Chapter 16

$$ \begin{array}{l}{\cdot \text { A coal-fired power plant that operates at an efficiency of }} \\ {38 \% \text { generates } 750 \mathrm{MW} \text { of electric power. How much heat }} \\ {\text { does the plant discharge to the environment in one day? }}\end{array} $$

\(\cdot\) 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.

A diesel engine performs 2200 \(\mathrm{J}\) of mechanical work and discards 4300 \(\mathrm{J}\) of heat each cycle. (a) How much heat must be supplied to the engine in each cycle? (b) What is the thermal efficiency of the engine?

\(\cdot\) An aircraft engine takes in 9000 \(\mathrm{J}\) of heat and discards 6400 \(\mathrm{J}\) each cycle. (a) What is the mechanical work output of the engine during one cycle? (b) What is the thermal efficiency of the engine?

\(\cdot\) 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 gasoline engine has a power output of 180 \(\mathrm{kW}\) (about 241 \(\mathrm{hp}\) ). Its thermal efficiency is 28.0\(\% .\) (a) How much heat must be supplied to the engine per second? (b) How much heat is discarded by the engine per second?

A certain nuclear power plant has a mechanical power out- put (used to drive an electric generator) of 330 \(\mathrm{MW}\) . Its rate of heat input from the nuclear reactor is 1300 \(\mathrm{MW}\) . (a) What is the thermal efficiency of the system? (b) At what rate is heat discarded by the system?

\(\cdot\) What compression ratio \(r\) must an Otto cycle have to achieve an ideal efficiency of 65.0\(\%\) if the gas used in the chamber has \(\gamma=1.40 ?\)

\(\bullet\) For an Otto engine with a compression ratio of \(7.50,\) you have your choice of using an ideal monatomic or ideal diatomic gas. Which one would give you greater efficiency? Calculate the efficiency in both cases to find out.

\(\bullet\) (a) Calculate the theoretical efficiency for an Otto cycle engine with \(\gamma=1.40\) and \(r=9.50\) . (b) If this engine takes in \(10,000\) J of heat from burning its fuel, how much heat does it discard to the outside air?

\(\cdot\) 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?

A refrigerator has a coefficient of performance of 2.10 . Each cycle, it absorbs \(3.40 \times 10^{4} \mathrm{J}\) of heat from the cold reservoir. (a) How much mechanical energy is required each cycle to operate the refrigerator? (b) During each cycle, how much heat is discarded to the high-temperature reservoir?

\(\cdot\) A window air-conditioner unit absorbs \(9.80 \times 10^{4} \mathrm{J}\) of heat per minute from the room being cooled and in the same period deposits \(1.44 \times 10^{5} \mathrm{J}\) of heat into the outside air. What is the power consumption of the unit in watts?

\(\bullet\) A cooling unit for chilling the water of an aquarium gives specifications of 1\(/ 10\) hp and 1270 \(\mathrm{Btu} / \mathrm{h}\) . Assuming the unit produces its 1\(/ 10\) hp at 70.0\(\%\) efficiency, calculate its perform- ance coefficient.

A Carnot engine whose high-temperature reservoir is at 620 \(\mathrm{K}\) takes in 550 \(\mathrm{J}\) of heat at this temperature in each cycle and gives up 335 \(\mathrm{J}\) to the low-temperature reservoir. (a) How much mechanical work does the engine perform during each cycle? (b) What is the temperature of the low-temperature reservoir' (c) What is the thermal efficiency of the cycle"

A heat engine is to be built to extract energy from the tem- perature gradient in the ocean. If the surface and deepwater temperatures are \(25^{\circ} \mathrm{C}\) and \(8^{\circ} \mathrm{C}\) , respectively, what is the maxi- mum efficiency such an engine can have?

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?

A Carnot engine has an efficiency of 59\(\%\) and performs \(2.5 \times 10^{4} \mathrm{J}\) of work in each cycle. (a) How much heat does the engine extract from its heat source in each cycle? (b) Suppose the engine exhausts heat at room temperature \(\left(20.0^{\circ} \mathrm{C}\right) .\) What is the temperature of its heat source?

. 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?

\(\bullet\) A Carnot freezer that runs on electricity removes heat from the freezer compartment, which is at \(-10^{\circ} \mathrm{C}\) , and expels it into the room at \(20^{\circ} \mathrm{C}\) . You put an ice-cube tray con- taining 375 \(\mathrm{g}\) of water at \(18^{\circ} \mathrm{C}\) into the freezer. (a) What is the coefficient of performance of this freezer? (b) How much energy is needed to freeze this water? (c) How much electrical energy must be supplied to the freezer to freeze the water? (d) How much heat does the freezer expel into the room while freezing the ice?

(. A large factory furnace maintained at \(175^{\circ} \mathrm{C}\) at its outer surface is wrapped in an insulating blanket of thermal conduc- tivity 0.055 \(\mathrm{W} /(\mathrm{m} \cdot \mathrm{K})\) which is thick enough that the outer surface of the insulation is at \(42^{\circ} \mathrm{C}\) while heat escapes from the furnace at a steady rate of 125 W for each square meter of surface area. By how much does each square meter of the furnace change the entropy of the factory every second?

You decide to take a nice hot bath but discover that your thoughtless roommate has used up most of the hot water. You fill the tub with 270 kg of \(30.0^{\circ} \mathrm{C}\) water and attempt to warm it further by pouring in 5.00 \(\mathrm{kg}\) of boiling water from the stove. (a) Is this a reversible or an irreversible process? Use physical reasoning to explain. (b) Calculate the final temperature of the bath water. (c) Calculate the net change in entropy of the system (bath water + boiling water), assuming no heat exchange with the air or the tub itself.

Three moles of an ideal gas undergo a reversible isother- mal compression at \(20.0^{\circ} \mathrm{C}\) . During this compression, 1850 \(\mathrm{J}\) of work is done on the gas. What is the change in entropy of the gas?

Entropy change due to driving. Premium gasoline pro- duces \(1.23 \times 10^{8} \mathrm{J}\) of heat per gallon when it is burned at a temperature of approximately \(400^{\circ} \mathrm{C}\) (although the amount can vary with the fuel mixture). If the car's engine is 25\(\%\) efficient, three- fourths of that heat is expelled into the air, typically at \(20^{\circ} \mathrm{C}\) . If your car gets 35 miles per gallon of gas, by how much does the car's engine change the entropy of the world when you drive 1.0 mile? Does it decrease or increase it?

\(\bullet\) Entropy of metabolism. An average sleeping person metabolizes at a rate of about 80 \(\mathrm{W}\) by digesting food or burn- ing fat. Typically, 20\(\%\) of this energy goes into bodily func- tions, such as cell repair, pumping blood, and other uses of mechanical energy, while the rest goes to heat. Most people get rid of all of this excess heat by transferring it (by conduc- tion and the flow of blood) to the surface of the body, where it is radiated away. The normal internal temperature of the body (where the metabolism takes place) is \(37^{\circ} \mathrm{C}\) , and the skin is typically 7 \(\mathrm{C}^{\circ}\) cooler. By how much does the person's entropy change per second due to this heat transfer?

. Entropy change from digesting fat. Digesting fat pro- duces 9.3 food calories per gram of fat, and typically 80\(\%\) of this energy goes to heat when metabolized. The body then moves all this heat to the surface by a combination of thermal conductivity and motion of the blood. The internal temperature of the body (where digestion occurs) is normally \(37^{\circ} \mathrm{C}\) , and the surface is usually about \(30^{\circ} \mathrm{C}\) . By how much does the digestion and metabolism of a 2.50 \(\mathrm{g}\) pat of butter change your body's entropy? Does it increase or decrease?

\(\bullet\) Solar collectors. A well-insulated house of moderate size in a temperate climate requires an average heat input rate of 20.0 \(\mathrm{kW}\) . If this heat is to be supplied by a solar collector with an average (night and day) energy input of 300 \(\mathrm{W} / \mathrm{m}^{2}\) and a collection efficiency of \(60.0 \%,\) what area of solar collector is required?

Solar power. A solar power plant is to be built with an aver- age power output capacity of 2500 \(\mathrm{MW}\) in a location where the average power from the sun's radiation is 200 \(\mathrm{W} / \mathrm{m}^{2}\) at the earth's surface. What land area \(\left(\mathrm{in} \mathrm{km}^{2}\) and \(\mathrm{mi}^{2}\right)\) must the solar collectors occupy if they are (a) photocells with 42\(\%\) efficiency. (b) mirrors that generate steam for a turbine- generator unit with an overall efficiency of 21\(\%\) ?

An experimental power plant at the Natural Energy Labo- ratory of Hawaii generates electricity from the temperature gradient of the ocean. The surface and deep-water tempera- tures are \(27^{\circ} \mathrm{C}\) and \(6^{\circ} \mathrm{C}\) , respectively. (a) What is the maximum theoretical efficiency of this power plant? (b) If the power plant is to produce 210 \(\mathrm{kW}\) of power, at what rate must heat be extracted from the warm water? At what rate must heat be absorbed by the cold water? Assume the maximum theoretical efficiency. (c) The cold water that enters the plant leaves it at a temperature of \(10^{\circ} \mathrm{C}\) . What must be the flow rate of cold water through the system? Give your answer in \(\mathrm{kg} / \mathrm{h}\) and \(\mathrm{L} / \mathrm{h}\) .

Solar water heater. A solar water heater for domestic hot- water supply uses solar collecting panels with a collection efficiency of 50\(\%\) in a location where the average solar-energy input is 200 \(\mathrm{W} / \mathrm{m}^{2} .\) If the water comes into the house at \(15.0^{\circ} \mathrm{C}\) and is to be heated to \(60.0^{\circ} \mathrm{C},\) what volume of water can be heated per hour if the area of the collector is 30.0 \(\mathrm{m}^{2} ?\)

.. You are designing a Carnot engine that has 2 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 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 heat rejected by the engine?

A Carnot engine operates between two heat reservoirs at temperatures \(T_{\mathrm{H}}\) and \(T_{\mathrm{C}}\) . An inventor proposes to increase the efficiency by running one engine between \(T_{\mathrm{H}}\) and an intermediate temperature \(T^{\prime}\) and a second engine between \(T^{\prime}\) and \(T_{\mathrm{C}}\) using as input the heat expelled by the first engine. Compute the efficiency of this composite system, and compare it to that of the original engine.

\bullet A cylinder contains oxygen gas \(\left(\mathrm{O}_{2}\right)\) 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 sys- tem back to state I. (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) Cal- culate \(Q\) 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 heat engine? How does this efficiency compare with that of a Carnot-cycle engine operating between the same minimum and maximum temperatures of 250 \(\mathrm{K}\) and 450 \(\mathrm{K}\) ?

A typical coal-fired power plant generates 1000 \(\mathrm{MW}\) of usable power at an overall thermal efficiency of 40\(\%\) . (a) What is the rate of heat input to the plant? (b) The plant burms anthracite coal, which has a heat of combustion of \(2.65 \times 10^{7} \mathrm{J} / \mathrm{kg} .\) How much coal does the plant use per day, if it operates continuously? (c) At what rate is heat ejected into the cool reservoir, which is the nearby river? (d) The river's temperature is \(18.0^{\circ} \mathrm{C}\) before it reaches the power plant and \(18.5^{\circ} \mathrm{C}\) after it has received the plant's waste heat. Calculate the river's flow rate, in cubic meters per second. (e) By how much does the river's entropy increase each second?

. 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.