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Chapter 20: Problem 38

20.38. You are designing a Carnot engine that has 2 \(\mathrm{mol}\) of \(\mathrm{CO}_{2}\) as its working substance; the gas may be treated as ideal. The gas is to have a maximum temperature of \(527^{\circ} \mathrm{C}\) and a maximum pressure of 5.00 \(\mathrm{atm}\) . With a heat input of 400 \(\mathrm{J}\) per cycle, you want 300 \(\mathrm{J}\) of useful work (a) Find the temperature of the cold reservoir. (b) For how many cycles must this engine run to melt completely a 10.0 kg block of ice originally at \(0.0^{\circ} \mathrm{C}\) , using only the beat rejected by the engine?

Short Answer

Expert verified
(a) 200.0375 K (b) 33,400 cycles

Step by step solution

01

Analyze the Carnot Engine Efficiency

The efficiency of a Carnot engine is given by \( \eta = 1 - \frac{T_c}{T_h} \), where \( T_c \) is the temperature of the cold reservoir and \( T_h \) is the temperature of the hot reservoir. We convert the maximum temperature from Celsius to Kelvin: \( T_h = 527 + 273.15 = 800.15 \text{ K} \). The efficiency can also be expressed in terms of work and heat input: \( \eta = \frac{W}{Q_h} \). Given \( W = 300 \text{ J} \) and \( Q_h = 400 \text{ J} \), we find \( \eta = \frac{300}{400} = 0.75 \).
02

Calculate Cold Reservoir Temperature

From the efficiency formula \( \eta = 1 - \frac{T_c}{T_h} \), we can rearrange it to solve for \( T_c \): \( T_c = T_h (1 - \eta) \). Substituting in the known values: \( T_c = 800.15 (1 - 0.75) = 800.15 \times 0.25 = 200.0375 \text{ K} \).
03

Determine Heat Rejected

The heat rejected \( Q_c \) can be found using the formula \( Q_c = Q_h - W \). Given \( Q_h = 400 \text{ J} \) and \( W = 300 \text{ J} \), we have \( Q_c = 400 - 300 = 100 \text{ J} \).
04

Calculate Number of Cycles to Melt Ice

To melt the ice, we use the latent heat of fusion \( L_f = 334,000 \text{ J/kg} \) for ice. The heat required to melt 10.0 kg of ice is \( Q_{ice} = m L_f = 10 \times 334,000 = 3,340,000 \text{ J} \). The number of cycles needed is \( n = \frac{Q_{ice}}{Q_c} = \frac{3,340,000}{100} = 33,400 \) cycles.

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Key Concepts

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

Ideal Gas
Ideal gases are theoretical models that help us understand the behavior of gas molecules under different conditions. An ideal gas behaves predictably using certain assumptions, particularly at low pressures and high temperatures.

Key characteristics of an ideal gas include:
  • Molecules move randomly and continuously.
  • Molecules are point particles, meaning they have negligible volume.
  • No intermolecular forces act between them except during collisions.
  • Collisions are perfectly elastic, conserving both momentum and energy.
The ideal gas law, expressed as \( PV = nRT \), relates pressure \( P \), volume \( V \), number of moles \( n \), the universal gas constant \( R \), and temperature \( T \). This allows us to predict how a gas will respond to changes in these variables under the assumption of ideal behavior.
Carnot Cycle
The Carnot cycle is a theoretical model that represents the most efficient way to convert heat into work and vice versa. Formulated by Sadi Carnot, it is an iterative process involving four reversible steps: two isothermal processes and two adiabatic processes.

The cycle operates as follows:
  • Isothermal Expansion: The gas absorbs heat from a hot reservoir at a constant high temperature \( T_h \), expanding and doing work.
  • Adiabatic Expansion: The gas expands further without heat exchange, lowering its temperature to that of the cold reservoir \( T_c \).
  • Isothermal Compression: The gas expels heat to the cold reservoir at a constant temperature \( T_c \), doing work on the gas.
  • Adiabatic Compression: The gas is compressed further without heat exchange, raising its temperature back to \( T_h \).
Carnot's theorem states that no real engine operating between two heat reservoirs can be more efficient than a Carnot engine.
Heat Engine Efficiency
Heat engine efficiency refers to the ratio of the work output of a heat engine to the heat input. It is a measure of how well an engine converts heat into useful work. For real-world engines, many factors such as friction, heat losses, and imperfect processes lead to less than maximum efficiency. However, the Carnot engine sets an idealized benchmark for efficiency.

The formula for heat engine efficiency is:\[\eta = \frac{W}{Q_h} = 1 - \frac{T_c}{T_h}\]Where:
  • \( W \) is the work done by the engine.
  • \( Q_h \) is the heat absorbed from the hot reservoir.
  • \( T_c \) and \( T_h \) are the temperatures of the cold and hot reservoirs, respectively.
Perfect efficiency is impossible due to the second law of thermodynamics, which implies that some heat will always be expelled during the process. The Carnot cycle serves as the upper limit to which real engines can aspire.

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Most popular questions from this chapter

20.42. Heat Pump. A heat pump is a heat engine run in reverse. In winter it pumps heat from the cold air outside into the warmer air inside the building, maintaining the building at a comfortable temperature. In summer it pumps heat from the cooler air inside the building to the warmer air outside, acting as an air conditioner. (a) If the outside temperature in winter is \(-5.0^{\circ} \mathrm{C}\) and the inside temperature is \(17.0^{\circ} \mathrm{C}\) , how many joules of heat will the heat pump deliver to the inside for each joule of electrical energy used to run the unit, assuming an ideal Carnot cycle? ( b) Suppose you have the option of using electrical resistance heating rather than a heat pump. How much electrical energy would you need in order to deliver the same amount of heat to the inside of the house as in part (a)? Consider a Carnot heat pump delivering heat to the inside of a house to maintain it at \(68^{\circ} \mathrm{F}\) . Show that the beat pump delivers less heat for each joule of electrical energy used to operate the unit as the outside temperature decreases. Notice that this behavior is opposite to the dependence of the efficiency of a Carnot heat engine on the difference in the reservoir temperatures. Explain why this is so.

20.63. An object of mass \(m_{1}\) , specific heat capacity \(c_{1}\) , and temperature \(T_{1}\) is placed in contact with a second object of mass \(m_{2},\) specific heat capacity \(c_{2},\) and temperature \(T_{2}>T_{1} .\) As a result, the temperature of the first object increases to \(T\) and the temperature of the second object decreases to \(T^{\prime} .\) (a) Show that the entropy increase of the system is $$ \Delta S=m_{1} c_{1} \ln \frac{T}{T_{1}}+m_{2} c_{2} \ln \frac{T^{\prime}}{T_{2}} $$ and show that energy conservation requires that $$ m_{1} c_{1}\left(T-T_{1}\right)=m_{2} c_{2}\left(T_{2}-T^{\prime}\right) $$ (b) Show that the entropy change \(\Delta S\) , considered as a function of \(T,\) is a maximum if \(T=T,\) which is just the condition of thermodynamic equilibrium. (c) Discuss the result of part (b) in terms of the idea of entropy as a measure of disorder.

20.10. A room air conditioner has a coefficient of performance of 29 on a hot day, and uses 850 \(\mathrm{W}\) of electrical power. (a) How many joules of heat does the air conditioner remove from the room in one minute? (b) How many joules of heat does the air conditioner deliver to the hot outside air in one minute? (c) Explain why your answers to parts (a) and (b) are not the same.

20.42 A cylinder contains oxygen at a pressure of 2.00 atm. The volume is 4.00 \(\mathrm{L}\) , and the temperature is 300 \(\mathrm{K}\) . Assume that the oxygen may be treated as an ideal gas. The oxygen is carried through the following processes: (i) Heated at constant pressure from the initial state (state 1) to state \(2,\) which has \(T=450 \mathrm{K}\) . (ii) Cooled at constant volume to 250 \(\mathrm{K}\) (state 3). (iii) Compressed at constant temperature to a volume of 4.00 \(\mathrm{L}\) (state 4\()\) . (iv) Heated at constant volume to 300 \(\mathrm{K}\) , which takes the system back to state 1. (a) Show these four processes in a \(p V\) -diagram, giving the numerical values of \(p\) and \(V\) in each of the four states. (b) Calculate \(Q\) and \(W\) for each of the four processes. (c) Calculate the net work done by the oxygen. (d) What is the efficiency of this device as a heal engine? How does this compare to the efficiency of a Carnot-cycle engine operating between the same minimum and maximum tem- peratures of 250 \(\mathrm{K}\) and 450 \(\mathrm{K} ?\)

20.25. A sophomore with nothing better to do adds heat to 0.350 \(\mathrm{kg}\) of ice at \(0.0^{\circ} \mathrm{C}\) until it is all melted. (a) What is the change in entropy of the water? (b) The source of heat is a very massive body at a temperature of \(25.0^{\circ} \mathrm{C}\) . What is the change in entropy of this body? (c) What is the total change in entropy of the water and the heat source?
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