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Chapter 15: Problem 33

(a) Calculate the efficiency of a reversible engine that operates between the temperatures \(600.0^{\circ} \mathrm{C}\) and \(300.0^{\circ} \mathrm{C} .\) (b) If the engine absorbs \(420.0 \mathrm{kJ}\) of heat from the hot reservoir, how much does it exhaust to the cold reservoir?

Short Answer

Expert verified
Answer: The engine exhausts 275.86 kJ of heat to the cold reservoir.

Step by step solution

01

Convert temperatures to Kelvin

To use the efficiency formula, we first need to convert the given Celsius temperatures to Kelvin. To do so, simply add 273.15 to the Celsius temperature: \(T_{hot} = 600.0 + 273.15 = 873.15 \mathrm{K}\) \(T_{cold} = 300.0 + 273.15 = 573.15 \mathrm{K}\)
02

Calculate the efficiency of the engine

We can use the formula for the efficiency of a reversible heat engine: Efficiency = \(1 - \frac{T_{cold}}{T_{hot}}\) Plug in the temperatures in Kelvin: Efficiency = \(1 - \frac{573.15}{873.15} = 1 - 0.6568 = 0.3432 = 34.32\%\)
03

Calculate the heat exhausted to the cold reservoir

Now that we have the efficiency, we can use the formula to find the heat exhausted to the cold reservoir: \(Q_{cold} = Q_{hot} * (1 - \text{Efficiency})\) Given that the engine absorbs \(420.0 \mathrm{kJ}\) of heat from the hot reservoir: \(Q_{cold} = 420.0 * (1 - 0.3432) = 420.0 * 0.6568 = 275.86 \mathrm{kJ}\) So, the engine exhausts \(275.86 \mathrm{kJ}\) of heat to the cold reservoir.

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Most popular questions from this chapter

Calculate the maximum possible efficiency of a heat engine that uses surface lake water at \(18.0^{\circ} \mathrm{C}\) as a source of heat and rejects waste heat to the water \(0.100 \mathrm{km}\) below the surface where the temperature is \(4.0^{\circ} \mathrm{C}\).
An air conditioner whose coefficient of performance is 2.00 removes $1.73 \times 10^{8} \mathrm{J}$ of heat from a room per day. How much does it cost to run the air conditioning unit per day if electricity costs \(\$ 0.10\) per kilowatt-hour? (Note that 1 kilowatt-hour \(=3.6 \times 10^{6} \mathrm{J} .\) )
List these in order of increasing entropy: (a) 0.01 mol of \(\mathrm{N}_{2}\) gas in a \(1-\mathrm{L}\) container at \(0^{\circ} \mathrm{C} ;\) (b) 0.01 mol of \(\mathrm{N}_{2}\) gas in a 2 -L container at \(0^{\circ} \mathrm{C} ;\) (c) 0.01 mol of liquid \(\mathrm{N}_{2}\).
An engincer designs a ship that gets its power in the following way: The engine draws in warm water from the ocean, and after extracting some of the water's internal energy, returns the water to the ocean at a temperature \(14.5^{\circ} \mathrm{C}\) lower than the ocean temperature. If the ocean is at a uniform temperature of \(17^{\circ} \mathrm{C},\) is this an efficient engine? Will the engineer's design work?
An electric power station generates steam at \(500.0^{\circ} \mathrm{C}\) and condenses it with river water at \(27^{\circ} \mathrm{C} .\) By how much would its theoretical maximum efficiency decrease if it had to switch to cooling towers that condense the steam at \(47^{\circ} \mathrm{C} ?\)
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