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Chapter 14: Problem 75

Two heat engines \(A\) and \(B\) have their sources at \(1000 \mathrm{~K}\) and \(1100 \mathrm{~K}\) and their sinks are at \(500 \mathrm{~K}\) and \(400 \mathrm{~K}\) respectively. What is true about their efficiencies? (a) \(\eta_{A}=\eta_{B}\) (b) \(\eta_{A}>\eta_{B}\) (c) \(\eta_{A}<\eta_{B}\) (d) Cannot say

Short Answer

Expert verified
(c) \(\eta_{A}<\eta_{B}\)

Step by step solution

01

Understand the Efficiency Formula

The efficiency \(\eta\) of a heat engine is determined by the formula \(\eta = 1 - \frac{T_C}{T_H}\), where \(T_C\) is the temperature of the cold reservoir (sink) and \(T_H\) is the temperature of the hot reservoir (source). Both temperatures should be in Kelvin (K).
02

Calculate Efficiency of Engine A

For Engine A, the source temperature \(T_{H,A} = 1000\text{ K}\) and the sink temperature \(T_{C,A} = 500\text{ K}\). Apply the formula: \[\eta_A = 1 - \frac{500}{1000} = 1 - 0.5 = 0.5.\] The efficiency of Engine A is 0.5 or 50%.
03

Calculate Efficiency of Engine B

For Engine B, the source temperature \(T_{H,B} = 1100\text{ K}\) and the sink temperature \(T_{C,B} = 400\text{ K}\). Apply the formula: \[\eta_B = 1 - \frac{400}{1100} = 1 - \frac{4}{11}.\] Simplify to compute this: \(1 - 0.3636 \approx 0.6364.\) The efficiency of Engine B is approximately 0.6364 or 63.64%.
04

Compare Efficiencies

Now, compare the efficiencies: Engine A has an efficiency of 50% and Engine B has an efficiency of approximately 63.64%. Since \(0.5 < 0.6364\), Engine A is less efficient than Engine B.
05

Determine the Correct Answer

Since \(\eta_A = 0.5\) and \(\eta_B \approx 0.6364\), \(\eta_{A}<\eta_{B}\). Therefore, the correct answer is option (c), \(\eta_{A}<\eta_{B}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Heat Engines
Heat engines are fascinating devices found in so many of the machines we use daily! They work by absorbing heat from a high-temperature source, converting part of this energy into work, and then releasing the remaining heat to a low-temperature sink. The key to their operation is a cycle of processes that a working substance undergoes, gaining and releasing energy.
  • In the case of heat engines, the primary focus is on converting thermal energy into mechanical work efficiently.
  • The engines typically include components like pistons, turbines, or other mechanisms that convert energy into work throughout the cycle.
  • It's important to grasp that no heat engine can convert 100% of heat energy into work due to inherent limitations like friction and resistance.
Therefore, understanding how efficiently these engines operate is crucial for tasks like designing new engines or improving existing technologies.
Efficiency Calculation
Efficiency plays a significant role in understanding how well heat engines perform their task of converting heat into work. Efficiency, denoted by \(\eta\), is calculated using the formula: \[ \eta = 1 - \frac{T_C}{T_H}.\] Here, \(T_C\) represents the temperature of the cold reservoir, and \(T_H\) represents the temperature of the hot reservoir. All temperatures should be in Kelvin to maintain accuracy.
  • The efficiency of a heat engine tells us what fraction of the absorbed heat is converted into useful work.
  • A higher efficiency signifies a more effective engine that loses less energy to the surroundings.
  • Practically, perfect efficiency is never achievable due to energy losses from factors like friction and irreversible processes.
You can use this formula to compare two heat engines, as shown in the example, where Engine A has an efficiency of 50% and Engine B around 63.64%. The comparison illustrates how small differences in temperature can significantly impact the engine's performance.
Carnot Cycle
The Carnot cycle represents an idealized heat engine cycle proposed by Sadi Carnot in the 19th century. It's a theoretical model that sets the upper limit on the efficiency that any heat engine can achieve. This cycle includes four stages: two isothermal processes and two adiabatic processes.
  • The cycle begins with an isothermal expansion where the engine absorbs heat from the hot reservoir.
  • This is followed by an adiabatic expansion where the engine continues to do work without exchanging heat.
  • Subsequently, an isothermal compression occurs where the engine releases heat to the cold reservoir.
  • Finally, during adiabatic compression, the cycle is completed without any heat exchange, bringing the system back to its original state.
Using the Carnot cycle as a reference, we see that real engines constantly strive to reach this ideal efficiency. This cycle fundamentally illustrates why increasing the temperature difference between the heat source and sink can lead to more efficient heat engines.

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Most popular questions from this chapter

In a thermodynamic process, pressure of a fixed mass of gas is changed in such a manner that the gas releases \(20 \mathrm{~J}\) of heat and \(8 \mathrm{~J}\) of work is done on the gas. If internal energy of the gas was \(30 \mathrm{~J}\), then the final internal energy will be (a) \(42 \mathrm{~J}\) (b) \(18 \mathrm{~J}\) (c) \(12 \mathrm{~J}\) (d) \(60 \mathrm{~J}\)

The pressure inside a tyre is \(4 \mathrm{~atm}\) at \(27^{\circ} \mathrm{C}\). If the tyre bursts suddenly, new temperature will be \((\gamma=7 / 5)\) (a) \(300(4)^{7 / 2}\) (b) \(300(4)^{2 / 7}\) (c) \(300(2)^{7 / 2}\) (d) \(300(4)^{-2 / 7}\)

A Carnot engine has same efficiency between (i) \(100 \mathrm{~K}\) and \(500 \mathrm{~K}\), (ii) \(T \mathrm{~K}\) and \(900 \mathrm{~K}\). The value of \(T\) is (a) \(180 \mathrm{~K}\) (b) \(90 \mathrm{~K}\) (c) \(270 \mathrm{~K}\) (d) \(360 \mathrm{~K}\)

If the heat of \(110 \mathrm{~J}\) is added to a gaseous system and it acquires internal energy of \(40 \mathrm{~J}\), then the amount of internal work done is (a) \(40 \mathrm{~J}\) (b) \(70 \mathrm{~J}\) (c) \(150 \mathrm{~J}\) (d) \(110 \mathrm{~J}\)

What is the temperature of source in Carnot cycle of \(10 \%\) efficiency when heat exhausts at \(270 \mathrm{~K}\) ? (a) \(400 \mathrm{~K}\) (b) \(500 \mathrm{~K}\) (c) \(300 \mathrm{~K}\) (d) \(600 \mathrm{~K}\)
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