatures, but are less than predicted by the theorem at low temperatures. The discrepancy is caused by the failure to take into account the fact that vibrational and rotational energy in molecules is quantized. When quantum theory is used to model the vibrational and rotational motion of gas molecules, the values calculated for specific heat are in general agreement with experimental results at all temperatures. For ideal gases, the difference between constant pressure and constant volume specific heats per mole of substance is equal to R, the gas constant, independent of the type of gas involved.

For solids, added heat is used to increase the amount of vibrational energy of the particles making up the solid. The equipartition theorem predicts that the specific heat per mole of solid, measured at constant volume, should be equal to three times the gas constant. As with gases, the equipartition theorem correctly predicts the specific heat for solids at high temperatures but fails for low temperatures. When quantum theory is used to model the vibrational motion of the particles making up a solid, agreement is again achieved between theory and experiment at all temperatures. Constant pressure and constant volume specific heats for solids are usually close in value, but can in some cases differ from each other by as much as 50 percent. The difference between the constant pressure and constant volume specific heat can be calculated from general thermodynamic relationships; the difference depends on the compressibility of the solid and its change in volume, or thermal expansion, with temperature.

The change in the specific heat of a substance with temperature is also used as a means of classifying different types of phase transitions, or transformations to different forms of matter.

For a first-order phase transition, such as occurs when a substance is converted from the solid to liquid phase, the specific heat of the substance measured at constant pressure changes discontinuously, and is infinitely large at the temperature at which the phase transition occurs.

This results from the fact that all the heat being added is used to convert the substance from the initial to the final phase. Other types of phase transitions, such as the conversion of the normal liquid phase of helium to the superfluid liquid phase, show different behavior for the specific heat at the point where the phase transition occurs.

A second thermal property of substances is the change in the volume of a substance that occurs when heat is added. This volume change is usually expressed in terms of the coefficient of thermal expansion for the substance, defined as the relative change in the volume of the substance with temperature, measured under conditions of constant applied pressure. For an ideal gas, the coefficient of thermal expansion is equal to 1/T, where T is the temperature of the gas measured in Kelvins. Real gases can show deviations from this simple behavior. For solids and liquids, the coefficient of thermal expansion tends to be much smaller than for gases. Occasionally, substances will over some ranges of temperature decrease in volume as the temperature of the substance increases, as is the case for liquid water below 4 degrees Celsius.

Substances also differ from one another in the rate that they conduct heat. For a solid, the rate of heat flow can be determined by measuring the movement of heat through a rod of the substance whose two ends are held at different temperatures. The conduction of heat through such a rod is observed to be directly proportional to the temperature difference between the two ends of the rod, and to depend also on the physical properties of the substance composing the rod. Conduction of heat by liquids and gases is measured by observing the rate at which heat passes through a sample of the fluid when placed between two walls maintained at different temperatures. The ability of a substance to conduct heat is expressed in terms of the coefficient of thermal conductivity for the substance, which is the constant of proportionality that appears in the equation describing the heat flow through the substance.

Solids differ widely in their ability to conduct heat. Solids can be qualitatively divided into two categories: thermal insulators (which have low values for the coefficient of thermal conductivity and conduct heat slowly) and thermal conductors (which have large values for the coefficient of thermal conductivity and are efficient conductors of heat). The presence of trace amounts of impurities in an otherwise pure solid can have a large effect on the ability of the solid to conduct heat, particularly at low temperatures. Thermal conduction in solids takes place by two mechanisms: conduction by electrons in the solid and conduction by vibrational motion of the particles making up the substance. For crystals, these vibrations are called lattice vibrations.

Since thermal conduction by electrons is in general more efficient than conduction through lattice vibrations, metals--which possess a large number of free electrons--tend to be good conductors of heat, while solids without large numbers of free electrons, such as salts, are thermal insulators. The theory used to describe electrical conduction in metals can be modified to describe thermal conduction by electrons as well.

For gases, thermal conductivity is observed to be independent of gas pressure except at extremely low pressures, when the thermal conductivity is directly proportional to the pressure of the gas. This surprising result can be understood by consideration of the mechanism by which heat flow occurs. The rate of heat flow through a gas depends both on the number of gas molecules available to transport heat through the gas and the average distance a gas molecule moves through the gas before undergoing a collision, and giving up part of its energy to another molecule. As the pressure of a gas increases, the number of molecules available to carry thermal energy also increases. Nevertheless, because the density of a gas increases with pressure, the average distance a molecule in a gas can travel before colliding with another molecule decreases with increasing pressure. For most pressures, the two effects cancel, resulting in no net change in the thermal conductivity of the gas. At very low pressures, however, the probability of a gas molecule colliding with another molecule becomes negligible, and the rate of heat conduction then becomes proportional to the density of molecules in the gas. Vacuum represents an extremely efficient thermal insulator, since heat can move through a vacuum only by radiative processes.

Applications

Substances respond in different ways to addition of heat and changes in temperature.

Substances also differ in their ability to conduct heat. These differences can be exploited for practical benefits. Substances with a large value for specific heat can be used as coolants to remove heat from systems. One common coolant is water, which has an unusually large specific heat on a per gram basis, approximately ten times that of iron. This makes water an excellent substance for the removal of heat, and it is used as a coolant in everything from automobile engines to some types of nuclear power reactors. The large quantity of water found in the environment has a moderating effect on the temperature of the earth, tending to minimize variations of temperature with latitude or time of day. Other liquids with a large value for specific heat, such as sodium (which melts at 97 degrees Celsius) and ammonia (a gas that can be liquefied by the application of pressure), are also used as coolants in specialized applications.

Knowledge of the specific heat of materials is used in scientific applications as well. A common method for determining the heat evolved or taken up during a chemical reaction is calorimetry. In bomb calorimetry, for example, a sample of a compound is burned in a closed container in the presence of several atmospheres of pure oxygen. If the specific heat of the system is known, the change in temperature observed when combustion takes place can be used to calculate the energy of combustion for the reaction. Calorimetry represents the most common method for obtaining precise information on the thermodynamic properties of matter.

The conducting and insulating properties of substances have a variety of practical applications. In many cases, such as in electronic chips and cutting tools, it is important to remove heat as quickly as possible. While metals are in general good thermal conductors, other substances are even better conductors of heat. Diamond, for example, is four times more efficient at conducting heat at room temperature than copper and has been used in cases where rapid removal of heat is of critical importance. In other cases, the desire is to prevent the flow of heat into or out of a system. Materials such as Styrofoam are used to hold hot and cold beverages, or to keep foods warm, while foam insulation is used to minimize heat flow between a building and its surroundings. The insulating properties of vacuum are also used to minimize heat flow, as in a Thermos bottle. Specially designed insulating tiles are used in the space shuttle to shield it from the intense heat generated upon reentry into Earth's atmosphere.

The change in the volume of a substance with temperature can also be put to practical use. A mercury thermometer makes use of the fact that the change in volume for a fixed quantity of mercury is, to a first approximation, proportional to the difference between the temperature of the mercury and some arbitrary reference temperature. By confining the mercury to a narrow tube, small changes in the volume occupied by the mercury can be detected easily. Mercury thermometers were among the first devices developed to measure temperature and are still the most common devices for measuring temperature. Since solid metals also expand with increasing temperature, metal coils are often used in thermostats for regulating the temperature of a room or building. The change in the length of the coil with temperature can be used as a switch to turn a heater or air conditioner on and off.

Context

The study of the thermal properties of matter has played a central role in the development of thermodynamics, statistical mechanics, kinetic theory of gases, quantum theory, and solid-state physics. In case after case, the failure of theory to explain experimental observations of thermal properties has resulted in advances in the understanding of the laws of nature.

Qualitative observations of the relationship between heat, temperature, and volume date back to ancient times. In the seventeenth century, Galileo developed the first thermometer, using the change in the volume of a gas to measure temperature. Improvements in the thermometer occurred when liquids replaced gases as the working fluid. In 1714, Daniel Gabriel Fahrenheit devised the first mercury thermometer.

In the early nineteenth century, two French scientists, Pierre-Louis Dulong and Alexis-Therese Petit, developed a procedure for the determination of the molecular weight of an element based on measurements of specific heat. Dulong and Petit observed that for elemental solids with known molecular weights, the specific heat per mole of element was always approximately 25 joules per Kelvin. Assuming that this number was a universal constant, the molecular weight for a newly discovered element could be determined by measurement of its specific heat. This method was in fact used to determine the molecular weight of a number of elements.

At about the same period of time, consideration of the thermal properties of matter was leading to the general relationships summarized in the laws of thermodynamics. One consequence of this development was the derivation of exact expressions for the difference between the specific heat of a substance measured at constant volume and that measured at constant external pressure. When statistical thermodynamics was developed by James Clerk Maxwell, Ludwig Boltzmann, and others in the late 1800's, one of the results was the equipartition theorem, which made it possible to predict the value for the specific heat of a substance from theory.

While some substances were found to have the value of heat capacity predicted from statistical mechanics, discrepancies between prediction and experiment soon appeared. The specific heat for diatomic and polyatomic gases was far below the predicted value, although agreement between theory and experiment improved at high temperatures. For solids, it was found that at low temperatures, the specific heat was also smaller than calculated from theory.

Also, as noted by the American chemist Josiah Willard Gibbs, there seemed to be no contribution to the heat capacity of a metal from its free electrons.

In 1907, Albert Einstein recalculated the specific heat for a solid, assuming that the vibrational energy for the particles making up the solid was quantized, that is, that it could only take on particular values. The quantization of energy had previously been used by Max Planck to explain black body radiation and by Einstein to explain the photoelectric effect. Using the assumption of quantized values for vibrational energy, Einstein was able to predict the specific heat of a solid even at low temperatures. The success of Einstein's model in predicting the specific heats of solids was one factor leading to the development of quantum mechanics.

Modification of Einstein's model by Peter J. W. Debye and others soon led to better agreement between theory and experiment.

Following the development of quantum mechanics in the 1920's, the combination of quantum mechanics and statistical mechanics was used to advance the understanding of thermal properties of matter. At temperatures approaching absolute zero, the spin properties of the particles making up a substance become an important factor in determining the specific heat and thermal conductivity. In the 1940's, Lars Onsager and others developed general methods for describing heat flow through matter. These methods have made it possible to understand heat flow in different types of matter under a variety of conditions of temperature and pressure.

Principal terms

CONDUCTOR: a substance, such as a metal, through which heat can flow at a rapid rate

EQUIPARTITION THEOREM: a classical theorem that predicts the value of the specific heat for substances, which is obeyed for most substances in the limit of high temperature

HEAT: the energy that flows between substances because of their difference in temperature

HEAT CAPACITY: the ratio of the heat added to a system divided by the resulting temperature change of the system

INSULATOR: a substance that is a poor conductor of heat, such as an ionic solid

MOLE: a unit for quantity equal to the number of atoms in a 12-gram sample of isotopically pure carbon, approximately 6 x 10²³

SPECIFIC HEAT: the heat capacity for a specified amount of substance, either 1 gram of substance (specific heat per gram) or 1 mole of substance (specific heat per mole)

TEMPERATURE: a measure of the magnitude of thermal energy contained in a quantity of matter

THERMAL CONDUCTIVITY: the flow of heat through a substance, caused by a difference of temperature

Bibliography

Hemminger, Wolfgang, and Gunther Hoehne. CALORIMETRY. Translated by Y. Goldman. Deerfield Beach, Fla.: Verlag Chemie, 1984. A detailed survey of the theory and application of calorimetry to problems in physics and chemistry. There is a good discussion of how heat capacity is related to calorimetry.

McClintock, P. V. E., D. J. Meredith, and J. K. Wigmore. MATTER AT LOW TEMPERATURES. New York: Wiley, 1984. The introductory chapters of this book give the basic theoretical background for the heat capacity and thermal conductivity of matter. The discussion is well organized, although at a fairly sophisticated level. The remainder of the book is a survey of low-temperature properties of matter, with emphasis on the unusual behavior of liquid helium at low temperatures.

Mott-Smith, Morton. HEAT AND ITS WORKINGS. New York: Dover Press, 1963. A general introduction to thermodynamics for nonscientists. There are a large number of illustrations, and the discussion is supplemented with examples from everyday experience. Chapter 4 focuses on heat and heat capacity.

Yates, Bernard. THERMAL EXPANSION. New York: Plenum Press, 1972. A discussion of the thermal properties of solids. The emphasis is on the change in volume of solids with temperature, and includes sections on the properties of metals, salts, glasses, polymers, and semiconductors. There is also a brief discussion of specific heat and its connection to thermal expansion.

Zemansky, Mark W. HEAT AND THERMODYNAMICS. New York: McGraw-Hill, 1968. One of the best introductory texts for thermodynamics. The book presents both classical and statistical thermodynamic theory. Chapter 4 is devoted to heat, and includes a discussion of heat capacity and its measurement. Heat capacity and other thermal properties of matter as applied to thermodynamics also appear in other sections of the book.

The Behavior of Gases

Thermal Properties of Solids