Problem 10.98 Challenge
หัวข้อ: ทฤษฎีบทของคาร์โนต์ (Carnot’s Theorem)
\(e_{D}< e_{C}\;\blacksquare\)
ดูวิธีทำ
จาก \(Q = nc_{P}\Delta T; Isobaric Process\)
\(Q = nc_{P}(T_{3} - T_{2})\)
\(Q = nc_{P}(\frac{P_{2}V_{3}}{nR}-\frac{P_{2}V_{2}}{nR})\)
\(Q = c_{P}\frac{P_{2}}{R}(V_{3} - V_{2})\)
\(|Q| = c_{P}\frac{P_{2}V_{2}}{R}(\frac{V_{3}}{V_{2}}- 1)\cdots \cdots (1)\)
จาก \(q = nc_{V}\Delta T; Isovolumetric Process\)
\(q = nc_{V}(T_{1} - T_{4})\)
\(q = nc_{V}(\frac{P_{1}V_{1}}{nR}-\frac{P_{4}V_{1}}{nR})\)
\(q = c_{V}\frac{V_{1}}{R}(P_{1} - P_{4})\)
\(|q| = c_{V}\frac{P_{1}V_{1}}{R}(\frac{P_{4}}{P_{1}}- 1)\cdots \cdots (2)\)
พิจารณาช่วง \(1 \to 2\)
\(_{\gamma\gamma}\)
จะได้\(P_{2}V_{2}=P_{1}V_{1}\)
\(_{\gamma}\)
\(\frac{P_{1}}{P_{2}}= (\frac{V_{2}}{V_{1}})\cdots \cdots (3)\)
พิจารณาช่วง \(3 \to 4\)
\(_{\gamma\gamma}\)
จะได้\(P_{2}V_{3}=P_{4}V_{1}\)
\(_{\gamma}\)
\(\frac{P_{4}}{P_{2}}= (\frac{V_{3}}{V_{1}})\cdots \cdots (4)\)
\(_{\gamma}\)
\((4)/(3)\) จะได้\(\frac{P_{4}}{P_{1}}= (\frac{V_{3}}{V_{2}})\cdots \cdots (5)\)
\(_{\gamma}\)
\((5)\) แทนใน \((2)\) จะได้ \(|q| = c_{V}\frac{P_{1}V_{1}}{R}((\frac{V_{3}}{V_{2}})-1)\cdots \cdots (6)\)
\(_{\gamma}\)
\((6)/(1)\) จะได้ \(\frac{|q|}{|Q|}=\frac{P_{1}V_{1}}{\gamma P_{2}V_{2}}(((\frac{V_{3}}{V_{2}})-1)/(\frac{V_{3}}{V_{2}}- 1))\)
\(_{\gamma\gamma}\)
\(\frac{|q|}{|Q|}=\frac{V_{1}}{\gamma V_{2}}(\frac{V_{2}}{V_{1}}) (((\frac{V_{3}}{V_{2}})-1)/(\frac{V_{3}}{V_{2}}- 1)) ; (1)\)
\(_{1 - \gamma\gamma}\)
\(\frac{|q|}{|Q|}=\frac{1}{\gamma}(\frac{V_{1}}{V_{2}})(((\frac{V_{3}}{V_{2}})-1)/(\frac{V_{3}}{V_{2}}- 1))\)
กำหนดให้ \(r = \frac{V_{1}}{V_{2}}\) และ \(α = \frac{V_{3}}{V_{2}}\)
\(_{1-\gamma\gamma}\)
จะได้ \(\frac{|q|}{|Q|}=\frac{1}{\gamma}r((\alpha -1)/(\alpha -1))\)
\(_{1-\gamma\gamma}\)
\(1 -\frac{|q|}{|Q|}=1-\frac{1}{\gamma}r((\alpha -1)/(\alpha -1))\)
\(_{1-\gamma\gamma}\)
\(e_{D}=1-\frac{1}{\gamma}r((\alpha -1)/(\alpha -1))\;\blacksquare\)
\(_{\gamma}\)
จาก \(α> \alpha ; γ > 1\) และ \(α > 1\)
\(_{\gamma}\)
\(\alpha -1> \alpha - 1\)
\(_{\gamma}\)
\((\alpha -1)/(\alpha -1)>1\)
\(_{1-\gamma}\)
จาก \(r< 1; γ > 1\) และ \(r > 1\)
\(_{1-\gamma}\)
\(-r> -1\)
จาก \(< 1\frac{1}{\gamma}; \gamma > 1\)
\(-\frac{1}{\gamma}>-1\)
\(_{1-\gamma\gamma}\)
\(\therefore (-\frac{1}{\gamma})(-r)((\alpha -1)/(\alpha -1))>(-1)(-1)(1)\)
\(_{1-\gamma\gamma}\)
\(r\frac{1}{\gamma}(\alpha -1)/(\alpha -1)>1\)
\(_{1-\gamma\gamma}\)
\(r\frac{1}{\gamma}(\alpha -1)/(\alpha -1)>\frac{t}{T}\)
\(_{1-\gamma\gamma}\)
\(-\frac{1}{\gamma}r(\alpha -1)/(\alpha -1)<-\frac{t}{T}\)
\(_{1-\gamma\gamma}\)
\(1-\frac{1}{\gamma}r(\alpha -1)/(\alpha -1)<1-\frac{t}{T}\)
\(e_{D}< e_{C}\;\blacksquare\)