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PROBLEM 4-3 Using the definition of pressure (force per unit area) and the definition of work (force times distance), show that P DV from Eq (4-7) is just the threedimensional work done by the system
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Work, by definition, is force times distance moved: w 5 F Dx where Dx is the distance moved (change in position) along the same direction the force was applied Pressure, by definition, is force per unit area, or P 5 F/A To simplify the problem, consider a cylinder closed at one end, with a piston moving up or down inside the cylinder from the other end; see Fig 4-1 The volume inside the cylinder (between the piston and the
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FigurE 4-1 A piston in a cylinder allows us to simplify the case of work associated with pressure and a volume change
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closed end) changes as the piston moves up and down The surface area of the piston is constant, so the volume change is just the area times the distance the piston is moved DV 5 A Dx So for the cylinder, we have F F P V 5 V 5 ( A x ) A A The A are canceled, leaving P DV 5 F Dx 5 w In the case where work is done in multiple directions, we can imagine the work done in each direction as analogous to a piston in a cylinder aligned with that direction Take, for example, the work done in compressing or expanding a balloon We can imagine that at each infinitesimally small part of the balloon s surface there is a cylinder of constant area and the expansion or contraction of the balloon is equivalent to moving a piston up or down inside the cylinder The total change in volume for the balloon is then the sum of all of these individual volume changes given by DV = A Dx Similarly the pressure against the surface of the balloon can be broken down, at each infinitesimally small part of the balloon s surface, as a force in a direction perpendicular to the balloon s surface at that point, divided by the area of the infinitesimal piston for which we are considering the volume change The volume change for any three-dimensional shape can, in this way, be broken down into a sum of many simple volume changes, each of which can be expressed as a constant area times a distance And the pressure that causes that volume change is just, at each point on the surface, the force divided by that small constant area So we see that a constant pressure times a change in volume P DV is just the three-dimensional equivalent of force times distance, or work PD V = F (AD x) = F D x A (4-10)
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The idea of entropy was developed by Rudolf Clausius in the mid-1800s It was well known at the time that heat engines of the day converted only a small portion of their heat into useful work Typically only about 1% of the heat was
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transformed The rest of the heat was lost, dissipated, merely warming up the surroundings Clausius s goal was to account quantitatively for this loss He originally called this lost energy the equivalence value of all uncompensated transformations, meaning that it was equivalent to the portion of the original heat that could not be compensated for by transforming the resulting work back into heat A few years later he coined the term entropy Clausius himself said he took the Greek word trope meaning transformation and put it in a form intended to resemble the word energy So, from trope and energy he got entropy The change in entropy as the result of a process is defined as DS 5 Q/T where S is the entropy and Q is the heat irreversibly lost to the surroundings at absolute temperature T Notice that the units on entropy are energy units divided by temperature units, for example, joules per degree celsius If a process or transformation involves passing the heat through a working body (for example, a heat engine) so that the heat is passing from a place at temperature T1 to a place at temperature T2, then the entropy change is defined as 1 1 D S = Q T1 T2 Although Clausius explained how he came up with the term entropy, he never indicated why he chose the letter S for entropy Some have speculated that it was in honor of another famous early researcher of thermodynamics, Sadi Carnot Figure 4-2 shows a diagram by Sadi Carnot illustrating the flow of heat and work in a heat engine (4-12) (4-11)
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