Problem 10.97 Challenge
หัวข้อ: ทฤษฎีบทของคาร์โนต์ (Carnot’s Theorem)
\(e_{B}< e_{C}\;\blacksquare ; (7)\)
ดูวิธีทำ
T
q
\(^{Adiabatic}1\)
\(\frac{_{Process}t}{V_{2}V_{1}}V\)
จาก \(Q = nc_{V}(T_{3} - T_{2}); Isovolumetric Process\)
\(|Q| = nc_{V}(T_{3} - T_{2})\cdots \cdots (1)\)
จาก \(q = nc_{V}(T_{1} - T_{4}); Isovolumetric Process\)
\(|q| = nc_{V}(T_{4} - T_{1})\cdots \cdots (2)\)
พิจารณาช่วง \(1 \to 2\)
\(_{\gamma -1\gamma -1}\)
จะได้ \(T_{2}V_{2}= T_{1}V_{1}; Adiabatic Process\)
\(_{\gamma -1}\)
\(T_{2}= T_{1}(\frac{V_{1}}{V_{2}})\cdots \cdots (3)\)
พิจารณาช่วง \(3 \to 4\)
\(_{\gamma -1\gamma -1}\)
จะได้ \(T_{4}V_{1}= T_{3}V_{2}; Adiabatic Process\)
\(_{\gamma -1}\)
\(T_{4}= T_{3}(\frac{V_{2}}{V_{1}})\cdots \cdots (4)\)
\((3)\) แทนใน \((1)\) จะได้
\(_{\gamma -1}\)
\(|Q| = nc_{V}(T_{3} - T_{1}(\frac{V_{1}}{V_{2}})) \cdots \cdots (5)\)
\((4)\) แทนใน \((2)\) จะได้
\(_{\gamma -1}\)
\(|q| = nc_{V}(T_{3}(\frac{V_{2}}{V_{1}})- T_{1})\)
\(_{\gamma -1\gamma -1}\)
\(|q| = nc_{V}(T_{3} - T_{1}(\frac{V_{1}}{V_{2}}))(\frac{V_{2}}{V_{1}})\)
\(_{\gamma -1}\)
\(|q| = |Q|(\frac{V_{2}}{V_{1}}); (5)\)
\(_{\gamma -1}\)
\(\frac{|q|}{|Q|}= (\frac{V_{2}}{V_{1}})\)
\(_{\gamma -1}\)
\(1-\frac{|q|}{|Q|}= 1 - (\frac{V_{2}}{V_{1}})\)
\(_{\gamma -1}\)
\(e_{B}= 1 - (\frac{V_{2}}{V_{1}})\)
\(_{1-\gamma}\)
\(e_{B}= 1 - (\frac{V_{1}}{V_{2}})\;\blacksquare\)
พิจารณาช่วง \(3 \to 4\)
\(_{\gamma -1\gamma -1}\)
จะได้ \(T_{4}V_{1}= TV_{2}; Adiabatic Process\)
\(_{\gamma -1}\)
\(\frac{T_{4}}{T}= (\frac{V_{2}}{V_{1}})\)
\(_{\gamma -1}\)
\(1 -\frac{T_{4}}{T}= 1 - (\frac{V_{2}}{V_{1}})\)
\(1 -\frac{T_{4}}{T}= e_{B}; (7)\)
จาก \(T_{4} > t\)
\(\frac{T_{4}}{T}>\frac{t}{T}\)
\(-\frac{T_{4}}{T}< -\frac{t}{T}\)
\(1 -\frac{T_{4}}{T}< 1 -\frac{t}{T}\)
\(e_{B}< e_{C}\;\blacksquare ; (7)\)