A sta6s6cal view of entropy
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1 A sta6s6ca view of entropy 20-4 A Sta&s&ca View of Entropy The entropy of a system can be defined in terms of the possibe distribu&ons of its moecues. For iden&ca moecues, each possibe distribu&on of moecues is caed a microstate of the system. A equivaent microstates are grouped into a configura&on of the system. The number of microstates in a configura&on is the mu6picity W of the configura&on. For a system of N moecues that may be distributed between the two haves of a box, the mu&picity is given by Here n 1 is the number of moecues in one haf of the box and n 2 is the number in the other haf. A basic assump&on of sta&s&ca mechanics is that a the microstates are equay probabe John Wiey & Sons, Inc. A rights reserved.
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3 20-4 A Sta&s&ca View of Entropy Probabiity and Entropy The mu&picity W of a configura&on of a system and the entropy S of the system in that configura&on are reated by Botzmann s entropy equa6on: Here k = J/K is the Botzmann constant. When N is very arge (the usua case), we can approximate n N! with S6ring s approxima6on: 2014 John Wiey & Sons, Inc. A rights reserved.
4 Chapter summary 20 Summary Irreversibe (one-way) Process If an irreversibe process occurs in a cosed system, the entropy of the system aways increases. Second Law of Thermodynamics If a process occurs in a cosed system, the entropy of the system increases for irreversibe processes and remains constant for reversibe processes. Eq Entropy Change Entropy change for reversibe process is given by Eq Entropy Change The efficiency ε of any engine Efficiency of Carnot engine Eq Eq & John Wiey & Sons, Inc. A rights reserved.
5 2014 John Wiey & Sons, Inc. A rights reserved.
6 14-5 Archimedes' Principe 14-5 Archimedes' Principe A bock of wood in static equiibrium is foating: This is expressed: Eq. (14-17) Because of Eq we know: Which means: Eq. (14-18) 2014 John Wiey & Sons, Inc. A rights reserved.
7 14-5 Archimedes' Principe 14-5 Archimedes' Principe The apparent weight of a body in a fuid is reated to the actua weight of the body by: (apparent weight) = (actua weight) (buoyant force) We write this as: Eq. (14-19) Answer: (a) a the same (b) 0.95ρ 0, 1ρ 0, 1.1ρ John Wiey & Sons, Inc. A rights reserved.
8 14-6 The Equation of Continuity 14-6 The Equation of Continuity Learning Objectives Describe steady fow, incompressibe fow, nonviscous fow, and irrotationa fow Expain the term streamine Identify and cacuate voume fow rate Identify and cacuate mass fow rate Appy the equation of continuity to reate the crosssectiona area and fow speed at one point in a tube to those quantities at a different point John Wiey & Sons, Inc. A rights reserved.
9 14-6 The Equation of Continuity 14-6 The Equation of Continuity Motion of rea fuids is compicated and poory understood (e.g., turbuence) We discuss motion of an idea fuid 1. Steady fow: Laminar fow, the veocity of the moving fuid at any fixed point does not change with time 2. Incompressibe fow: The idea fuid density has a constant, uniform vaue 3. Nonviscous fow: Viscosity is, roughy, resistance to fow, fuid anaog of friction. No resistive force here 4. Irrotationa fow: May fow in a circe, but a dust grain suspended in the fuid wi not rotate about com 2014 John Wiey & Sons, Inc. A rights reserved.
10 14-6 The Equation of Continuity 14-6 The Equation of Continuity Visuaize fuid fow by adding a tracer Each bit of tracer (see figure 14-13) foows a streamine A streamine is the path a tiny eement of fuid foows Veocity is tangent to streamines, so they can never intersect (then 1 point woud experience 2 veocities) Figure John Wiey & Sons, Inc. A rights reserved. Figure 14-14
11 14-6 The Equation of Continuity 14-6 The Equation of Continuity Fuid speed depends on cross-sectiona area Because of incompressibiity, the voume fow rate through any cross-section must be constant We write the equation of continuity: Hods for any tube of fow whose boundaries consist of streamines Fuid eements cannot cross streamines Eq. (14-23) 2014 John Wiey & Sons, Inc. A rights reserved. Figure 14-15
12 14-6 The Equation of Continuity 14-6 The Equation of Continuity We can rewrite the equation as: Eq. (14-24) Where RV is the voume fow rate of the fuid (voume passing a point per unit time) If the fuid density is uniform, we can mutipy by the density to get the mass fow rate: Eq. (14-25) 2014 John Wiey & Sons, Inc. A rights reserved.
13 14-7 Bernoui's Equation 14-7 Bernoui's Equation Learning Objectives Cacuate the kinetic energy density in terms of a fuid's density and fow speed Identify the fuid pressure as being a type of energy density Cacuate the gravitationa potentia energy density Appy Bernoui's equation to reate the tota energy density at one point on a streamine to the vaue at another point Identify that Bernoui's equation is a statement of the conservation of energy John Wiey & Sons, Inc. A rights reserved.
14 14-7 Bernoui's Equation 14-7 Bernoui's Equation Figure represents a tube through which an idea fuid fows Appying the conservation of energy to the equa voumes of input and output fuid: Eq. (14-28) The ½ρv 2 term is caed the fuid's kinetic energy density Figure John Wiey & Sons, Inc. A rights reserved.
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16 14-7 Bernoui's Equation 14-7 Bernoui's Equation Equivaent to Eq , we can write: Eq. (14-29) These are both forms of Bernoui's Equation Appying this for a fuid at rest we find Eq Appying this for fow through a horizonta pipe: Eq. (14-30) 2014 John Wiey & Sons, Inc. A rights reserved.
14-6 The Equation of Continuity
14-6 The Equation of Continuity 14-6 The Equation of Continuity Motion of rea fuids is compicated and poory understood (e.g., turbuence) We discuss motion of an idea fuid 1. Steady fow: Laminar fow, the
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