One mole of a monoatomic gas behaving ideally is used as working substance in an engine working in the cycle as shown in the figure. The process AB , BC , CD and DA are respectively reversible isobari

Question

One mole of a monoatomic gas behaving ideally is used as working substance in an engine working in the cycle as shown in the figure. The process $AB, BC, CD$ and DA are respectively reversible isobaric, adiabatic, isochoric and isothermal. The ratio of maximum to minimum volume and temperature during the cycle is $82$ and 4 respectively. If the maximum T is 800 K and $γ = 5/3$ . Calculate $Δ U$ (in kJ) for the process BC. Given $R = 8.3 J / K - mol$ .

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Identify Temperatures: Since T ma x ​ = 800 K and T ma x ​ / T min ​ = 4 , the minimum temperature is T B ​ = T min ​ = 200 K , and T A ​ = T D ​ = 800 K (isothermal). Find Volume Relationship: For the isobaric process AB, T A ​ V A ​ ​ = T B ​ V B ​ ​ ⟹ 800 V A ​ ​ = 200 V B ​ ​ ⟹ V B ​ = 4 1 ​ V A ​ . Determine T C ​ : Given V min ​ V ma x ​ ​ = V C ​ V A ​ ​ = 8 2 ​ , we get V C ​ V B ​ ​ = V A ​ / ( 8 2 ​ ) V A ​ /4 ​ = 2 2 ​ = 2 3/2 . Calculate T C ​ : For the adiabatic process BC ( γ − 1 = 3 2 ​ ): T C ​ = T B ​ ( V C ​ V B ​ ​ ) 2/3 = 200 × ( 2 3/2 ) 2/3 = 400 K . Calculate Δ U BC ​ : Δ U BC ​ = n C v ​ ( T C ​ − T B ​ ) = 1 × 2 3 ​ × 8.3 × ( 400 − 200 ) = 2490 J = 2.49kJ .

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