Question

A heat engine with 0.500 mol of a monatomic ideal gas initially fills a 3000 cm3 cylinder at 800 K. The gas goes through the following closed cycle: - Isothermal expansion to 4000 cm3. - Isochoric cooling to 300 K. - Isothermal compression to 3000 cm3. - Isochoric heating to 800 K.

Answer 1

Step-by-step explanation:

`\text{For isothermal expansion:}`

`W_1 = nRT_1 In ((V_2)/(V_1)) \\ \\ W_1 = 0.5 * 8.314 * 800 * In ((4000)/(3000))\\ \\ W _1 = 956.72 \ J \\ \\ Q_1 = W_1 = 956.72 \ J \ since \ (dU=0) \\ \\ \\ \text{For isochoric cooling,} W_2 = 0} \\ \\ Q_2 = nCr \Delta T \\ \\ Q_2 = 0.5 ((3R)/(2))(T_2-T_1) \\ \\ Q_2 = 0.5 * (3* 8.314)/(2)(-500)= -3117.75`

`\text{For Isothermal compression:}\\\\ W_3 = nRT_2 \ In ((V_4)/(V_3)) \\ \\ 0.5 * 8.314 * 300 * In ((3000)/(4000)) \\ \\ W_3 =- 358.77 \ J \\ \\ Q_3=W_3= - 358.77 \ J`

`\text{For isochoric heating; }W_4 =0} \\ \\ Q_4 = nC_v\Delta T \\ \\ = 0.5 * (3)/(2)* 8.314 * 500 \\ \\ Q_4 = 3117.5 \ J`

`\text{Total workdone W}= W_1 + W_2+W_3+W_4 \\ \\ W = 956.71 \ J + 0 + (-358.77 \ J) +0 \\ \\ \mathbf{W = 597.94 J} \\ \\ \\ \eta = (Work \ done)/(heat \ taken ) \\ \\ \eta = (W)/(Q_1+Q_4) \\ \\ \eta = (597.94 \ J)/(956.71 \ J + 3117.5 \ J) \\ \\ \eta = 0.1468 \\ \\ \mathbf{\eta = 14.68\%}`