thermodynamics
thermodynamics
[¦thər·mō·dī′nam·iks]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 (seeJOULE-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 ΔQ to the system and with the performance of work ΔA by the system on external bodies. When the initial and final states are specified, experience shows that ΔQ and ΔA 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 ΔQ – ΔA = ΔU, 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, ΔU 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 ΔA = 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 cv = (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: cp = (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 cp – cV = 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 = T1), the working substance receives the quantity of heat ΔQ1. In the other isothermal process of the cycle, where the working substance is in contact with the heat sink (T = T2), the working substance gives up the quantity of heat ΔQ2. The ratio ΔQ2/ΔQ1 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 ΔQ2/ΔQ1 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 = (T1 – T2)/T1. 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 ΔQ to the system is ΔS = ΔQ/T. The net change in S in the Carnot cycle is then zero; in the adiabatic processes of the cycle ΔS = 0 (since ΔQ = 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 cp and cV, the isobaric thermal expansion coefficient (∂V/∂T)p, and the isothermal compressibility (∂V/∂p)T:
cp – cV = – T(∂V/∂T)2p/(∂V/∂P)T
and we can obtain the following relation between the isothermal and adiabatic compressibilities:
(∂V/∂p)S = (cp/cV)(∂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 cp > cV0.
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 cp and cV vanish as 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
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G. M. ELIASHBERG