Final answer:
To find the minimum possible temperature of the hot reservoir for an engine with given work and waste heat values, we use the efficiency of a Carnot engine. The calculations lead to a minimum temperature for the hot reservoir of approximately 246.4 °C.
Step-by-step explanation:
The question asks for the minimum possible temperature of the hot reservoir for an engine performing 25 J of work and exhausting 30 J of waste heat per cycle, with a cold-reservoir temperature of 10 °C. We calculate this using the laws of thermodynamics and the efficiency equation for a Carnot engine.
To find the minimum possible temperature of the hot reservoir (Th), we need to convert the cold reservoir temperature from Celsius to Kelvin by adding 273.15, which gives us Tc = 283.15 K. The efficiency (η) of the engine can be calculated as the work done divided by the total heat input (work done + waste heat): η = 25 J / (25 J + 30 J) = 25/55 ≈ 0.4545. Now, using the efficiency formula for a Carnot engine: η = 1 - (Tc/Th), we can solve for Th.
Rearranging the Carnot efficiency equation and solving for Th yields: Th = Tc / (1 - η). Substituting the known values we get: Th = 283.15 K / (1 - 0.4545) ≈ 519.5 K. Converting this back to Celsius by subtracting 273.15 gives us: Th ≈ 246.4 °C.
Therefore, the minimum possible temperature of the hot reservoir is 246.4 °C.