Reversibility, Heat, and Work

In Chapter 1, equilibrium was seen as the most probable way in which a system exists. An equation of state, as introduced in Chapter 2, gives the thermodynamic relationship for a gas system at equilibrium. Our molecular level view, based on statistical arguments, allows that a system does not have to be at equilibrium at any given instant. There can be small fluctuations from equilibrium, which over time average to the equilibrium behavior. There may also be sudden changes, external perturbations, that disrupt equilibrium. 

Let us consider a reversible isothermal expansion of the same gas. If for the moment we take the gas to be ideal, then reversibility means that PV = nRT must hold at every instant of the expansion process. With temperature being held constant, this means that P will vary inversely and continuously with volume. For each infinitesimal change in volume, there will be an infinitesimal change in pressure. It is difficult to envision a means of accomplishing this type of expansion in fhe laboratory. We would need to remove mass continuously from the top of the piston, and the rate at which mass was removed would have to be slow enough for the system to remain at equilibrium at every instant. The only rate that could guarantee this is one that is infinitesimally close to zero. Though we may not be able to achieve this in practice—it would take forever— it is of interest to consider the work done in such a reversible process. 

We obtain the work done in the reversible expansion by integration of the differential of work, which is -PdV. However, we must realize that the pressure is not constant in this type of expansion but is instead a function of the volume. It is determined by the equation of state since equilibrium is being maintained throughout the expansion. 

For a given system, 5xv is negative if fhe system does work on fhe surroundings and fhereby gives up energy. 5q is negative if fhe system fransfers energy away as heat. It is important 

Expansion of an ideal gas at a specific temperature. The curve is the reversible path for the given temperature. Expansion from to V2 follows this path only if equilibrium is maintained throughout. Then, the area under the segment from V- to V2 is iPdV, which is the work associated with the reversible process. Several irreversible expansions are indicated. A one-step irreversible process is for the external pressure to go instantaneously from Pi to 2, and then the gas expands from to V2 against a pressure of P2. The amount of work expended by the gas in this irreversible process is the area of the shaded rectangle. It is clear that the area under any of the irreversible paths is less than the area under the path that follows the isotherm where the gas pressure is always equal to the external pressure. The amount of reversible work is approached through a sequence of irreversible steps as the steps are made smaller and smaller. 

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Figure 7.9 Schematic diagram of heat and work sources in which the hot gas in an engine partly converts to work, and partly converts to colder exhaust gas. The ellipses indicate reversible heat and work sources. The arrows indicate processes. A physical realization is shown in Figure 7.10. 

T0 compute the maximum work, we need tw o other idealizations. A reversible work source can change volume or perform work of any other kind quasi-statically, and is enclosed in an impermeable adiabatic waU, so 6g = TdS = 0 and dU = S w. A reversible heat source can exchange heat quasi-statically, and is enclosed in a rigid wall that is impermeable to matter but not to heat flow, so = pdV = 0 and dU = 6q = TdS. A reversible process is different from a reversible heat or work source. A reversible heat source need not have AS = 0. A reversible process refers to changes in a whole system, in w-hich a collection of reversible heat plus work sources has AS = 0. The frictionless weights on pulleys and inclined planes of Newtonian mechanics are reversible w ork sources, for example. The maximum possible work is achieved w hen reversible processes are performed with reversible heat and work sources. 

We see from this that if both T and V are held constant dT = 0 and dV = 0, dA = 0 so that under those conditions (U TS) = 0 and so the Helmholtz energy A indicates an equilibrium a balanced trade-off between increasing entropy and decreasing energy. That is very interesting and a mathematical truth under reversible heat and work conditions. However, it turns out that it is not very useful in the laboratory, since it implies that pressures must be the only variable if T and V are held constant. Certain experiments can be designed to meet these conditions, but more likely the pressure... 

Heat, like work, is energy in transit and is not a function of the state of a system. Heat and work are interconvertible. A steam engine is an example of a machine designed to convert heat into work.h The turning of a paddle wheel in a tank of water to produce heat from friction represents the reverse process, the conversion of work into heat. 

Consider any reversible cyclic process that involves the exchange of heat and work. Again, the net area enclosed by the cycle on a p-V plot gives the work. This work can be approximated by taking the areas enclosed within a series of Carnot cycles that overlap the area enclosed by the cycle as closely as possible as shown in Figure 2.10b. For each of the Carnot cycles, the sum of the q/T terms... 

In this expression consistent units must be used. In the SI system each of the terms in equation 2.1 is expressed in Joules per kilogram (J/kg). In other systems either heat units (e g. cal/g) or mechanical energy units (e.g. erg/g) may be used, dU is a small change in the internal energy which is a property of the system it is therefore a perfect differential. On the other hand, Sq and SW are small quantities of heat and work they are not properties of the system and their values depend on the manner in which the change is effected they are, therefore, not perfect differentials. For a reversible process, however, both Sq and SW can be expressed in terms of properties of the system. For convenience, reference will be made to systems of unit mass and the effects on the surroundings will be disregarded. 

Heat is different than all the other types of energy. All other types of energy can be completely changed into heat, but the reverse is not true heat cannot be changed completely into any other kind of energy. In this chapter we will classify energy as heat and work, where work includes all other types of energy besides heat (Table 18-1). 

The thermodynamic changes for reversible, free, and intermediate expansions are compared in the first column of Table 5.1. This table emphasizes the difference between an exact differential and an inexact differential. Thus, U and H, which are state functions whose differentials are exact, undergo the same change in each of the three different paths used for the transformation. They are thermodynamic properties. However, the work and heat quantities depend on the particular path chosen, even though the initial and final values of the temperature, pressure, and volume, respectively, are the same in all these cases. Thus, heat and work are not thermodynamic properties rather, they are energies in transfer between system and surroundings. 

As we noticed in Table 5.1, AC/ = 0 both for the free expansion and for the reversible expansion of an ideal gas. We used an ideal gas as a convenient example because we could calculate easily the heat and work exchanged. Actually, for any gas, AC/has the same value for a free and a reversible expansion between the corresponding initial and final states. Furthermore, AC/ for a compression is equal in magnitude and opposite in sign to AC/ for an expansion no indication occurs from the first law of which process is the spontaneous one. 

If we designate the heat and work quantities for the reverse cycle by primed symbols, the reverse and forward relationships are... 

The Carnot cycle is a reversible cycle. Reversing the cycle will also reverse the directions of heat and work interactions. The reversed Carnot heat engine cycles are Carnot refrigeration and heat pump cycles. Therefore, a reversed Carnot vapor heat engine is either a Carnot vapor refrigerator or a Carnot vapor heat pump, depending on the function of the cycle. 

As the water splitting process is cyclic (reversible behavior shown by dotted lines in Fig. 2.6) it has limitations imposed by second law of thermodynamics [60]. Hence the operating temperatures Tr and To are crucial in determining the thermal efficiency of the process. In a water splitting process using both heat and work inputs, the thermal efficiency in general is defined as [60,61]... 

Reversible Isothermal Expansion Let us consider the heat and work of ideal gas expansion from V to V2 under isothermal conditions (AT = 0). We recognize from (3.74a) that... 

The clockwise direction in C corresponds to the clockwise direction in the Carnot cycle, with heat and work input/output as shown in Fig. 4.2. We can similarly envision a reverse Carnot engine ( heat pump ) C, which is obtained by reversing the directions of heat and work arrows and traversing the Carnot cycle in counterclockwise direction ... 

Why does heat flow from a warm body into a cold one Why doesn t it ever flow in the reverse direction We can see that differences in temperature control the direction of flow of heat, but this observation raises still another question What is temperature Reflection on these questions, and on the interconversion of heat and work, led to the discovery of the second law of thermodynamics and to the definition of a new thermodynamic function, the entropy S. 

Since H, Cp, and T are all state functions, Eq. (2.26) applies to any process for which P2 = Pi whether or not it is actually carried out at constant pressure. However, it is only for the mechanically reversible, constant-pressure path that heat and work can be calculated by the equations Q = n AH, Q = n f CPdT, and W = PnAV,... 

The work of Carnot, published in 1824, and later the work of Clausius (1850) and Kelvin (1851), advanced the formulation of the properties of entropy and temperature and the second law. Clausius introduced the word entropy in 1865. The first law expresses the qualitative equivalence of heat and work as well as the conservation of energy. The second law is a qualitative statement on the accessibility of energy and the direction of progress of real processes. For example, the efficiency of a reversible engine is a function of temperature only, and efficiency cannot exceed unity. These statements are the results of the first and second laws, and can be used to define an absolute scale of temperature that is independent of ary material properties used to measure it. A quantitative description of the second law emerges by determining entropy and entropy production in irreversible processes. 

Determine the heat and work needed to reversibly and isothermally separate an equimolar binary mixture into its pure species if the excess Gibbs free energy for the mixture is... 

Entrojiy and probability. The recognition of the universal applicability of the law of the conservation of energy is partly based on the mechanical conception of heat as motion of the ultimate particles of matter. If heat, energy, and kinetic energy of the molecules are essentially of the same nature, and are differentiated from one another only by the units in which we measure them, the validity of the law of the equivalence of heat and work is explained. At first sight, however, it is not easy to understand why heat cannot be converted completely into work, or, in other words, why the conversion of heat into work is an irreversible process (second law of thermodynamics). In pure mechanics we deal only with perfectly reversible processes. By the principles of mechanics the complete conversion of heat into work should be just as possible as the conversion... 

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