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Chapter 16: Problem 21

A Carnot engine is operated between two heat reservoirs at temperatures of 520 \(\mathrm{K}\) and 300 \(\mathrm{K}\) . (a) If the engine receives 6.45 \(\mathrm{kJ}\) of heat energy from the reservoir at 520 \(\mathrm{K}\) in each cycle, how many joules per cycle does it reject to the reservoir at 300 \(\mathrm{K} ?\) (b) How much mechanical work is performed by the engine during each cycle? (c) What is the thermal efficiency of the engine?

Short Answer

Expert verified
(a) The engine rejects approximately 3723.8 J per cycle. (b) It performs about 2726.2 J of work per cycle. (c) The thermal efficiency is 42.31%.

Step by step solution

01

Calculate the Carnot Efficiency

The efficiency of a Carnot engine depends only on the temperatures of the two heat reservoirs. The efficiency \( \eta \) is given by: \[ \eta = 1 - \frac{T_C}{T_H} \]Where \( T_C = 300 \, \text{K} \) and \( T_H = 520 \, \text{K} \).Substituting the given values, we have:\[ \eta = 1 - \frac{300}{520} = 1 - 0.5769 \approx 0.4231 \]So, the efficiency \( \eta \) is approximately 0.4231 or 42.31%.
02

Determine the Heat Rejected

To find the heat rejected \( Q_C \), use the relationship between the efficiency, heat absorbed \( Q_H \), and heat rejected:\[ \eta = \frac{W}{Q_H} = \frac{Q_H - Q_C}{Q_H} \]Rearrange the equation to find:\[ Q_C = Q_H (1 - \eta) \]Substitute \( Q_H = 6.45 \, \text{kJ} = 6450 \, \text{J} \) and \( \eta = 0.4231 \):\[ Q_C = 6450 \, \text{J} \times (1 - 0.4231) = 6450 \, \text{J} \times 0.5769 \approx 3723.8 \, \text{J} \]
03

Compute the Mechanical Work Done

Using the efficiency \( \eta \) we can also calculate the work done \( W \):\[ W = Q_H \times \eta \]Substitute the known values:\[ W = 6450 \, \text{J} \times 0.4231 \approx 2726.2 \, \text{J} \]
04

Verify Thermal Efficiency

Finally, verify the calculations for thermal efficiency by comparing with the work done and heat absorbed:\[ \eta = \frac{W}{Q_H} = \frac{2726.2}{6450} \approx 0.4231 \text{ or } 42.31\% \]This confirms the previous calculation for efficiency.

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Key Concepts

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

Thermal Efficiency
Thermal efficiency is a measure of how well a heat engine converts the thermal energy it receives into useful work. For a Carnot engine, thermal efficiency is determined solely by the temperatures of the hot and cold reservoirs, following the equation: \[ \eta = 1 - \frac{T_C}{T_H} \]where \( T_C \) is the temperature of the cold reservoir and \( T_H \) is the temperature of the hot reservoir.
  • The efficiency indicates the fraction of heat energy that is converted into mechanical work.
  • A higher efficiency means less energy wasted as heat.
  • Carnot engines, being idealized, represent the maximum possible efficiency that any heat engine can achieve.
Understanding thermal efficiency helps in grasping how energy is utilized in practical systems and the limits imposed by temperature differences.
Mechanical Work
Mechanical work in a Carnot engine is the amount of energy it produces to perform tasks. The work done per cycle is directly linked to the efficiency of the engine. In the Carnot cycle, the mechanical work \( W \) is given by: \[ W = Q_H \times \eta \]where \( Q_H \) is the heat absorbed from the hot reservoir. This relationship highlights:
  • How work is derived from the absorbed heat and the thermal efficiency.
  • That improvements in thermal efficiency increase the work output for the same heat input.
  • The importance of optimizing temperatures to maximize work.
Mechanical work helps in understanding the practical applications of energy conversion in engines and machinery.
Heat Reservoirs
Heat reservoirs are large bodies capable of absorbing or supplying infinite amounts of heat without changing their temperatures significantly. In the Carnot cycle, two reservoirs are involved:
  • The hot reservoir at a higher temperature \( T_H \), from which the engine absorbs heat \( Q_H \).
  • The cold reservoir at a lower temperature \( T_C \), to which the engine rejects heat \( Q_C \).
These reservoirs enable the cyclic process of the Carnot engine:
  • The engine absorbs heat from the hot reservoir to perform work.
  • It then expels some residual heat to the cold reservoir, completing the cycle.
The concept of heat reservoirs is key to understanding the flow of energy in engines and the limits of efficiency.
Carnot Cycle
The Carnot cycle is a theoretical model that describes an idealized heat engine cycle, achieving maximum efficiency due to its reversible processes. The cycle consists of four distinct stages:
  • Isothermal Expansion: The gas absorbs heat \( Q_H \) from the hot reservoir, doing work on its surroundings while maintaining a constant temperature.
  • Adiabatic Expansion: The gas continues to expand without heat exchange, doing work and lowering its temperature.
  • Isothermal Compression: Heat \( Q_C \) is expelled to the cold reservoir as the gas is compressed at a constant, lower temperature.
  • Adiabatic Compression: The gas is compressed further, raising its temperature back to the initial state without heat exchange.
This cycle helps visualize how the theoretical maximum efficiency of heat engines can be achieved and why real engines fall short of this ideal due to irreversibilities.

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Most popular questions from this chapter

. An ice-making machine operates in a Carnot cycle. It takes heat from water at \(0.0^{\circ} \mathrm{C}\) and rejects heat to a room at \(24.0^{\circ} \mathrm{C}\) . Suppose that 85.0 \(\mathrm{kg}\) of water at \(0.0^{\circ} \mathrm{C}\) are converted to ice at \(0.0^{\circ} \mathrm{C}\) (a) How much heat is rejected to the room? (b) How much energy must be supplied to the device?

\(\cdot\) In one cycle, a freezer uses 785 \(\mathrm{J}\) of electrical energy in order to remove 1750 \(\mathrm{J}\) of heat from its freezer compartment at \(10^{\circ} \mathrm{F}\) . (a) What is the coefficient of performance of this freezer? (b) How much heat does it expel into the room during this cycle?

\(\cdot\) Fach cycle, a certain heat engine expels 250 \(\mathrm{J}\) of heat when you put in 325 \(\mathrm{J}\) of heat. Find the efficiency of this engine and the amount of work you get out of the 325 \(\mathrm{J}\) heat input.

\(\cdot\) A gasoline engine. A gasoline engine takes in \(1.61 \times 10^{4} \mathrm{J}\) of heat and delivers 3700 \(\mathrm{J}\) of work per cycle. The heat is obtained by burning gasoline with a heat of combustion of \(4.60 \times 10^{4} \mathrm{J} / \mathrm{g}\) . (a) What is the thermal efficiency of the engine? (b) How much heat is discarded in each cycle? (c) What mass of fuel is burned in each cycle? (d) If the engine goes through 60.0 cycles per second, what is its power output in kilowatts? In horsepower?

. A human engine. You decide to use your body as a Carnot heat engine. The operating gas is in a tube with one end in your mouth (where the temperature is \(37.0^{\circ} \mathrm{C}\) ) and the other end at the surface of your skin, at \(30.0^{\circ} \mathrm{C}\) . (a) What is the maximum efficiency of such a heat engine? Would it be a very use- ful engine? (b) Suppose you want to use this human engine to lift a 2.50 \(\mathrm{kg}\) box from the floor to a tabletop 1.20 \(\mathrm{m}\) above the floor. How much must you increase the gravitational potential energy and how much heat input is needed to accomplish this? (c) How many 350 calorie (those are food calories, remember) candy bars must you eat to lift the box in this way? Recall that 80\(\%\) of the food energy goes into heat.
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