Problem 10.78 Challenge
หัวข้อ: กฎข้อที่หนึ่งของอุณหพลศาสตร์ (The First Law of Thermodynamics)
\(W = -P_{1}(V_{2} - V_{1})\ln (\frac{P_{2}}{P_{1}}) \;\blacksquare\)
ดูวิธีทำ
พิจารณาช่วง A → B
จาก \(W_{AB}= -nRT_{1}\ln (\frac{V}{V_{1}}); Isothermal Process\)
\(W_{AB}= -P_{1}V_{1}\ln (\frac{V}{V_{1}}); P_{1}V_{1} = nRT_{1}\)
\(W_{AB}= -P_{1}V_{1}\ln (\frac{P_{1}}{P_{2}}); P_{2}V = P_{1}V_{1}\)
\(W_{AB}= P_{1}V_{1}\ln (\frac{P_{2}}{P_{1}})\)
พิจารณาช่วง B → C
จาก \(W_{BC}= -P_{2}\Delta V; Isobaric Process\)
\(W_{BC}= -P_{2}(V_{C}- V_{B})\)
\(W_{BC}= -P_{2}V_{C}+ P_{2}V_{B}\)
\(W_{BC}= -P_{2}V_{C}+ P_{1}V_{1}; P_{2}V_{B}= P_{1}V_{1}\)
\(W_{BC}= -P_{1}V_{2} + P_{1}V_{1}; P_{2}V_{C}= P_{1}V_{2}\)
\(W_{BC}= -P_{1}(V_{2} - V_{1})\)
พิจารณาช่วง C → D
จาก \(W_{CD}= -nRTln(\frac{V_{2}}{V}); Isothermal Process\)
\(W_{CD}= -P_{2}V_{C}\ln (\frac{V_{2}}{V}); P_{2}V_{C}= nRT\)
\(W_{CD}= -P_{1}V_{2}\ln (\frac{V_{2}}{V}); P_{2}V_{C}= P_{1}V_{2}\)
\(W_{CD}= -P_{1}V_{2}\ln (\frac{P_{2}}{P_{1}}); P_{2}V= P_{1}V_{2}\)
พิจารณาช่วง D → A
จาก \(W_{DA}= -P_{1}\Delta V; Isobaric Process\)
\(W_{DA}= -P_{1}(V_{1} - V_{2})\)
\(W_{DA}= P_{1}(V_{2} - V_{1})\)
จาก \(W = W_{AB}+ W_{BC}+ W_{CD}+ W_{DA}\)
\(W = P_{1}V_{1}\ln (\frac{P_{2}}{P_{1}}) - P_{1}(V_{2} - V_{1}) - P_{1}V_{2}\ln (\frac{P_{2}}{P_{1}}) + P_{1}(V_{2} - V_{1})\)
\(W = P_{1}V_{1}\ln (\frac{P_{2}}{P_{1}}) - P_{1}V_{2}\ln (\frac{P_{2}}{P_{1}})\)
\(W = P_{1}(V_{1} - V_{2})\ln (\frac{P_{2}}{P_{1}})\)
\(W = -P_{1}(V_{2} - V_{1})\ln (\frac{P_{2}}{P_{1}}) \;\blacksquare\)