The First Law and Other Basic Concepts
2.6 THE REVERSIBLE PROCESS
another at a rate such that the piston’s rise is continuous, with minute oscillation only at the end of the process.
The limiting case of removal of a succession of infinitesimal masses from the piston is approximated when the masses m in Fig. 2.1 are replaced by a pile of powder, blown in a very fine stream from the piston. During this process, the piston rises at a uniform but very slow rate, and the powder collects in storage at ever higher levels. The system is never more than differentially displaced from internal equilibrium or from equilibrium with its surroundings. If the removal of powder from the piston is stopped and the direction of transfer of powder is reversed, the process reverses direction and proceeds backwards along its original path. Both the system and its surroundings are ultimately restored to virtually their initial conditions. The original process approaches reversibility.
Without the assumption of a frictionless piston, we cannot imagine a reversible process.
If the piston sticks, a finite mass must be removed before the piston breaks free. Thus the equi- librium condition necessary to reversibility is not maintained. Moreover, friction between two sliding parts is a mechanism for the dissipation of mechanical energy into internal energy.
This discussion has centered on a single closed-system process, the expansion of a gas in a cylinder. The opposite process, compression of a gas in a cylinder, is described in exactly the same way. There are, however, many processes that are driven by an imbalance of other than mechanical forces. For example, heat flow occurs when a temperature difference exists, electricity flows under the influence of an electromotive force, and chemical reactions occur in response to driving forces that arise from differences in the strengths and configurations of chemical bonds in molecules. The driving forces for chemical reactions and for transfer of substances between phases are complex functions of temperature, pressure, and composition, as will be described in detail in later chapters. In general, a process is reversible when the net force driving it is infinitesimal in size. Thus heat is transferred reversibly when it flows from a finite object at temperature T to another such object at temperature T − dT.
Figure 2.1: Expansion of a gas. The nature of reversible processes is illustrated by the expansion of gas in an idealized piston/cylinder arrangement. The apparatus shown is imagined to exist in an evacuated space. The gas trapped inside the cylinder is chosen as the system; all else is the surroundings. Expansion results when mass is removed from the piston. For simplicity, assume that the piston slides within the cylinder without friction and that the piston and cylinder neither absorb nor transmit heat. Moreover, because the density of the gas in the cylinder is low and its mass is small, we ignore the effects of gravity on the contents of the cylinder. This means that gravity-induced pressure gradients in the gas are very small relative to its total pressure and that changes in potential energy of the gas are negligible in comparison with the potential-energy changes of the piston assembly.
m
∆l
2.6. The Reversible Process 37
Some chemical reactions can be carried out in an electrolytic cell, and in this case they may be held in balance by an applied potential difference. For example, when a cell consisting of two electrodes, one of zinc and the other of platinum, is immersed in an aqueous solution of hydrochloric acid, the reaction that occurs is:
Zn + 2HCl ⇌ H 2 + ZnCl 2
The cell is held under fixed conditions of temperature and pressure, and the electrodes are connected externally to a potentiometer. If the electromotive force (emf) produced by the cell is exactly balanced by the potential difference of the potentiometer, the reaction is held in equilibrium. The reaction can be made to proceed in the forward direction by a slight decrease in the opposing potential difference, and it can be reversed by a corresponding increase in the potential difference above the emf of the cell. For a real energy storage device, finite
Reversible Chemical Reaction
The concept of a reversible chemical reaction is illustrated by the decomposition of solid cal- cium carbonate to form solid calcium oxide and carbon dioxide gas. At equilibrium, this sys- tem exerts a specific decomposition pressure of CO2 for a given temperature. The chemical reaction is held in balance (in equilibrium) by the pressure of the CO2. Any change of condi- tions, however slight, upsets the equilibrium and causes the reaction to proceed in one direc- tion or the other.
If the mass m in Fig. 2.2 is minutely increased, the CO2 pressure rises, and CO2 com- bines with CaO to form CaCO3, allowing the weight to fall. The heat given off by this reaction raises the temperature in the cylinder, and heat flows to the bath. Decreasing the weight sets off the opposite chain of events. The same results are obtained if the temperature of the bath is raised or lowered. Raising the bath temperature slightly causes heat transfer into the cylinder, and calcium carbonate decomposes. The CO2 generated causes the pressure to rise, which in turn raises the piston and weight. This continues until the CaCO3 is completely decomposed.
A lowering of the bath temperature causes the system to return to its initial state. The imposi- tion of differential changes causes only minute displacements of the system from equilibrium, and the resulting process is exceedingly slow and reversible.
Figure 2.2: Reversibility of a chemical reaction. The cylinder is fitted with a frictionless piston and contains CaCO3, CaO, and CO2 in equilibrium. It is immersed in a constant-temperature bath, and thermal equilibrium assures equality of the system temperature with that of the bath. The temperature is adjusted to a value such that the decomposition pressure is just sufficient to balance the weight on the piston, a condition of mechanical equilibrium.
CO2 T
CaCO3 + CaO Thermostat
m
differences in emf are required to drive the electrochemical reactions. Thus, a charger may apply up to about 4.2 V to charge a familiar lithium ion battery, but that battery might then supply only 3.7 V to a device that it powers. The difference arises from irreversibilities in the charging and discharging processes within the battery.
Summary Remarks on Reversible Processes
A reversible process:
∙ Can be reversed at any point by an infinitesimal change in external conditions ∙ Is never more than minutely removed from equilibrium
∙ Traverses a succession of equilibrium states ∙ Is frictionless
∙ Is driven by forces whose imbalance is infinitesimal in magnitude ∙ Proceeds infinitely slowly
∙ When reversed, retraces its path, restoring the initial state of system and surroundings
Computing Work for Reversible Processes
Equation (1.3) gives the work of compression or expansion of a gas caused by the displace- ment of a piston in a cylinder:
dW = − P dV t (1.3)
The work done on the system is in fact given by this equation only when certain characteristics of the reversible process are realized. The first requirement is that the system be no more than infinitesimally displaced from a state of internal equilibrium, characterized by uniformity of temperature and pressure. The system then has an identifiable set of properties, including pressure P. The second requirement is that the system be no more than infinitesi- mally displaced from mechanical equilibrium with its surroundings. In this event, the internal pressure P is never more than minutely out of balance with the external force, and we can make the substitution F = PA that transforms Eq. (1.2) into Eq. (1.3). Processes for which these requirements are met are said to be mechanically reversible, and for such processes Eq. (1.3) can be integrated:
W = − ∫ V 1V t 2t
P dVt (1.4)
This equation gives the work for the mechanically reversible expansion or compression of a fluid in a piston/cylinder arrangement. Its evaluation clearly depends on the relation between P and Vt, i.e., on the “path” of the process, which must be specified. To find the work of an irreversible process for the same change in Vt, one must apply an efficiency, which relates the actual work to the reversible work.