Technology & Engineering
Entropy Change for Ideal Gas
The entropy change for an ideal gas refers to the measure of disorder or randomness in the system as it undergoes a process. When an ideal gas expands, its entropy increases, as the molecules become more dispersed and the system becomes more disordered. Conversely, when an ideal gas is compressed, its entropy decreases, as the molecules become more ordered and the system becomes less disordered.
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6 Key excerpts on "Entropy Change for Ideal Gas"
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
The backward direction of time (a video run 615 20.1 ENTROPY backwards) would correspond to the exploded popcorn re-forming the original kernel. Because this backward process would result in an entropy decrease, it never happens. There are two equivalent ways to define the change in entropy of a system: (1) in terms of the system’s temperature and the energy the system gains or loses as heat, and (2) by counting the ways in which the atoms or molecules that make up the system can be arranged. We use the first approach in this module and the second in Module 20.4. Change in Entropy Let’s approach this definition of change in entropy by looking again at a pro- cess that we described in Modules 18.5 and 19.9: the free expansion of an ideal gas. Figure 20.1.1a shows the gas in its initial equilibrium state i, confined by a closed stopcock to the left half of a thermally insulated container. If we open the stopcock, the gas rushes to fill the entire container, eventually reaching the final equilibrium state f shown in Fig. 20.1.1b. This is an irreversible process; all the molecules of the gas will never return to the left half of the container. The p-V plot of the process, in Fig. 20.1.2, shows the pressure and volume of the gas in its initial state i and final state f. Pressure and volume are state properties, properties that depend only on the state of the gas and not on how it reached that state. Other state properties are temperature and energy. We now assume that the gas has still another state property—its entropy. Furthermore, we define the change in entropy S f − S i of a system during a process that takes the system from an initial state i to a final state f as ΔS = S f − S i = t f dQ ____ T (change in entropy defined). (20.1.1) Here Q is the energy transferred as heat to or from the system during the pro- cess, and T is the temperature of the system in kelvins.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
Other state properties are temperature and energy. We now assume that the gas has still another state property — its entropy. Furthermore, we define the change in entropy S f − S i of a system during a process that takes the system from an initial state i to a final state f as ΔS = S f − S i = ∫ f i dQ T (change in entropy defined). (20-1) Here Q is the energy transferred as heat to or from the system during the pro- cess, and T is the temperature of the system in kelvins. Thus, an entropy change depends not only on the energy transferred as heat but also on the temperature at which the transfer takes place. Because T is always positive, the sign of ΔS is the same as that of Q. We see from Eq. 20-1 that the SI unit for entropy and entropy change is the joule per kelvin. There is a problem, however, in applying Eq. 20-1 to the free expansion of Fig. 20-1. As the gas rushes to fill the entire container, the pressure, temperature, and volume of the gas fluctuate unpredictably. In other words, they do not have a sequence of well-defined equilibrium values during the intermediate stages of the change from initial state i to final state f. Thus, we cannot trace a pressure – volume path for the free expansion on the p-V plot of Fig. 20-2, and we cannot find a rela- tion between Q and T that allows us to integrate as Eq. 20-1 requires. However, if entropy is truly a state property, the difference in entropy between states i and f must depend only on those states and not at all on the way the system went from one state to the other. Suppose, then, that we replace the irreversible free expansion of Fig. 20-1 with a reversible process that connects states i and f. With a reversible process we can trace a pressure – volume path on a p-V plot, and we can find a relation between Q and T that allows us to use Eq. 20-1 to obtain the entropy change. We saw in Module 19-9 that the temperature of an ideal gas does not change during a free expansion: T i = T f = T.
- Kevin Dahm, Donald Visco(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
Once the state is defined, all other intensive properties (e.g., S ) have unique values. Consequently, if one defines two states ( T 1 , P 1 and T 2 , P 2 ), one can evaluate the change in a state property ( S in the example) along any path be-tween these two states, and know that the result is valid for all other paths. This problem- solving approach is of particular importance for entropy because the definition of entropy (Equation 4.14) only applies directly to reversible processes carried out in closed systems. In steps 7–9 of Example 4-6, we modeled the change in entropy of an ideal gas. Since no assumptions (beyond ideal gas behavior) were made in these steps, the result is correct for ideal gases in general. One can carry out an analogous derivation on a mass basis and show that dS ˆ 5 C ˆ * V T dT 1 R V ˆ dV ˆ (4.55) A common conceptual error is proceeding directly from Equation 4.54 to the inte-grated form of S 2 2 S 1 5 C * V ln T 2 T 1 1 R ln V 2 V 1 (4.56) Equations 4.54 and 4.55 are valid for any process involving ideal gases. In the definition of entropy (equation 4.14), T must be ex-pressed on an absolute scale. Since Equation 4.51 is directly derived from Equation 4.14, the given temperature must be converted into Kelvin. PITFALL PREVENTION Copyright 2015 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 165 C H A P T E R 4 Entropy The integration of Equation 4.54 only yields Equation 4.56 when C V * is constant with respect to temperature. If C V * is a function of temperature, as is generally true, a more complex integration results.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
20.08 For stretched rubber, relate the elastic force to the rate at which the rubber’s entropy changes with the change in the stretching distance. Key Ideas ● An irreversible process is one that cannot be reversed by means of small changes in the environment. The direc- tion in which an irreversible process proceeds is set by the change in entropy ΔS of the system undergoing the process. Entropy S is a state property (or state function) of the system; that is, it depends only on the state of the system and not on the way in which the system reached that state. The entropy postulate states (in part): If an irreversible process occurs in a closed system, the entropy of the system always increases. ● The entropy change ΔS for an irreversible process that takes a system from an initial state i to a final state f is exactly equal to the entropy change ΔS for any reversible process that takes the system between those same two states. We can compute the latter (but not the former) with ΔS = S f − S i = ∫ f i dQ T . Here Q is the energy transferred as heat to or from the system during the process, and T is the temperature of the system in kelvins during the process. ● For a reversible isothermal process, the expression for an entropy change reduces to ΔS = S f − S i = Q T . ● When the temperature change ΔT of a system is small relative to the temperature (in kelvins) before and after the process, the entropy change can be approximated as ΔS = S f − S i ≈ Q T avg , where T avg is the system’s average temperature during the process. ● When an ideal gas changes reversibly from an initial state with temperature T i and volume V i to a final state with temperature T f and volume V f , the change ΔS in the entropy of the gas is ΔS = S f − S i = nR ln V f V i + nC V ln T f T i .
eBook - PDF- Guy Deutscher(Author)
- 2008(Publication Date)
- World Scientific(Publisher)
53 Chapter 4 Entropy in Thermodynamics and Our Energy Needs As the name indicates, thermodynamics deals with heat and motion. This branch of physics developed after the invention of the steam engine which transforms the heat of the steam coming from the boiler into mechanical energy. This transformation from heat into mechanical energy occurs via the pressure that steam exercises on a piston which it gets moving inside a cylinder. After it does its work the steam is released and condenses back into water. In the process, heat has been transferred from the hot boiler into the colder environment, and work has been performed by the moving piston This immediately poses the question of the equivalence between these two forms of energy, heat and mechanical energy, and of the efficiency of the engine in converting the first into the second. Here entropy plays a decisive role, which we will discuss in the first part of this chapter. In particular, we shall see how the increase of entropy in the course of the conversion process must be compensated by an input of energy, a notion that we introduced already, qualitatively, in Chapter 1. 4.1. Entropy in thermodynamics 4.1.1. Heat and mechanical work as two forms of energy: the first law of thermodynamics While the transformation of heat into mechanical energy is a relatively recent invention with the discovery of the heat engine (and later the internal combustion engine), it has been known to mankind since the dawn of civilization that mechanical work can be transformed into heat. Today we light a match by rubbing it against a rough surface, the heat produced by friction raising 54 The Entropy Crisis sufficiently the temperature of the tip of the match to put it on fire. Life was more difficult for our ancestors, but they used the same physical principle apparently by rotating rapidly a sharpened tip of hard wood against another piece of wood.
eBook - PDFPhysical Principles of Chemical Engineering
International Series of Monographs in Chemical Engineering
- Peter Grassmann, H. Sawistowski(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
% This section does not describe the derivation of the second law, which can be found in the textbooks of thermodynamics (cf. also § 8.7), but attempts to aid the reader in under-standing the concept of entropy. J. D. Fast, Entropy {Entropie), Philips Technische Bibliothek, Eindhoven, 1960 (the significance of the concept of entropy and its use in science and engineer-ing). 53 54 Concept and Use of Entropy [§2.1 could be entirely converted into work. If, for example, the molecules of a gas obeyed a dictator and he ordered that, in accordance with Fig. 1 b : (1) the energy should be distributed uniformly among all molecules; (2) only motion vertically upwards and downwards was permissible; (3) no energy should be used for rotation round any axis of the molecule or for any oscillations; then the energy of this ordered pile of molecules could be turned entirely into mechanical work, but we should then have to deal with the kinetic energy of two groups of molecules moving in opposite directions to each other which, like the kinetic energy of two solid bodies, have to be considered no longer as heat but as mechanical energy. In general, whenever work is to be produced from energy—by work we denote all completely ordered energy—a certain state of order must be reached which is higher than the state of maximum disorder consistent with the given external conditions. Usually it is said that for heat to be turned into work there must be a temperature difference. A temperature difference always presupposes a certain state of order. However, the molecules with the—on the average— greater kinetic energy are located in the area of higher temperatures, while those with the lower kinetic energy are in the area of lower temperatures. Corresponding to this partial order, part of the heat content of the two bodies in question can be turned into work. So far as we know at present, this also applies to the world as a whole. Its brisk movement, life, and passage of time is only possible when at the begin-ning a state of order is present and when light and darkness, heat and cold are separated from each other in the first act of creation. It is these opposites which maintain the passage of time. In fact, in a chaotic world without contrasts it makes no sense to
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