►1.3 MEASURED THERMODYNAMIC PROPERTIES
EXAMPLE 1.1 Determination of
Ideal Gas Law from the Defi nition of P and T
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1.4 Equilibrium ◄ 15
A large part of thermodynamics deals with predicting the state that systems will reach at equilibrium. Equilibrium refers to a condition in which the state neither changes with time nor has a tendency to spontaneously change. At equilibrium, there is no net driving force for change. In other words, all opposing driving forces are counterbalanced. We use driving force as a generic term that represents some type of infl uence for a system to change. If the equilibrium state is stable, the system will return to that state when a small disturbance is imposed upon it. A system that has mass being supplied or removed cannot be at equilibrium, since a net driving force must exist to move the species about.
Hence, equilibrium can only occur in a closed system. In general, any system subject to net fl uxes cannot be in equilibrium.
We can distinguish between a system in an equilibrium state and a process at steady-state. If the state of an open system does not change with time as it undergoes a process, it is said to be at steady-state; however, it is not at equilibrium since there must be a net driving force to get the mass into and out of the system. For example, consider the open system shown in Figure 1.2. At steady-state, the thermodynamic state of the system itself remains constant—that is, — Tsys, Psys and its other properties do not change with time.
However, the system’s properties may vary spatially. On the other hand, the fl uid enter-ing the system undergoes a transformation and exits in a different state; thus, when the fl uid enters the system, its properties (Tin, Pin, vin, etc.) have different values than when it leaves 1Tout, Pout, vout2. Since the state of the fl uid that fl ows through the system changes, we cannot say the system is at equilibrium.
The factor of 3 arises since there are three possible directions of motion. Plugging in Equation (E1.1E) into (E1.1D) gives:
P52N 3V ¢1
2mVS2≤ 52N
3V1eKmolecular2 where Equation (1.4) was used.
Finally substituting Equation (1.5) gives the ideal gas relation:
P5NkT V 5nRT
V where R5 kNA and NA is Avogadro’s number.
► 1.4 EQUILIBRIUM
Types of Equilibrium
If a system is in equilibrium with its surroundings, its properties will remain constant with time. On the other hand, a system that is not at equilibrium will change spontane-ously to progress toward its equilibrium state. If a pressure difference exists between the system and surroundings, the system will tend to expand or contract until the pressures balance. A system is said to be in mechanical equilibrium when there is no pressure difference and thus this tendency for change is eliminated. Therefore, to be in mechani-cal equilibrium,
Psys5 Psurr (1.10)
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16 ► Chapter 1. Measured Thermodynamic Properties and Other Basic Concepts
where the subscript “sys” refers to the system and is often omitted.
Similarly, if the system is hotter (or colder) than the surroundings, there is a ther-mal driving force for change. Energy will transfer via heat until the temperatures bal-ance. The system is in thermal equilibrium when there is no temperature difference between it and the surroundings:
Tsys5 Tsurr (1.11)
A system is in chemical equilibrium when there is no tendency for a species to change phases or chemically react. In Chapter 6, we will develop the analogous criterion to Equations (1.10) and (1.11) for chemical equilibrium. To be in thermodynamic equi-librium, a system must be in mechanical, thermal, and chemical equilibrium simultane-ously, so that there is no net driving force for any type of change.10
We can refer to equilibrium between different phases or chemical species within the system as well. A system is said to be in phase equilibrium if it has more than one phase present with no tendency to change. For example, a two-phase liquid–vapor sys-tem is in phase equilibrium when there is no tendency for the liquid to boil or the vapor to condense. To be complete, we must also have mechanical and thermal equilibrium between the liquid (l) and vapor (v) phases, that is,
Pl5 Pv and,
Tl5 Tv
Likewise, when liquid and solid phases or vapor and solid phases are in equilibrium, they will have equal temperature and equal pressure.
Remarkably, P and T are unique among the thermodynamic properties in that they both exist only in the intensive form and that they are equal across the different phases that coexist at equilibrium. Other thermodynamic properties (such as volume) can be written in both extensive and intensive forms, and most of these properties differ between two phases that coexist at equilibrium.
A system undergoing chemical reactions is in chemical reaction equilibrium only when the reactions have no more tendency to react. The second half of this text (Chapters 6–9) exclusively treats phase and chemical reaction equilibrium.
10 If effects such as surface tension or gravitational, electric, or magnetic fi elds are important, the system is confronted with other driving forces that the criteria for equilibrium must also include.
Molecular View of Equilibrium
Phase equilibrium can be viewed as a dynamic process on the molecular level. We will discuss this perspective by considering a system containing a pure species in vapor–
liquid equilibrium, but the principles can be applied to liquid–solid, vapor–solid, and even solid–solid phase equilibria.
At a given temperature, a species exists in the liquid phase if the potential energy of attraction between the molecules is greater than their kinetic energy. Temperature is representative of the average molecular kinetic energy of the species in the system;
however, the species have a distribution of energies. A certain fraction of species will have enough kinetic energy to overcome the attractive forces keeping them in the liquid phase. Thus, they will escape into the vapor phase. If housed in a closed container, the vapor that leaves will exert a pressure on the container’s walls.
Vapor–liquid equilibrium depends on two counteracting processes occurring at the phase boundary marked by the liquid surface. The liquid-phase molecules with enough
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1.5 Independent and Dependent Thermodynamic Properties ◄ 17 kinetic energy break free and go into the vapor phase. Conversely, molecules from the vapor phase can strike the surface and be contained by attraction to the other mol-ecules in the liquid. This process causes them to condense. As the pressure of the vapor increases, more molecules strike the surface and condense. When the rates of vaporiza-tion and condensavaporiza-tion match, both phases can coexist. On a molecular level, we have a dynamic process where the number of molecules leaving the surface is exactly balanced by the number arriving. If we followed a single molecule, however, it could go back and forth between liquid and vapor.
When the temperature is too high or the pressure too low, all molecules will eventu-ally escape to the vapor and only that phase will exist at equilibrium. On the other hand, if the temperature is too low or the pressure too high, only liquid will exist. For a pure species, the dynamic process at which the rate of molecules that vaporize equals the rate at which they condense occurs at a unique pressure for a given temperature and is called the saturation pressure, Psat. As the temperature increases, more molecules enter the vapor phase, and the saturation pressure increases. Since the fraction of species at a given kinetic energy depends exponentially on temperature,11 the saturation pressure increases exponentially with temperature.
We can consider the energetics of the evaporation and condensation processes as well. A molecule leaves the liquid only when it has greater kinetic energy than the poten-tial energy of attraction keeping it in the liquid; this energy is much larger than the aver-age kinetic energy of all the molecules in the liquid. Thus, the higher-energy molecules preferentially depart from the liquid into the vapor phase. Consequently, the average kinetic energy of the molecules that remain will be lower, and the liquid will cool off dur-ing evaporation. Conversely, durdur-ing condensation, the condensed molecule is stabilized by the attractive force between it and the other molecules in the liquid, which causes the liquid to heat up.
Similarly, chemical reaction equilibrium represents a dynamic process on the molec-ular scale. Macroscopically, a reaction can proceed in the forward direction from reac-tants to products or in the reverse direction from products to reacreac-tants. A given reaction is said to be at “chemical reaction equilibrium” when there is no net reaction in either direction. However, again there is a dynamic process on a molecular scale. Reactant mol-ecules will react to form products at the same rate that the product molmol-ecules form reac-tants. If we followed an individual molecule, it might indeed react. However, for each molecule that reacts in the forward direction, another molecule will be reacting in the reverse direction. On the other hand, if an excess of reactants is present, there will be a net macroscopic reaction in the forward direction, since more individual molecules will react in this direction than in the reverse direction. Reaction will occur until equilibrium is reached and there is no more tendency to react on a macroscopic scale. Conversely, if an excess of products is present, macroscopic reaction will occur in the reverse direction until the same equilibrium state is reached.
11 Through a Boltzmann distribution.