31, 37 and 39. thank you! **Transcribed Text for Educational Website**35. The work done by an engine equals one-fourth the energy it absorbs from a reservoir. (a) What is its thermal efficiency? (b) What fraction of the energy absorbed is expelled to the cold reservoir?36. In each cycle of its operation, a heat engine expels 2400 J of energy and performs 1800 J of mechanical work. (a) How much thermal energy must be added to the engine in each cycle? (b) Find the thermal efficiency of the engine.37. One of the most efficient engines ever built is a coal-fired steam turbine engine in the Ohio River Valley, driving an electric generator as it operates between 1870°C and 430°C. (a) What is its maximum theoretical efficiency? (b) Its actual efficiency is 42.0%. How much mechanical power does the engine deliver if it absorbs 1.40 × 10^5 J of energy each second from the hot reservoir.38. A lawnmower engine ejects 1.00 × 10^4 J each second while running with an efficiency of 0.200. Find the engine’s horsepower rating, using the conversion factor 1 hp = 746 W.39. An engine absorbs 1.70 kJ from a hot reservoir at 277°C and expels 1.20 kJ to a cold reservoir at 27°C in each cycle. (a) What is the engine’s efficiency? (b) How much work is done by the engine in each cycle? (c) What is the power output of the engine if each cycle lasts 0.300 s?40. A heat pump has a coefficient of performance of 3.80 and operates with a power consumption of 7.03 × 10^3 W. (a) How much energy does the heat pump deliver into a home during---**Note:**The image contains five problems related to thermodynamics and heat engines. Each exercise involves calculations related to energy, efficiency, and power. The descriptions provide foundational problem scenarios for engineering or physics students to apply thermodynamic principles. **12.4 Heat Engines and the Second Law of Thermodynamics****31.** A gas increases in pressure from 2.00 atm to 6.00 atm at a constant volume of 1.00 m³ and then expands at constant pressure to a volume of 3.00 m³ before returning to its initial state as shown in Figure P12.31. How much work is done in one cycle?**32.** An ideal gas expands at constant pressure of 6.00 x 10³ Pa from a volume of 1.00 m³ to a volume of 4.00 m³ and then is compressed to one-third that pressure and a volume of 2.00 m³ as shown in Figure P12.32. How much work is done in one cycle?### Figure Descriptions:- **Figure P12.31**: This is a pressure-volume (P-V) diagram with pressure (P) in atm on the y-axis and volume (V) in m³ on the x-axis. The cycle forms a triangle with vertices at (1.00, 2.00), (1.00, 6.00), and (3.00, 2.00).- **Figure P12.32**: This is another pressure-volume (P-V) diagram with pressure (P) in kPa on the y-axis and volume (V) in m³ on the x-axis. The cycle forms a triangle with vertices at (1.00, 6.00), (4.00, 6.00), and (2.00, 2.00).These diagrams visually represent thermodynamic cycles where work done is calculated as the area enclosed by the triangular paths on the P-V diagram.

(31) Work done by the gas is the area under the curve W=12bh=124 atm2 m31.013×105 Pa1 atm=4.05×105 J

37. One of the most efficient engines ever built is a coal-fired steam turbine engine in the Ohio River valley, driving an electric generator as it operates between 1 870°C and 430°C. (a) What is its maximum theoretical efficiency? (b) Its actual efficiency is 42.0%. How much mechanical power does the engine deliver if it absorbs 1.40 × 10° J of energy each second from the hot reservoir.

37. One of the most efficient engines ever built is a coal-fired steam turbine engine in the Ohio River valley, driving an electric generator as it operates between 1 870°C and 430°C. (a) What is its maximum theoretical efficiency? (b) Its actual efficiency is 42.0%. How much mechanical power does the engine deliver if it absorbs 1.40 × 10° J of energy each second from the hot reservoir.