III. Energy vs. Free Enerky A) Consider two pistons cach containing an equal amooret of ideal gas and reach in thermal equi Litrium with the same exvironment. The ga in Piston \( \mathrm{A} \) is compresed to a velume V/2. and the gas in Pistos B is compresed to 2 volume \( V / 4 \). The pistons are held in pluce uttil the moment of release, at which point ewc. is allowerd to pash on two identical blocks po sitioned next to it as show 1. After the phstons are relesand, describo the motion of the blocks. Does one move further than the other, snd if so, which oae? 2. Before the pistons sre released, how docs the cnergy of the gas in piston A cotrpare to the energy of the gas in piaton \( B \) ? How do you lanow? What type of energy is it? 3. Are your answers to questions 1 and 2 coasistent? If so, explain why. If not, don't mak them consistent just to answer the question-for now just record your observations Now let's think about what happens to the entropy and free energy of the gas duri the expansion process described in Part (A). For clarity, let's focus on just the gas in piston and ignore the block for now. The plston in both the before and after states is thermal equilibrium with the same environment. Piston A (before) Piston A (after) 1. How does the enetgy \( U \) of the gas after the expansion compare to the energy of the gas before the piston is released? How do you know? 2. How does the entropy \( S \) of the gas after the expansion compare to the entropy of the gas before the piston is released? Explain. 3. How does the free eucrgy \( C \) of the gas after the expansion compare to the free energy of the gas before the piston is relcased? Rocall that \( H-U+P V \) and \( G=H-T S \). 4. Is your answer to question 3 comsi-tent with whether or not the piston will mov outward spontaneously? Explain. Now, let's put all the picces together 1. In question A.2, you compared the energy of the gas in each cylinder before releas the pistons. Now, use the result of question B. 3 to compare the free ensergy of the in each cylinder before the pistons are relcased. 2. Considering the effects the pistons had on the blocks, which of the following is true? (a) "Energy is the capacity of a system to perform mochanical work." (b) "Free energy is the capacity of a system to perform morhnnical work." 3. Describe the role that entropy plays in relating free cnergy to energy. 4. Finally, resulve the original issue of question A. 3 (if you did! noi do so originally). [Hiut: Remnmber, evon though we hawo o lot of differeni. labels for eneryy types, the tolal amount of elergy must Ix accounted for. Whirh law addresses total quentily of cmorgy?]
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The gasoline internal combustion engine operates in a cycle consisting of six parts. Four of these parts involve, among other things, friction, heat exchange through finite temperature differences, and accelerations of the piston; it is irreversible. Nevertheless, it is represented by the ideal reversible Otto cycle, which is illustrated below. The working substance of the cycle is assumed to be air. The six steps of the Otto cycle are as follows: i. Isobaric intake stroke (OA). A mixture of gasoline and air is drawn into the combustion chamber at atmospheric pressure $p_{0}$ as the piston expands, increasing the volume of the cylinder from zero to $V_{A}$ ii. Adiabatic compression stroke $(A B) .$ The temperature of the mixture rises as the piston compresses it adiabatically from a volume $V_{\mathrm{A}}$ to $V_{\mathrm{B}}$ iii. Ignition at constant volume (BC). The mixture is ignited by a spark. The combustion happens so fast that there is essentially no motion of the piston. During this process, the added heat $Q_{1}$ causes the pressure to increase from $p_{B}$ to $p_{C}$ at the constant volume $V_{\mathrm{B}}\left(=V_{\mathrm{C}}\right)$ iv. Adiabatic expansion (CD). The heated mixture of gasoline and air expands against the piston, increasing the volume from $V_{C}$ to $V_{D}$ This is called the power stroke, as it is the part of the cycle that delivers most of the power to the crankshaft. v. Constant-volume exhaust $(D A)$. When the exhaust valve opens, some of the combustion products escape. There is almost no movement of the piston during this part of the cycle, so the volume remains constant at $V_{A}\left(=V_{D}\right)$ Most of the available energy is lost here, as represented by the heat exhaust $Q_{2}$ vi. Isobaric compression (AO). The exhaust valve remains open, and the compression from $V_{A}$ to zero drives out the remaining combustion products. (a) Using (i) $e=W / Q_{1} ;$ (ii) $W=Q_{1}-Q_{2} ;$ and (iii) $Q_{1}=n C_{\nu}\left(T_{C}-T_{B}\right), Q_{2}=n C_{\nu}\left(T_{D}-T_{A}\right),$ show that $e=1-\frac{T_{D}-T_{A}}{T_{C}-T_{B}}$ (b) Use the fact that steps (ii) and (iv) are adiabatic to show that $e=1-\frac{1}{r^{\gamma-1}}$ where $r=V_{A} / V_{B}$ The quantity $r$ is called the compression ratio of the engine. (c) In practice, $r$ is kept less than around 7 . For larger values, the gasoline-air mixture is compressed to temperatures so high that it explodes before the finely timed spark is delivered. This preignition causes engine knock and loss of power. Show that for $r=6$ and $\gamma=1.4$ (the value for air), $e=0.51,$ or an efficiency of $51 \%$ Because of the many irreversible processes, an actual internal combustion engine has an efficiency much less than this ideal value. A typical efficiency for a tuned engine is about $25 \%$ to $30 \%$
20.50. A stirling-cycle Engine. the Otto cycle, except that the compression and expansion of the gas are done at constant temperature, not adiabatically as in the Otto cycle. The Stirling cycle is used in external combustion engines (in fact, burning fuel is not necessary; any way of producing a temperature difference will do -solar, geothermal, ocean temperature gradient, etc. $.$ which means that the gas inside the cylinder is not used in the combustion process. Heat is supplied by burning fuel steadily outside the cylinder, instead of explosively inside the cylinder as in the Otto cycle. For this reason Stirling-cycle engines are quieter than Otto-cycle engines, since there are no intake and exhaust valves (a major source of engine noise). While small Stirling engines are used for a variety of purposes, Stiring engines for automobiles have not been successful because they are larger, heavier, and more expensive than conventional automobile engines. In the cycle, the working fluid goes through the following sequence of steps (Fig. 20.30$)$ : (i) Compressed isothermally at temperature $T_{1}$ from the initial state $a$ to state $b$ , with a compression ratio $r .$ (ii) Heated at constant volume to state $c$ at temperature $T_{2}$ . (iii) Expanded isothermally at $T_{2}$ to state $d$ . (iv) Cooled at constant volume back to the initial state $a$ . Assume that the working fluid is $n$ moles of an ideal gas (for which $C_{V}$ is independent of temperature). (a) Calculate $Q, W,$ and $\Delta U$ for each of the processes $a \rightarrow b, b \rightarrow c, c \rightarrow d,$ and $d \rightarrow a$ . (b) In the Stirling cycle, the heat transfers in the processes $b \rightarrow c$ and $d \rightarrow a$ do not involve external heat sources but rather use regeneration: The same substance that transfers heat to the gas inside the cylinder in the process $b \rightarrow c$ also absorbs heat back from the gas in the process $d \rightarrow a$ . Hence the heat transfers $Q_{b \rightarrow c}$ and $Q_{d \rightarrow a}$ do not play a role in determining the efficiency of the engine. Explain this last statement by comparing the expressions for $Q_{b \rightarrow c}$ and $Q_{d \rightarrow a}$ calculated in part (a). (c) Calculate the efficiency of a Stirling-cycle engine in terms of the temperatures $T_{1}$ and $T_{2}$ . How does this compare to the efficiency of a Carnot-cycle engine operating between these same two temperatures? (Historically, the Stirling cycle was devised before the Carnot cycle.) Does this result violate the second law of thermodynamics? Explain. Unfortunately, actual Stirling-cycle engines cannot achieve this efficiency due to problems with the heat-transfer processes and pressure losses in the engine.
A Stirling-Cycle Engine. The Stirling cycle is similar to the Otto cycle, except that the compression and expansion of the gas are done at constant temperature, not adiabatically in the Otto cycle. The Stirling cycle is used in external combustion engines (in fact, burning fuel is not necessary; any way of producing a temperature difference will do-solar, geothermal, ocean temperature gradient, etc.), which means that the gas inside the cylinder is not used in the combustion process. Heat is supplied by burning fuel steadily outside the cylinder, instead of explosively inside the cylinder as in the Otto cycle. For this reason Stirling-cycle engines are quieter than Otto-cycle engines, since there are no intake and exhaust valves (a major source of engine noise). While small Stirling engines are used for a variety of purposes, Stirling engines for automobiles have not been successful because they are larger, heavier, and more expensive than conventional automobile engines. In the cycle, the working fluid goes through the following sequence of steps (Fig. P20.52): \begin{equation}\begin{array}{l}{\text { (i) Compressed isothermally at temperature } T_{1} \text { from the ir }} \\ {\text { state } a \text { to state } b, \text { with a compression ratio } r .} \\ {\text { (ii) Heated at constant volume to state } c \text { at temperature } T_{2}} \\ {\text { (iii) Expanded isothermally at } T_{2} \text { to state } d .} \\ {\text { (iv) Cooled at constant volume back to the initial state } a \text { . }}\end{array}\end{equation} Assume that the working fluid is $n$ moles of an ideal gas (for which $C_{V}$ is independent of temperature). (a) Calculate $Q, W,$ and $\Delta U$ for each of the processes $a \rightarrow b, b \rightarrow c, c \rightarrow d,$ and $d \rightarrow a$ . (b) In the Stirling cycle, the heat transfers in the processes $b \rightarrow c$ and $d \rightarrow a$ do not involve external heat sources but rather use regeneration: The same substance that transfers heat to the gas inside the cylinder in the process $b \rightarrow c$ also absorbs heat back from the gas in the process $d \rightarrow a .$ Hence the heat transfers $Q_{b \rightarrow c}$ and $Q_{d \rightarrow a}$ do not play a role in determining the efficiency of the engine. Explain this last statement by comparing the expressions for $Q_{b \rightarrow c}$ and $Q_{d \rightarrow a}$ calculated in part (a). (c) Calculate the efficiency of a Stirlingcycle engine in terms of the temperatures $T_{1}$ and $T_{2} .$ How does this compare to the efficiency of a Carnot-cycle engine operating between these same two temperatures? (Historically, the Stirling cycle was devised before the Carnot cycle.) Does this result violate the second law of thermodynamics? Explain. Unfortunately, actual Stirling-cycle engines cannot achieve this efficiency due to problems with the heat-transfer processes and pressure losses in the engine.
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