# thermodynamics

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## thermodynamics,

branch of science concerned with the nature of heat**heat,**

nonmechanical energy in transit, associated with differences in temperature between a system and its surroundings or between parts of the same system.

**Measures of Heat**

**.....**Click the link for more information. and its conversion to mechanical, electric, and chemical energy

**energy,**

in physics, the ability or capacity to do work or to produce change. Forms of energy include heat, light, sound, electricity, and chemical energy. Energy and work are measured in the same units—foot-pounds, joules, ergs, or some other, depending on the system of

**.....**Click the link for more information. . Historically, it grew out of efforts to construct more efficient heat engines—devices for extracting useful work from expanding hot gases.

### The Thermodynamic System and Its Environment

In thermodynamics, one usually considers both the thermodynamic system and its environment. The environment often contains one or more idealized heat reservoirs—heat sources with infinite heat capacity enabling them to give up or absorb heat without changing their temperature. (An ocean or other large body of water approximates a heat reservoir.) A typical thermodynamic system is a definite quantity of gas enclosed in a cylinder with a sliding piston that allows the volume to vary. In general, a thermodynamic system is defined by its temperature, volume, pressure, and chemical composition. A system is in equilibrium when these variables have the same value at all points.

A mathematical statement that links the variables to show their interdependence is called an equation of state; the gas laws**gas laws,**

physical laws describing the behavior of a gas under various conditions of pressure, volume, and temperature. Experimental results indicate that all real gases behave in approximately the same manner, having their volume reduced by about the same proportion of the**.....** Click the link for more information. are simple examples of such equations. Equations of state take on their simplest form when the Kelvin temperature scale**Kelvin temperature scale,**

a temperature scale having an absolute zero below which temperatures do not exist. Absolute zero, or 0°K;, is the temperature at which molecular energy is a minimum, and it corresponds to a temperature of −273.**.....** Click the link for more information. is used; on this scale 0° corresponds to the lowest temperature theoretically possible.

When the external conditions are altered, a thermodynamic system will respond by changing its state; the temperature, volume, pressure, and chemical composition will adjust to a new equilibrium. The most important kinds of changes are adiabatic and isothermal changes. An adiabatic change is one that occurs without any flow of heat. The system is thermally insulated from the environment, and the first law of thermodynamics requires that the work done by or on the system be equal to the loss or gain of the system's internal energy. An isothermal change occurs when the system is in contact with a heat reservoir, so that the system remains at the temperature of the reservoir. In the isothermal process, heat flows from the reservoir if the system is expanding and into the reservoir if the system is being compressed. For an ideal gas the internal energy depends only on the temperature; hence the internal energy remains constant during an isothermal change, and the heat absorbed from or by the reservoir is equal to the work done on or by the environment.

### The First Law of Thermodynamics

Toward the middle of the 19th cent. heat was recognized as a form of energy associated with the motion of the molecules of a body (see kinetic-molecular theory of gases**kinetic-molecular theory of gases,**

physical theory that explains the behavior of gases on the basis of the following assumptions: (1) Any gas is composed of a very large number of very tiny particles called molecules; (2) The molecules are very far apart compared to their sizes,**.....** Click the link for more information. ). Speaking more strictly, heat refers only to energy that is being transferred from one body to another. The total energy a body contains as a result of the positions and motions of its molecules is called its internal energy; in general, a body's temperature**temperature,**

measure of the relative warmth or coolness of an object. Temperature is measured by means of a thermometer or other instrument having a scale calibrated in units called degrees. The size of a degree depends on the particular temperature scale being used.**.....** Click the link for more information. is a direct measure of its internal energy. All bodies can increase their internal energies by absorbing heat (see heat capacity**heat capacity**

or **thermal capacity,**

ratio of the change in heat energy of a unit mass of a substance to the change in temperature of the substance; like its melting point or boiling point, the heat capacity is a characteristic of a substance.**.....** Click the link for more information. ). However, mechanical work done on a body can also increase its internal energy; e.g., the internal energy of a gas increases when the gas is compressed. Conversely, internal energy can be converted into mechanical energy; e.g., when a gas expands it does work on the external environment. In general, the change in a body's internal energy is equal to the heat absorbed from the environment minus the work done on the environment. This statement constitutes the first law of thermodynamics, which is a general form of the law of conservation of energy (see conservation laws**conservation laws,**

in physics, basic laws that together determine which processes can or cannot occur in nature; each law maintains that the total value of the quantity governed by that law, e.g., mass or energy, remains unchanged during physical processes.**.....** Click the link for more information. ).

### The Second Law of Thermodynamics

A cyclic process is one that returns the system, but not the environment, to its original state. A closed cycle consisting of two isothermal and two adiabatic transformations is called a **Carnot cycle** after the French physicist Sadi Carnot**Carnot, Sadi**

, 1837–94, French statesman, president of the Third Republic (1887–94); son of Hippolyte Carnot. As minister of public works (1880–85) and of finance (1886), he remained untainted by the financial scandals of the time.**.....** Click the link for more information. , who first discussed the implications of such cycles. During the Carnot cycle occurring in the operation of a heat engine, a definite quantity of heat is absorbed from a reservoir at high temperature; part of this heat is converted into useful work, but the balance is expelled into a low-temperature reservoir and thus "wasted." The greater the temperature difference between the two reservoirs, which in a steam engine are represented by the boiler and the condenser, the greater the fraction of absorbed heat that is converted into useful work. It is, however, theoretically impossible to convert all the heat extracted from the reservoir into useful work.

In general it is impossible to perform a transformation whose only final result is to convert into useful work heat extracted from a source that is at the same temperature throughout. This statement is Lord Kelvin's version of the second law of thermodynamics. Another version of this law, formulated by R. J. E. Clausius, states that a transformation is impossible whose only final result is to transfer heat from a body at a given temperature to a body at higher temperature; in other words, the spontaneous flow of heat from hot to cold bodies is reversible only with the expenditure of mechanical or other nonthermal energy. These two versions of the second law of thermodynamics can be shown to be entirely equivalent.

The second law is expressed mathematically in terms of the concept of entropy**entropy**

, quantity specifying the amount of disorder or randomness in a system bearing energy or information. Originally defined in thermodynamics in terms of heat and temperature, entropy indicates the degree to which a given quantity of thermal energy is available for doing**.....** Click the link for more information. . When a body absorbs an amount of heat *Q* from a reservoir at temperature *T,* the body gains and the reservoir loses an amount of entropy *S*=*Q/T.* Thus, in a reversible adiabatic process (no heat change) there is no change in the total entropy. If an amount of heat *Q* flows from a hot to a cold body, the total entropy increases; because *S*=*Q/T* is larger for smaller values of *T,* the cold body gains more entropy than the hot body loses. The statement that heat never flows from a cold to a hot body can be generalized by saying that in no spontaneous process does the total entropy decrease.

In all real physical processes entropy increases; in ideal reversible processes entropy remains constant. Thus, in the Carnot cycle, which is reversible, there is no change in the total entropy. The engine itself experiences no net change in entropy because it is returned to its original state at the end of the cycle. The entropy gained by the low temperature reservoir is equal to the entropy lost by the high temperature reservoir. However, according to the formula *S*=*Q/T,* less heat need be expelled into the low temperature reservoir than is extracted from the high temperature reservoir for equal and opposite changes in entropy. In the Carnot cycle this difference in heat appears as useful mechanical work.

### The Third Law of Thermodynamics

A postulate related to but independent of the second law is that it is impossible to cool a body to absolute zero by any finite process. Although one can approach absolute zero as closely as one desires, one cannot actually reach this limit. The third law of thermodynamics, formulated by Walter Nernst and also known as the Nernst heat theorem, states that if one could reach absolute zero, all bodies would have the same entropy. In other words, a body at absolute zero could exist in only one possible state, which would possess a definite energy, called the zero-point energy. This state is defined as having zero entropy.

### Bibliography

See E. Fermi, *Thermodynamics* (1937); F. W. Sears, *Thermodynamics, the Kinetic Theory of Gases, and Statistical Mechanics* (2d ed. 1953); M. W. Zemansky, *Heat and Thermodynamics* (5th ed. 1968).

## Thermodynamics

the study of the most general properties of macroscopic systems in states of thermodynamic equilibrium and of the processes by which such systems pass from one equilibrium state to another. Thermodynamics is constructed on the basis of fundamental principles, or laws, that are generalizations from numerous observations and that are satisfied independently of the specific nature of the bodies forming a system. The regularities found by thermodynamics in the relations between physical quantities are therefore universal in character. The branch of physics known as statistical mechanics provides a substantiation of the laws of thermodynamics and gives their relation to the laws governing the motion of the particles that make up bodies. By means of statistical mechanics the limits of the applicability of thermodynamics can be ascertained.

** Equilibrium and nonequilibrium states.** An equilibrium state is, strictly speaking, the state arrived at by an isolated system after an infinitely long period of time. For practical purposes equilibrium is reached in a finite time (the relaxation time) that depends on the nature of the bodies, their interactions, and the initial non-equilibrium state. If a system is in a state of equilibrium, then its individual macroscopic parts are also in a state of equilibrium. Under constant external conditions, such a state does not vary with time. Invariance in time, however, is not a sufficient criterion for a state to be an equilibrium state. If, for example, a section of an electric circuit through which a direct current flows is placed in a thermostat, or heat reservoir, the section can remain in an unchanging, or steady, state for a practically unlimited time. This state, however, is not an equilibrium state, since the flow of the current is accompanied by the irreversible conversion of the energy of the electric current into heat that is transferred to the thermostat. A temperature gradient is present in the system. Open systems may also be in a steady nonequilibrium state.

The equilibrium state can be characterized completely by a small number of physical parameters. The most important of these parameters is temperature. For a system to be in thermodynamic equilibrium, all parts of the system must be at the same temperature. The existence of temperature—that is, a parameter that has the same value for all parts of a system in equilibrium—is often called the zeroth law of thermodynamics. The state of a homogeneous liquid or gas can be defined completely by specifying any two of the following three quantities: the temperature *T*, volume *V*, and pressure *p.* The relation between *p*, *V*, and *T* is characteristic of each given liquid or gas and is called the equation of state. Examples are the equation of state for an ideal gas and van der Waals’ equation. In more complex cases other parameters—such as the concentrations of the individual components of a mixture of gases, electric field strength, and magnetic induction—may be required to characterize completely an equilibrium state.

** Reversible (quasi-static) and irreversible processes.** A system may undergo a change from one equilibrium state to another under the influence of various external factors. In this process the system passes through a continuous series of states that generally speaking are nonequilibrium states. A process must occur sufficiently slowly in order for its properties to approach those of an equilibrium process. Slowness, however, is not by itself a sufficient condition for an equilibrium process. For example, the process of the discharge of a capacitor across a high resistance or the process of throttling, wherein a pressure drop causes a gas to flow through a porous barrier from one vessel to another (

*see*JOULE-THOMSON EFFECT), may be arbitrarily slow and at the same time essentially nonequilibrium processes. Since an equilibrium process is a continuous chain of equilibrium states, it is reversible; in other words, it can be performed in the reverse direction, so that both the system and the surroundings are restored to their original states. Thermodynamics provides a complete quantitative description of reversible processes; for irreversible processes, it establishes only certain inequalities and indicates the direction in which the processes occur.

** First law of thermodynamics.** The state of a system can be changed in two fundamentally different ways. In one way, the system does work on surrounding bodies so as to displace them over macroscopic distances, or work is performed by these bodies on the system. In the other way, heat is transferred to or from the system, and the positions of the surrounding bodies remain unchanged. In the general case the change of a system from one state to another is associated with the transfer of some amount of heat Δ

*Q*to the system and with the performance of work Δ

*A*by the system on external bodies. When the initial and final states are specified, experience shows that Δ

*Q*and Δ

*A*depend essentially on the path of the change of state. In other words, these quantities are characteristics not of an individual state of the system but of the process followed by the system. The first law of thermodynamics states that if a system follows a thermodynamic cycle (that is, ultimately returns to its initial state), then the total amount of heat transferred to the system over the course of the cycle is equal to the work performed by the system.

The first law of thermodynamics is essentially an expression of the law of conservation of energy for systems in which thermal processes play an important role. The energy equivalence of heat and work—that is, the possibility of measuring their quantities in the same units and thus the possibility of comparing them—was demonstrated in experiments carried out by J. R. von Mayer in 1842 and, especially, J. Joule in 1843. The first law of thermodynamics was formulated by Mayer; a much clearer formulation was provided by H. von Helmholtz in 1847. The statement of the first law given above is equivalent to the assertion that a perpetual motion machine of the first kind is impossible.

For a process where the system does not return to its initial state, it follows from the first law that the difference Δ*Q* – Δ*A* = Δ*U*, which is in general nonzero, does not depend on the path between the initial and final states. In fact, any two processes occurring in opposite directions between the same end states form a closed cycle for which the indicated difference vanishes. Thus, Δ*U* is the change in the quantity *U*, which has a well-defined value in every state and is said to be a function of state, or state variable, of the system. The quantity *U* is called the internal energy, or simply the energy, of the system. The first law of thermodynamics thus implies that there exists a characteristic function of the state of a system: its internal energy. In the case of a homogeneous body that is capable of performing work only upon a change in volume, we have Δ*A* = *p dV*, and the infinitesimal increment (differential) of *U* is

(1) *dU* = *dQ* – *p dV*

Here, *dQ* is an infinitesimal increment of heat; it is not, however, a differential of some function. For a fixed volume (*dV* = 0), the heat supplied to the body goes to an increase in internal energy. The heat capacity of a body at constant volume is therefore *c _{v}* = (

*dU/dT*)

_{V}. Another state function is the enthalpy

*H*=

*U*+

*pV*with the differential

(2) *dH* = *dU* + *V dp*

The introduction of enthalpy makes it possible to obtain an expression for heat capacity measured at constant pressure: *c _{p}* = (

*dH/dT*)

_{p}. In the case of an ideal gas, which is described by the state equation

*pV*=

*nRT*(where

*n*is the number of moles of the gas in a volume

*V*and

*R*is the gas constant), both the free energy and the enthalpy of a certain mass of the gas depend only on

*T*. This assertion is confirmed, for example, by the absence of cooling in the Joule-Thomson process. Therefore, for an ideal gas

*c*–

_{p}*c*=

_{V}*nR*.

** Second law of thermodynamics.** Although it forbids the existence of a perpetual motion machine of the first kind, the first law of thermodynamics does not exclude the possibility of constructing a continuous-operation machine that would be capable of converting into useful work practically all of the heat supplied to it; such a device is called a perpetual motion machine of the second kind. Nevertheless, all the experience in designing heat engines that had been amassed by the early 19th century indicated that the efficiency of heat engines (the ratio of the heat expended to the work obtained) is always considerably less than unity: some of the heat is unavoidably dissipated to the surroundings. S. Carnot showed in 1824 that this fact is fundamental in character—that is, any heat engine must contain not only a heat source and a working substance, such as steam, that undergoes a thermodynamic cycle but also a heat sink whose temperature must be lower than that of the heat source.

In the second law of thermodynamics Carnot’s conclusion is generalized to arbitrary thermodynamic processes occurring in nature. In 1850, R. Clausius formulated the second law in the following way: heat cannot spontaneously pass from a system at a lower temperature to a system at a higher temperature. In 1851, W. Thomson (Lord Kelvin) set forth, independently, a slightly different statement of the law: it is impossible to construct a periodically operating machine whose activity reduces entirely to the raising of some load (the performance of mechanical work) and the corresponding cooling of a heat reservoir. Despite the qualitative character of this assertion, it has far-reaching quantitative consequences. For example, it permits the maximum efficiency of a heat engine to be determined.

Suppose a heat engine operates in a Carnot cycle. When the working substance is in isothermal contact with the heat source (*T* = *T*_{1}), the working substance receives the quantity of heat Δ*Q*_{1}. In the other isothermal process of the cycle, where the working substance is in contact with the heat sink (*T* = *T*_{2}), the working substance gives up the quantity of heat Δ*Q*_{2}. The ratio Δ*Q*_{2}/Δ*Q*_{1} cannot depend on the nature of the working substance and must be the same for all heat engines operating in a reversible Carnot cycle that have the same heat-source temperature and the same heat-sink temperature. If the opposite were the case, then an engine with a smaller ratio could be used to drive an engine with a larger ratio in the reverse direction (since the cycles are reversible). In this combined engine heat from the heat sink would be transferred to the heat source without the performance of work. Since this situation violates the second law of thermodynamics, the ratio Δ*Q*_{2}/Δ*Q*_{1} must be the same for both engines. In particular, it must be the same as in the case in which the working substance is an ideal gas. This ratio can be easily found. Thus, for all reversible Carnot cycles there holds the relation

which is sometimes called Carnot’s proportion. As a result, for all engines with a reversible Carnot cycle, the efficiency is at a maximum and is η = (*T*_{1} – *T*_{2})/*T*_{1}. In the case of irreversible cycles, the efficiency is less than this quantity. It must be emphasized that Carnot’s proportion and the efficiency of a Carnot cycle have the indicated form only when the temperature is measured on an absolute temperature scale. Carnot’s proportion has been made the basis for determining the absolute temperature scale (*see*).

The existence of entropy as a function of state is a consequence of the second law of thermodynamics (Carnot’s proportion). Let us introduce a quantity *S* such that the change in *S* upon an isothermal reversible transfer of the quantity of heat Δ*Q* to the system is Δ*S* = Δ*Q/T*. The net change in *S* in the Carnot cycle is then zero; in the adiabatic processes of the cycle Δ*S* = 0 (since Δ*Q* = 0), and the changes in the isothermal processes compensate for each other. The net change in *S* also turns out to be zero for an arbitrary reversible cycle. This statement can be proved by dividing the cycle into a sequence of infinitely small Carnot cycles with infinitesimal isothermal portions. It follows, as in the case of internal energy, that the entropy 5 is a function of the state of the system; that is, the change in entropy does not depend on the path of the changing states. Using the concept of entropy, Clausius showed in 1876 that the original statement of the second law of thermodynamics is equivalent to the following: there exists a state function of a system—the system’s entropy *S*—such that the change in 5 during a reversible transfer of heat to the system is

(4) *dS* = *dQ/T*

In real (irreversible) adiabatic processes, the entropy increases, reaching its maximum value at the state of equilibrium.

** Thermodynamic potentials.** The definition of entropy makes it possible to write the following expressions for the differentials of the internal energy and enthalpy:

(5) *dU* = *T ds* – *p dv dH* = *T ds* + *V dp*

It is evident that the pairs *S*, *V* and *S*, *p* are the natural independent state variables for the functions *U* and *H*, respectively. If, however, not entropy but temperature is used as an independent variable, then the system is more conveniently described by the Helmholtz free energy, or work function, *F* = *U* – *TS* (for the variables *T* and *V*) and the Gibbs free energy, or Gibbs function, *G* = *H* – *TS* (for the variables *T* and *p*). Their differentials are

(6) *dF* = – *S dT* – *p dV dG* = – *S dT* + *V dp*

The state functions *U*, *H*, *F*, and *G* are the thermodynamic potentials of the system for the corresponding pairs of independent variables. The method of thermodynamic potentials was developed by J. Gibbs between 1874 and 1878 and is based on the joint application of the first and second laws of thermodynamics. It permits the derivation of a number of important thermodynamic relations between various physical properties of a system. Because the mixed second derivatives are independent of the order of differentiation, we can obtain, for example, the following expression relating the heat capacities *c _{p}* and

*c*, the isobaric thermal expansion coefficient (∂

_{V}*V*/∂

*T*)

_{p}, and the isothermal compressibility (∂

*V*/∂

*p*)

_{T}:

*c _{p}* –

*c*= –

_{V}*T*(∂

*V*/∂

*T*)

^{2}

_{p}/(∂

*V*/∂

*P*)

_{T}

and we can obtain the following relation between the isothermal and adiabatic compressibilities:

(∂*V*/∂*p*)_{S} = (*c _{p}/c_{V}*)(∂

*V*/∂

*P*)

_{T}

Since the entropy of an isolated system in a state of equilibrium is a maximum, the thermodynamic potentials in the equilibrium state with respect to arbitrary small deviations from equilibrium, for fixed values of the corresponding independent variables, must be minima. This fact leads to important inequalities giving conditions for stability—in particular (∂*p*/∂*V*)_{S} < (∂*p*/∂*V*)_{T} < 0 and *c _{p}* >

*c*0.

_{V}** Third law of thermodynamics.** In accordance with the second law of thermodynamics, entropy is defined by the differential equation (4). This equation determines entropy to an accuracy of a constant term that is not dependent on temperature but could differ for different bodies in the equilibrium state. Thermodynamic potentials also have corresponding undefined terms. In 1906, W. Nernst concluded from electrochemical research he had conducted that these terms must be universal—that is, they are independent of pressure, state of aggregation, and other characteristics of a substance. This new experimentally based principle is usually called the third law of thermodynamics, or the Nernst heat theorem. In 1911, M. Planck showed that an equivalent statement of the principle is that the entropy of all bodies in equilibrium state tends toward zero as a temperature of absolute zero is approached, since the universal constant in entropy may be set equal to zero. It follows in particular from the third law of thermodynamics that the coefficient of thermal expansion, isochoric pressure coefficient (∂

*p*/∂

*T*)

_{V}, and heat capacities

*c*and

_{p}*c*vanish as

_{V}*T*→ 0.

It should be noted that the third law of thermodynamics and its consequences do not pertain to systems in metastable states. An example of such a system is a mixture of substances between which chemical reactions are possible but are retarded because the reaction rates at low temperatures are very low. Another example is a rapidly frozen solution, which should separate into phases at a low temperature; in practice, however, the separation process does not occur at low temperatures. Such states are similar to equilibrium states in many regards, but their entropy does not vanish when *T* = 0.

** Applications.** Important areas of application of thermodynamics include the theory of chemical equilibrium and the theory of phase equilibrium, in particular, the theory of equilibrium between different states of aggregation and equilibrium upon the phase separation of mixtures of liquids and gases. In these cases, the exchange of particles of matter between different phases plays an important role in the establishment of equilibrium, and the concept of chemical potential is used in formulating the equilibrium conditions. Constancy of chemical potential replaces the condition of constancy of pressure if the liquid or gas is located in an external field, such as a gravitational field. The methods of thermodynamics are used effectively in the study of natural phenomena in which heat effects play an important role. In thermodynamics a distinction is commonly made between branches that pertain to individual sciences and engineering, such as chemical thermodynamics and engineering thermodynamics, and branches dealing with different objects of investigation, such as the thermodynamics of elastic bodies, of dielectrics, of magnetic media, of superconductors, of plasmas, of radiation, of the atmosphere, and of water.

The establishment of the statistical nature of entropy led to the construction of the thermodynamic theory of fluctuations by A. Einstein in 1910 and to the development of nonequilibrium thermodynamics.

### REFERENCES

Sommerfeld, A.*Termodinamika i statisticheskaia fizika.*Moscow, 1955. (Translated from German.)

Leontovich, M. A.

*Vvedenie v termodinamiku*, 2nd ed. Moscow-Leningrad, 1952.

Landau, L. D., and E. M. Lifshits.

*Statisticheskaia fizika*, 2nd ed. (

*Teoreticheskaia fizika*, vol. 5.) Moscow, 1964.

*Vtoroe nachalo termodinamiki: Sb.*Moscow-Leningrad, 1934.

Epstein, P. S.

*Kurs termodinamiki.*Moscow-Leningrad, 1948. (Translated from English.)

Vander Waals, J. D., and F. Konstamm.

*Kurs termostatiki.*Moscow, 1936. (Translated from German.)

Kubo, R.

*Termodinamika.*Moscow, 1970. (Translated from English.)

*Termodinamika: Terminologiia: Sb.*Moscow, 1973.

G. M. ELIASHBERG