Thermodynamics, within physics, is the science that deals with work and heat, and the transformation of one into the other. It is a macroscopic theory, dealing with matter in bulk, disregarding the molecular nature of materials. The corresponding microscopic theory, based on the fact that materials are made up of a vast number of molecules, is called statistical mechanics.

Historical background

American physicist Benjamin Thompson (Count von Rumford) (1753–1814) recognized from observing the boring of cannons that the work (or mechanical energy) involved in the boring process was being

Table 1. Thermodynamics (Thomson Gale.)
Thermodynamics
Material	Material point °C	Heat of fusion cal/gm	Boiling point °C	Heat of vaporization cal/gm
Water	0	79.7	100	539
Ethyl alcohol	–114	24.9	78	204
Oxygen	–219	3.3	–183	51
Nitrogen	–210	6.1	–196	48
Mercury	–39	2.8	357	65

converted to heat by friction, causing the temperature of the cannon to rise. With the experiments of English physicist James Joule (1818–1889), it was recognized that heat is a form of energy that is transferred from one object to another, and that work can be converted to heat without limit. However, the opposite is found not to be true: that is, there are limiting factors in the conversion of heat to work. The research of French physicist Nicolas Leonard Sadi Carnot (1796–1832), of British mathematician and physicist William Thomson (Lord Kelvin) (1824–1907), and of German mathematical physicist Rudolph Clausius (1822–1888), among others, has led to an understanding of these limitations.

Temperature

the idea of temperature is well known to everyone, but the need to define it so that it can be used for measurements is far more complex than the simple concepts of hot and cold. If a rod of metal is placed in an ice-water bath and the length is measured, and then placed in a steam bath and the length again measured, it will be found that the rod has lengthened. This is an illustration of the fact that, in general, materials expand when heated, and contract when cooled (however, under some conditions rubber can do the opposite, while water is a very special case and is treated below). One could therefore use the length of a rod as a measure of temperature. However, that would not be useful, since different materials expand different amounts for the same increase in temperature. Therefore, everyone would need to have exactly the same type of rod to make certain that they obtained the same value of temperature under the same conditions.

However, it turns out that practically all gases, at sufficiently low pressures, expand in volume exactly the same amount with a given increase in temperature. This has given rise to the constant volume gas thermometer, which consists of a flask to hold the gas, attached to a system of glass and rubber tubes containing mercury. A small amount of any gas is introduced into the (otherwise empty) flask, and the top of the mercury in the glass column on the left is placed at some mark on the glass (by moving the right hand glass column up or down). The difference between the heights of the two mercury columns gives the difference between atmospheric pressure and the pressure of the gas in the flask. The gas pressure changes with a change in temperature of the flask, and can be used as a definition of the temperature by taking the temperature to be proportional to the pressure; the proportionality factor can be found in the following manner.

If the temperature at the freezing point of water is assigned the value 0° and that at the boiling point is called 100°, the temperature scale is called the Celsius scale (formerly called Centigrade); if those points are taken at 32° and 212°, it is known as the Fahrenheit scale. The relationship between them can be found as follows. If the temperature in the Celsius scale is T(°C), and that in the Fahrenheit scale is T(°F), they are related by T(°F) = (9/5) T(°C) + 32°. The importance of using the constant volume gas thermometer to define the temperature is that it gives the same value for the temperature no matter what gas is used (as long as the gas is used at a very low pressure), so that anyone at any laboratory would be able to find the same temperature under the same conditions. Of course, a variety of other types of thermometers are used in practice (mercury-in-glass, or the change in the electrical resistance of a wire, for example), but they all must be calibrated against a constant volume gas thermometer as the standard.

Expansion coefficients

An important characteristic of a material is how much it expands for a given increase in temperature. The amount that a rod of material lengthens is given by L= L₀ [1+α (T–T₀)], where L₀ is the length of the rod at some temperature T₀, and L is the length at some other temperature T;ψα (Greek alpha) is called the coefficient of linear expansion. Some typical values for α × 10⁶ (per °C) are: aluminum, 24.0; copper, 16.8; glass, 8.5; steel, 29.0 (this notation means that, for example, aluminum expands at a rate of 24.0/1,000,000 for each degree Celsius change in temperature). Volumes, of course, also expand with a rise in temperature, obeying a law similar to that for linear expansion; coefficients of volume expansion are approximately three times as large as that for linear expansion for the same material. It is interesting to note that, if a hole is cut in a piece of material, the hole expands just as if there were the same material filling it!

Thermostats

since various metals expand at different rates, a thermostat can be made to measure changes in temperature by securely fastening together two strips of metal with different expansion coefficients. If they are straight at one temperature, they will be bent at any other temperature, since one will have expanded or contracted more than the other. These are used in many homes to regulate the temperature by causing an electrical contact to be made or broken as temperature changes cause the end of the strips to move.

Water

water has the usual property of contracting when the temperature decreases, but only down to 39.2°F (4°C); below that temperature it expands until it reaches 32°F(0°C). It then forms ice at 0°C, expanding considerably in the process; the ice then behaves normally, contracting as the temperature decreases. Since the density of a substance varies inversely to the volume (as a given mass of a substance expands, its density decreases), this means that the density of water increases as the temperature decreases until 4°C, when it reaches its maximum density. The density of the water then decreases from 4 to 0°C; the formation of the ice also involves a decrease in density. The ice then increases its density as its temperature falls below 0°C. Thus, as a lake gets colder, the water at the top cools off and, since its density is increasing, this colder water sinks to the bottom. However, when the temperature of the water at the top becomes lower than 4°C, it remains at the top since its density is lower than that of the water below it. The pond then ices over, with the ice remaining at the top, while the water below remains at 4°C (until, if ever, the entire lake freezes). Fish are thus able to live in lakes even when ice forms at the top, since they have the 4°C water below it to live in.

Conservation of energy

The conservation of energy is well known from mechanics, where energy does not disappear but only changes its form. For example, the potential energy of an object at some height is converted to the kinetic energy of its motion as it falls. Thermodynamics is concerned with the internal energy of an object and those things that affect it; conservation of energy applies in this case, as well.

Heat

as noted in the introduction, doing work on an object (for example, by drilling a hole in a piece of metal, or by repeatedly bending it) causes its temperature to rise. If this object is placed in contact with a cooler object it is found that they eventually come to the same temperature, and remain that way as long as there are no outside influences (this is known as thermal equilibrium). This series of events is viewed as follows. Consistent with the concept of the conservation of energy, the energy due to the work done on the object is considered to be stored in the object as (what may be called) internal energy. In the particular example above, the increase in the internal energy of the object is recognized by the increase in temperature, but there are processes where the internal energy increases without a change in temperature. By then placing it in contact with an object of lower temperature, energy flows from the hotter to the colder one in the form of heat, until the temperatures become the same. Thus, heat should be viewed as a type of energy that can flow from one object to another by virtue of a temperature difference. It makes no sense to talk of an object having a certain amount of heat in it; whenever it is placed in contact with a lower-temperature object, heat will flow from the hotter to the cooler one.

The first law of thermodynamics

These considerations may be summarized in the first law of thermodynamics: the internal energy of an object is increased by the amount of work done on it, and by the amount of heat added to it. Mathematically, if U_f is the internal energy of an object at the end of some process, and U_i is the internal energy at the beginning of the process, then U_f– U_i= W + Q, where W is the amount of work done on the object, and Q is the amount of heat added to the object (negative values are used if work is done by the object, or heat is transferred from the object). As is usual for an equation, all quantities must be expressed in the same units; the usual mechanical unit for energy (in the International System of Units (SI)—formerly the MKS system) is the joule, where 1 joule equals 1 kg-m²/s².

Specific heats; the calorie

An important characteristic of materials is how much energy in the form of heat it takes to raise the temperature of some material by one degree. It depends upon the type of material being heated as well as its amount. The traditional basic unit, the calorie, is defined as the amount of heat that is needed to raise one gram of water by one degree Celsius. In terms of mechanical energy units, one calorie equals 4.186 joules (J).

The corresponding amount of heat necessary to raise the temperature of other materials is given by the specific heat capacity of a material, usually denoted by c. It is the number of kilojoules (kJ) needed to raise 1 kg of the material by 1°C. By definition, the value for water is 4.186 kilojoules. Typical values for c in kilojoules per kg (kJ/kg), at 0°C, are: ice, 2.11; aluminum, 0.88; copper, 0.38; iron, 0.45. It should be noted that water needs more heat to bring about a given rise in temperature than most other common substances.

Change of phase

The process of water changing to ice or to steam is a familiar one, and each is an example of a change in phase. Suppose a piece of ice were placed in a container and heated at a uniform rate; that is, a constant amount of heat per second is transferred to the material in the container. The ice (the solid phase of water) first rises in temperature at a uniform rate until its temperature reaches 32°F(0°C), when it begins to melt, that is, some of the ice changes to water (in its liquid phase); this temperature is called the melting point. It is important to note that the temperature of the ice-water mixture remains at 32°F(0°C) until all the ice has turned to water. The water temperature then rises until it reaches 212°F(100°C), when it begins to vaporize, that is, turns to steam (the gaseous phase of water); this temperature is called the boiling point. Again, the water-steam mixture remains at 212°F(100°C) until all the liquid water turns into steam. Thereafter, the temperature of the steam rises as more heat is transferred to the container. It is important to recognize that during a change in phase the temperature of the mixture remains constant. (The energy being transferred to the mixture goes into breaking molecular bonds rather than in increasing the temperature.)

Many substances undergo similar changes in phase as heat is applied, going from solid to liquid to gas, with the temperature remaining constant during each phase change. (Some substances, such as glass, do not have such a well-defined melting point.) The amount of heat needed to melt a gram of a material is known as the heat of fusion; that to vaporize it is the heat of vaporization. On the other hand, if steam is cooled at a uniform rate, it would turn to liquid water at the condensation temperature (equal to the boiling point, 212°F [100°C]), and then turn to ice at the solidification temperature (equal to the melting point, 32°F [0°C]). The heat of condensation is the amount of heat needed to be taken from a gram of a gas to change it to its liquid phase; it is equal to the heat of vaporization. Similarly, there is a heat of solidification that is equal to the heat of fusion. Some typical values are shown in Table 1.

It is interesting to note that water has much larger heats of fusion and of vaporization than many other usual substances. The melting and boiling points depend upon the pressure (the values given in the table are for atmospheric pressure). It is for this reason that water boils at a lower temperature in high-altitude Denver, Colorado, than at sea level in New Orleans, Louisiana.

Finally, below certain pressures it is possible for a substance to change directly from the solid phase to the gaseous one; this case of sublimation is best illustrated by the disappearance of dry ice when it is exposed to the atmosphere.

Equations of state; work

When an object of interest (usually called the system) is left alone for a sufficiently long time, and is subject to no outside influences from the surroundings, measurements of the properties of the object do not change with time; it is in a state of thermal equilibrium. It is found experimentally that there are certain measurable quantities that give complete information about the state of the system in thermal equilibrium (this is similar to the idea that measurements of the velocity and acceleration of an object give complete information about the mechanical state of a system). For each state, relationships can be found that hold true over a wide range of values of the quantities. These relationships are known as equations of state.

Equations of state

Thermodynamics applies to many different types of systems; gases, elastic solids (solids that can be stretched and return to their original form when the stretching force is removed), and mixtures of chemicals are all examples of such systems. Each system has its own equation of state, which depends upon the variables that need to be measured in order to describe its internal state. The relevant variables for a system can only be determined by experiment, but one of those variables will always be the temperature.

The system usually given as an example is a gas, where the relevant thermodynamic variables are the pressure of the gas (P), its volume (V), and, of course, the temperature (T). (These variables are the relevant ones for any simple chemical system, e.g., water, in any of its phases.) The amount of gas may be specified in grams or kilograms, but the usual way of measuring mass in thermodynamics (as well as in some other fields) is in terms of the number of moles. One kilo-mole (kmol) is defined as equal to M kilograms, where M is the molecular weight of the substance, with carbon-12 being taken as M = 12. (One mole of any substance contains 6.02× 10²³ molecules, known as Avogadro’s number.) Thus, one kilomole of oxygen has a mass of 70.56 lb (32 kg); of nitrogen, 61.76 lb (28.01 kg); the molar mass of air (which is, of course, actually a mixture of gases) is commonly taken as 63.87 lb (28.97 kg). It is found, by experiment, that most gases at sufficiently low pressures have an equation of state of the form: PV = NRT, where P is in Newtons/m², V is in m³, N is the number of kilomoles of the gas, T is the temperature in K, and R = 8.31 kJ/kmol-K is known as the universal gas constant. The temperature is in Kelvin (K), which is given in terms of the Celsius temperature as T(K)= T(°C)+ 273.15°C. It should be noted that real gases obey this ideal gas equation of state to within a few percent accuracy at atmospheric pressure and below.

The equation of state of substances other than gases is more complicated than the above ideal gas law. For example, an elastic solid has an equation of state that involves the length of the stretched material, the stretching force, and the temperature, in a relationship somewhat more complex than the ideal gas law.

Work

work is defined in mechanics in terms of force acting over a distance; that definition is exactly the same in thermodynamics. This is best illustrated by calculating the work done by a force F in compressing a volume of gas. If a volume (V) of gas is contained in a cylinder at pressure (P), the force needed on the piston is (by the definition of pressure) equal to PA, where A is the area of the piston. Let the gas now be compressed in a manner that keeps the pressure constant (by letting heat flow out, so that the temperature also decreases); suppose the piston moves a distance (d). Then the work done is W = Fd = PAd. However, Ad is the amount that the volume has decreased, V_i – V_f, where V_i is the initial volume and V_f is the final volume. (Note that this volume difference gives a positive value for the distance, in keeping with the fact that work done on a gas is taken as positive.) Therefore, the work done on a gas during a compression at constant pressure is P(V_i– V_f).

The first law thus gives a straightforward means to determine changes in the internal energy of an object (and it is only changes in the internal energy that can be measured), since the change in internal energy is just equal to the work done on the object in the absence of any heat flow. Heat flow to or from the object can be minimized by using insulating materials, such as fiberglass or, even better, styrofoam. The idealized process where there is zero heat flow is called an adiabatic process.

The second law of thermodynamics

One of the most remarkable facts of nature is that certain processes take place in only one direction. For example, if a high temperature object is placed in contact with one of lower temperature, heat flows from the hotter to the cooler until the temperatures become equal. In this case (where there is no work done), the first law simply requires that the energy lost by one object should be equal to that gained by the other object (through the mechanism of heat flow), but does not prescribe the direction of the energy flow. Yet, in a situation like this, heat never flows from the cooler to the hotter object. Similarly, when a drop of ink is placed in a glass of water that is then stirred, the ink distributes itself throughout the water. Yet no amount of stirring will make the uniformly distributed ink go back into a single drop. An open bottle of perfume placed in the corner of a room will soon fill the room with its scent, yet a room filled with perfume scent will never become scent-free with the perfume having gone back into the bottle. These are all examples of the second law of thermodynamics, which is usually stated in two different ways. Although the two statements appear quite different, it can be shown that they are equivalent and that each one implies the other.

Clausius statement of the second law

The Clausius statement of the second law is: No process is possible whose only result is the transfer of heat from a cooler to a hotter object. The most common example of the transfer of heat from a cooler object to a hotter one is the refrigerator (air conditioners and heat pumps work the same way). When, for example, a bottle of milk is placed in a refrigerator, the refrigerator takes the heat from the bottle of milk and transfers it to the warmer kitchen. (Similarly, a heat pump takes heat from the cool ground and transfers it to the warmer interior of a house.) An idealized view of the refrigerator is as follows. The heat transfer is accomplished by having a motor, driven by an electrical current, run a compressor. A gas is compressed to a liquid, a phase change that generates heat (heat is taken from the gas to turn it into its liquid state). This heat is dissipated to the kitchen by passing through tubes (the condenser) in the back of (or underneath) the refrigerator. The liquid passes through a valve into a low-pressure region, where it expands and becomes a gas, and flows through tubes inside the refrigerator. This change in phase from a liquid to a gas is a process that absorbs heat, thus cooling whatever is in the refrigerator. The gas then returns to the compressor where it is again turned into a liquid. The Clausius statement of the second law asserts that the process can only take place by doing work on the system; this work is provided by the motor that drives the compressor. However, the process can be quite efficient, and considerably more energy in the form of heat can be taken from the cold object than the work required to do it.

Kelvin-Planck statement of the second law

Another statement of the second law is due to Lord Kelvin and German physicist Maxwell Planck (1858– 1947): No process is possible whose only result is the conversion of heat into an equivalent amount of work. Suppose that a cylinder of gas fitted with a piston had heat added, which caused the gas to expand. Such an expansion could, for example, raise a weight, resulting in work being done. However, at the end of that process the gas would be in a different state (expanded) than the one in which it started, so that this conversion of all the heat into work had the additional result of expanding the working fluid (in this case, the gas).

If the gas were, on the other hand, then made to return to its original volume, it could do so in three possible ways: (a) the same amount of work could be used to compress the gas, and the same amount of heat as was originally added would then be released from the cylinder; (b) if the cylinder were insulated so that no heat could escape, then the end result would be that the gas is at a higher temperature than originally; (c) something in-between. In the first case, there is no net work output or heat input. In the second, all the work was used to increase the internal energy of the gas, so that there is no net work and the gas is in a different state from which it started. Finally, in the third case, the gas could be returned to its original state by allowing some heat to be transferred from the cylinder. In this case the amount of heat originally added to the gas would equal the work done by the gas plus the heat removed (the first law requires this). Thus, the only way in which heat could be (partially) turned into work and the working fluid returned to its original state is if some heat were rejected to an object having a temperature lower than the heating object (so that the change of heat into work is not the only result). This is the principle of the heat engine (an internal combustion engine or a steam engine are examples).

Heat engines

The working fluid (say, water for a steam engine) of the heat engine receives heat Q_h from the burning fuel (diesel oil, for example), which converts it to steam. The steam expands, pushing on the piston so that it does work, W; as it expands, it cools and the pressure decreases. It then traverses a condenser, where it loses an amount of heat Q_c to the coolant (cooling water or the atmosphere, for example), which returns it to the liquid state. The second law says that,

KEY TERMS

Adiabatic process —A process during which no heat is transferred between the system and surroundings is described as adiabatic.

Avogadro’s number —The number of molecules present in one mole of whatever the compound is always equal to 6.0229 × 10²³. It was named for Italian physicist Amedeo Avogadro.

Boiling point —The boiling point of a liquid is the temperature at which it boils, also the temperature at which its vapor condenses.

Calorie —The amount of heat necessary to increase the temperature of water by one degree Celsius.

Celsius temperature (°C) —The temperature scale on which the freezing point of water is 0° and the boiling point is 100°.

Change in phase —Change in the form and characteristics of a substance, e.g., changes from gas to liquid, or liquid to solid.

Coefficient of linear expansion —The fractional rate of change in length of an object with a change in temperature.

Condensation temperature —The temperature at which a gas changes into a liquid (equal to the boiling point).

Equation of state —Relationship among the (experimentally determined) variables, which give complete information about the state of a system.

Fahrenheit temperature (°F) —The temperature scale on which the freezing point of water is 32°and the boiling point is 212°.

First law of thermodynamics —The internal energy of a system is increased by the amount of work done on the system and the heat flow to the system (conservation of energy).

Heat of condensation —The amount of heat needed to be removed from a gas to change it to its liquid phase (equal to the heat of vaporization).

Heat of fusion —The amount of heat needed to be added to a solid to change it to its liquid phase.

Heat of solidification —The amount of heat needed to be removed from a liquid to change it to its solid phase (equal to the heat of fusion).

Heat of vaporization —The amount of heat needed to be added to a liquid to change it to its gaseous phase.

Ideal gas— A gas obeying the ideal gas equation of state, pV = nRT, where, e.g., p is in Newtons/meter², Visinm³, n is the number of kilomoles of the gas, T is the temperature in K, and R = 8.31 kJ/kmolK.

Internal energy —The change in the internal energy of a system is equal to the amount of adiabatic work done on the system.

Kelvin temperature (K) —The Celsius temperature plus 273.15°C.

Kilomole (kmol) —A quantity of matter equal to M kilograms, where M is the molecular weight of the substance, with carbon-12 being taken as M←= 12 (one kilomole equals 1,000 moles).

Macroscopic theory —A theory which ignores the molecular nature of matter.

Melting point —The temperature at which a solid changes into a liquid.

Microscopic theory —A theory that is based on the molecular nature of matter.

Second law of thermodynamics —No process is possible whose only result is the transfer of heat from a cooler to a hotter object (Clausius statement). No process is possible whose only result is the conversion of heat into an equivalent amount of work (Kelvin Planck statement).

Solidification temperature —The temperature at which a liquid changes into a solid (equal to the melting point).

Specific heat —The amount of heat needed to increase the temperature of a mass of material by one degree.

Statistical mechanics —The microscopic theory of matter for which the macroscopic theory is thermodynamics, or the molecular basis of thermodynamics.

Sublimation —The change of a material from its solid phase directly to its gaseous phase.

Temperature (T) —The (experimentally determined) variable which determines the direction of heat flow; the variable which is common to all equations of state.

Thermal equilibrium —A condition between two or more objects in direct thermal contact in which no energy as heat flows from one to the other. The temperatures of such objects are identical.

Universal gas constant (R) —The constant in the ideal gas equation of state (as well as elsewhere); equal to 8.31 kJ/kmolK.

if the working fluid (in this case the water) is to be returned to its original state so that the heat-work process could begin all over again, then some heat must be rejected to the coolant. Since the working fluid is returned to its original state, there is no change in its internal energy, so that the first law demands that Q_h – Q_c = W. The efficiency of the process is the amount of work obtained for a given cost in heat input: E = W/Q_h. Thus, combining the two laws, E = (Q_h – Q_c)/Q_h. It can be seen therefore that a heat engine can never run at 100% efficiency.

It is important to note that the laws of thermodynamics are of very great generality, and are of importance in understanding such diverse subjects as chemical reactions, very low temperature phenomena, and the changes in the internal structure of solids with changes in temperature, as well as engines of various kinds.

See also Gases, properties of.