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Chapter 20: Problem 22

With each cycle, a 2500.-W engine extracts 2100. J from a thermal reservoir at \(90.0^{\circ} \mathrm{C}\) and expels \(1500 .\) J into a thermal reservoir at \(20.0^{\circ} \mathrm{C}\). What is the work done for each cycle? What is the engine's efficiency? How much time does each cycle take?

Short Answer

Expert verified
Answer: In one cycle, the engine does 600 J of work, has an efficiency of 28.57%, and the cycle takes 0.24 seconds.

Step by step solution

01

Determine the work done during each cycle

Since this is a heat engine, we can use the first law of thermodynamics to determine the work done during each cycle: $$W = Q_h - Q_c$$ where \(W\) is the work done, \(Q_h\) is the heat extracted from the hot reservoir, \(Q_c\) is the heat expelled into the cold reservoir. In this case, \(Q_h = 2100\) J and \(Q_c = 1500\) J, so we have: $$W = 2100 \,\text{J} - 1500 \, \text{J} = 600 \,\text{J}$$ The work done during each cycle is 600 J.
02

Calculate the engine's efficiency

The efficiency of a heat engine is defined as the ratio of the work done to the heat extracted from the hot reservoir: $$\text{Efficiency} = \frac{W}{Q_h}$$ Using the values we found in Step 1, we can calculate the engine's efficiency: $$\text{Efficiency} = \frac{600 \, \text{J}}{2100\, \text{J}} = \frac{2}{7}$$ Converting this to a percentage, we have: $$\text{Efficiency} = \frac{2}{7} \times 100\% = 28.57\%$$ The engine's efficiency is 28.57%.
03

Determine the time taken for each cycle

To find the time taken for each cycle, we can use the given power of the engine and the work done during each cycle: $$P = \frac{W}{t}$$ where \(P\) is the power (2500 W), \(W\) is the work done during each cycle (600 J), \(t\) is the time taken for each cycle. Rearranging this formula to solve for \(t\), we get: $$t = \frac{W}{P}$$ Now, plug in the given values: $$t = \frac{600 \, \text{J}}{2500 \,\text{W}} = 0.24 \,\text{s}$$ The time taken for each cycle is 0.24 seconds.

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Most popular questions from this chapter

The burning of fuel transfers \(4.00 \cdot 10^{5} \mathrm{~W}\) of power into the engine of a \(2000 .-\mathrm{kg}\) vehicle. If the engine's efficiency is \(25.0 \%,\) determine the maximum speed the vehicle can achieve \(5.00 \mathrm{~s}\) after starting from rest.

Find the net change in entropy when \(100 . \mathrm{g}\) of water at \(0^{\circ} \mathrm{C}\) is added to \(100 . \mathrm{g}\) of water at \(100 .{ }^{\circ} \mathrm{C}\)

Is it a violation of the Second Law of Thermodynamics to capture all the exhaust heat from a steam engine and funnel it back into the system to do work? Why or why not?

What is the magnitude of the change in entropy when \(6.00 \mathrm{~g}\) of steam at \(100{ }^{\circ} \mathrm{C}\) is condensed to water at \(100{ }^{\circ} \mathrm{C} ?\) a) \(46.6 \mathrm{~J} / \mathrm{K}\) c) \(36.3 \mathrm{~J} / \mathrm{K}\) b) \(52.4 \mathrm{~J} / \mathrm{K}\) d) \(34.2 \mathrm{~J} / \mathrm{K}\)

An ideal gas undergoes an isothermal expansion. What will happen to its entropy? a) It will increase. c) It's impossible to determine. b) It will decrease. d) It will remain unchanged.
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