Solution: A copper rod and an aluminum rod of equal diameter

An ideal gas is compressed to half its initial volume by means of several possible processes. Which of the following processes results in the most work done on the gas? (a) isothermal (b) adiabatic (c) isobaric (d) The work done is independent of the process.

A poker is a stiff, nonflammable rod used to push burning logs around in a fireplace. For safety and comfort of use, should the poker be made from a material with (a) high specific heat and high thermal conductivity, (b) low specific heat and low thermal conductivity, (c) low specific heat and high thermal conductivity, or (d) high specific heat and low thermal conductivity?

Assume you are measuring the specific heat of a sample of originally hot metal by using a calorimeter containing water. Because your calorimeter is not perfectly insulating, energy can transfer by heat between the contents of the calorimeter and the room. To obtain the most accurate result for the specific heat of the metal, you should use water with which initial temperature? (a) slightly lower than room temperature (b) the same as room temperature (c) slightly higher than room temperature (d) whatever you like because the initial temperature makes no difference

An amount of energy is added to ice, raising its temperature from 210C to 25C. A larger amount of energy is added to the same mass of water, raising its temperature from 15C to 20C. From these results, what would you conclude? (a) Overcoming the latent heat of fusion of ice requires an input of energy. (b) The latent heat of fusion of ice delivers some energy to the system. (c) The specific heat of ice is less than that of water. (d) The specific heat of ice is greater than that of water. (e) More information is needed to draw any conclusion.

How much energy is required to raise the temperature of 5.00 kg of lead from 20.0C to its melting point of 327C? The specific heat of lead is 128 J/kg ? C. (a) 4.04 3 105 J (b) 1.07 3 105 J (c) 8.15 3 104 J (d) 2.13 3 104 J (e) 1.96 3 105 J

Ethyl alcohol has about one-half the specific heat of water. Assume equal amounts of energy are transferred by heat into equal-mass liquid samples of alcohol and water in separate insulated containers. The water rises in temperature by 25C. How much will the alcohol rise in temperature? (a) It will rise by 12C. (b) It will rise by 25C. (c) It will rise by 50C. (d) It depends on the rate of energy transfer. (e) It will not rise in temperature.

The specific heat of substance A is greater than that of substance B. Both A and B are at the same initial temperature when equal amounts of energy are added to them. Assuming no melting or vaporization occurs, which of the following can be concluded about the final temperature TA of substance A and the final temperature TB of substance B? (a) TA . TB (b) TA , TB (c) TA 5 TB (d) More information is needed.

Beryllium has roughly one-half the specific heat of water (H2O). Rank the quantities of energy input required to produce the following changes from the largest to the smallest. In your ranking, note any cases of equality. (a)raising the temperature of 1 kg of H2O from 20C to 26C (b) raising the temperature of 2 kg of H2O from 20C to 23C (c)raising the temperature of 2 kg of H2O from 1C to 4C (d)raising the temperature of 2 kg of beryllium from 21C to 2C (e) raising the temperature of 2 kg of H2O from 21C to 2C

A person shakes a sealed insulated bottle containing hot coffee for a few minutes. (i) What is the change in the temperature of the coffee? (a) a large decrease (b) a slight decrease (c) no change (d) a slight increase (e) a large increase (ii) What is the change in the internal energy of the coffee? Choose from the same possibilities.

A 100-g piece of copper, initially at 95.0C, is dropped into 200 g of water contained in a 280-g aluminum can; the water and can are initially at 15.0C. What is the final temperature of the system? (Specific heats of copper and aluminum are 0.092 and 0.215 cal/g ? C, respectively.) (a) 16C (b) 18C (c) 24C (d) 26C (e) none of those answers

Star A has twice the radius and twice the absolute surface temperature of star B. The emissivity of both stars can be assumed to be 1. What is the ratio of the power output of star A to that of star B? (a) 4 (b) 8 (c) 16 (d) 32 (e) 64

If a gas is compressed isothermally, which of the following statements is true? (a) Energy is transferred into the gas by heat. (b) No work is done on the gas. (c) The temperature of the gas increases. (d) The internal energy of the gas remains constant. (e) None of those statements is true.

If a gas is compressed isothermally, which of the following statements is true? (a) Energy is transferred into the gas by heat. (b) No work is done on the gas. (c) The temperature of the gas increases. (d) The internal energy of the gas remains constant. (e) None of those statements is true.

If a gas undergoes an isobaric process, which of the following statements is true? (a) The temperature of the gas doesnt change. (b) Work is done on or by the gas. (c) No energy is transferred by heat to or from the gas. (d) The volume of the gas remains the same. (e) The pressure of the gas decreases uniformly

How long would it take a 1 000 W heater to melt 1.00 kg of ice at 220.0C, assuming all the energy from the heater is absorbed by the ice? (a) 4.18 s (b) 41.8 s (c) 5.55 min (d) 6.25 min (e) 38.4 min

A 50.0-g copper calorimeter contains 250 g of water at 20.0C. How much steam at 100C must be condensed into the water if the final temperature of the system is to reach 50.0C?

A 75.0-kg cross-country skier glides over snow as in Figure P20.17. The coefficient of friction between skis and snow is 0.200. Assume all the snow beneath his skis is at 0C and that all the internal energy generated by friction is added to snow, which sticks to his skis until it melts. How far would he have to ski to melt 1.00 kg of snow?

How much energy is required to change a 40.0-g ice cube from ice at 210.0C to steam at 110C?

How much energy is required to change a 40.0-g ice cube from ice at 210.0C to steam at 110C?

A 3.00-g lead bullet at 30.0C is fired at a speed of 240 m/s into a large block of ice at 0C, in which it becomes embedded. What quantity of ice melts?

Steam at 100C is added to ice at 0C. (a) Find the amount of ice melted and the final temperature when the mass of steam is 10.0 g and the mass of ice is 50.0 g. (b) What If? Repeat when the mass of steam is 1.00 g and the mass of ice is 50.0 g.

A 1.00-kg block of copper at 20.0C is dropped into a large vessel of liquid nitrogen at 77.3 K. How many kilograms of nitrogen boil away by the time the copper reaches 77.3 K? (The specific heat of copper is 0.092 0 cal/g ? C, and the latent heat of vaporization of nitrogen is 48.0 cal/g.)

In an insulated vessel, 250 g of ice at 0C is added to 600 g of water at 18.0C. (a) What is the final temperature of the system? (b) How much ice remains when the system reaches equilibrium?

An automobile has a mass of 1 500 kg, and its aluminum brakes have an overall mass of 6.00 kg. (a) Assume all the mechanical energy that transforms into internal energy when the car stops is deposited in the brakes and no energy is transferred out of the brakes by heat. The brakes are originally at 20.0C. How many times can the car be stopped from 25.0 m/s before the brakes start to melt? (b) Identify some effects ignored in part (a) that are important in a more realistic assessment of the warming of the brakes.

An ideal gas is enclosed in a cylinder with a movable piston on top of it. The piston has a mass of 8 000 g and an area of 5.00 cm2 and is free to slide up and down, keeping the pressure of the gas constant. How much work is done on the gas as the temperature of 0.200 mol of the gas is raised from 20.0C to 300C?

An ideal gas is enclosed in a cylinder that has a movable piston on top. The piston has a mass m and an area A and is free to slide up and down, keeping the pressure of the gas constant. How much work is done on the gas as the temperature of n mol of the gas is raised from T1 to T2?

One mole of an ideal gas is warmed slowly so that it goes from the PV state (Pi , Vi) to (3Pi , 3Vi) in such a way that the pressure of the gas is directly proportional to the volume. (a) How much work is done on the gas in the process? (b) How is the temperature of the gas related to its volume during this process?

(a) Determine the work done on a gas that expands from i to f as indicated in Figure P20.28. (b) What If? How much work is done on the gas if it is compressed from f to i along the same path?

An ideal gas is taken through a quasi-static process described by P 5 aV 2, with a 5 5.00 atm/m6, as shown in Figure P20.29. The gas is expanded to twice its original volume of 1.00 m3. How much work is done on the expanding gas in this process?

A gas is taken through the cyclic process described in Figure P20.30. (a) Find the net energy transferred to the system by heat during one complete cycle. (b) What If? If the cycle is reversedthat is, the process follows the path ACBAwhat is the net energy input per cycle by heat?

Consider the cyclic process depicted in Figure P20.30. If Q is negative for the process BC and DE int is negative for the process CA, what are the signs of Q, W, and DE int that are associated with each of the three processes?

Why is the following situation impossible? An ideal gas undergoes a process with the following parameters: Q 5 10.0 J, W 5 12.0 J, and DT 5 22.00C.

A thermodynamic system undergoes a process in which its internal energy decreases by 500 J. Over the same time interval, 220 J of work is done on the system. Find the energy transferred from it by heat.

A sample of an ideal gas goes through the process shown in Figure P20.34. From A to B, the process is adiabatic; from B to C, it is isobaric with 345 kJ of energy entering the system by heat; from C to D, the process is isothermal; and from D to A, it is isobaric with 371 kJ of energy leaving the system by heat. Determine the difference in internal energy E int,B 2 E int,A

A 2.00-mol sample of helium gas initially at 300 K, and 0.400 atm is compressed isothermally to 1.20 atm. Noting that the helium behaves as an ideal gas, find (a) the final volume of the gas, (b) the work done on the gas, and (c) the energy transferred by heat.

(a) How much work is done on the steam when 1.00 mol of water at 100C boils and becomes 1.00 mol of steam at 100C at 1.00 atm pressure? Assume the steam to behave as an ideal gas. (b) Determine the change in internal energy of the system of the water and steam as the water vaporizes.

An ideal gas initially at 300 K undergoes an isobaric expansion at 2.50 kPa. If the volume increases from 1.00 m3 to 3.00 m3 and 12.5 kJ is transferred to the gas by heat, what are (a) the change in its internal energy and (b) its final temperature?

One mole of an ideal gas does 3 000 J of work on its surroundings as it expands isothermally to a final pressure of 1.00 atm and volume of 25.0 L. Determine (a) the initial volume and (b) the temperature of the gas

A 1.00-kg block of aluminum is warmed at atmospheric pressure so that its temperature increases from 22.0C to 40.0C. Find (a) the work done on the aluminum, (b) the energy added to it by heat, and (c) the change in its internal energy

In Figure P20.40, the change in internal energy of a gas that is taken from A to C along the blue path is 1800 J. The work done on the gas along the red path ABC is 2500 J. (a) How much energy must be added to the system by heat as it goes from A through B to C? (b) If the pressure at point A is five times that of point C, what is the work done on the system in going from C to D? (c) What is the energy exchanged with the surroundings by heat as the gas goes from C to A along the green path? (d) If the change in internal energy in going from point D to point A is 1500 J, how much energy must be added to the system by heat as it goes from point C to point D?

An ideal gas initially at Pi , Vi , and Ti is taken through a cycle as shown in Figure P20.41. (a) Find the net work done on the gas per cycle for 1.00 mol of gas initially at 0C. (b) What is the net energy added by heat to the gas per cycle?

An ideal gas initially at Pi , Vi , and Ti is taken through a cycle as shown in Figure P20.41. (a) Find the net work done on the gas per cycle. (b) What is the net energy added by heat to the system per cycle?

A glass windowpane in a home is 0.620 cm thick and has dimensions of 1.00 m 3 2.00 m. On a certain day, the temperature of the interior surface of the glass is 25.0C and the exterior surface temperature is 0C. (a) What is the rate at which energy is transferred by heat through the glass? (b) How much energy is transferred through the window in one day, assuming the temperatures on the surfaces remain constant?

A concrete slab is 12.0 cm thick and has an area of 5.00 m2. Electric heating coils are installed under the slab to melt the ice on the surface in the winter months. What minimum power must be supplied to the coils to maintain a temperature difference of 20.0C between the bottom of the slab and its surface? Assume all the energy transferred is through the slab.

A student is trying to decide what to wear. His bedroom is at 20.0C. His skin temperature is 35.0C. The area of his exposed skin is 1.50 m2. People all over the world have skin that is dark in the infrared, with emissivity about 0.900. Find the net energy transfer from his body by radiation in 10.0 min.

The surface of the Sun has a temperature of about 5 800 K. The radius of the Sun is 6.96 3 108 m. Calculate the total energy radiated by the Sun each second. Assume the emissivity of the Sun is 0.986.

The tungsten filament of a certain 100-W lightbulb radiates 2.00 W of light. (The other 98 W is carried away by convection and conduction.) The filament has a surface area of 0.250 mm2 and an emissivity of 0.950. Find the filaments temperature. (The melting point of tungsten is 3 683 K.)

At high noon, the Sun delivers 1 000 W to each square meter of a blacktop road. If the hot asphalt transfers energy only by radiation, what is its steady-state temperature?

Two lightbulbs have cylindrical filaments much greater in length than in diameter. The evacuated bulbs are identical except that one operates at a filament temperature of 2 100C and the other operates at 2 000C. (a) Find the ratio of the power emitted by the hotter lightbulb to that emitted by the cooler lightbulb. (b) With the bulbs operating at the same respective temperatures, the cooler lightbulb is to be altered by making its filament thicker so that it emits the same power as the hotter one. By what factor should the radius of this filament be increased?

The human body must maintain its core temperature inside a rather narrow range around 37C. Metabolic processes, notably muscular exertion, convert chemical energy into internal energy deep in the interior. From the interior, energy must flow out to the skin or lungs to be expelled to the environment. During moderate exercise, an 80-kg man can metabolize food energy at the rate 300 kcal/h, do 60 kcal/h of mechanical work, and put out the remaining 240 kcal/h of energy by heat. Most of the energy is carried from the body interior out to the skin by forced convection (as a plumber would say), whereby blood is warmed in the interior and then cooled at the skin, which is a few degrees cooler than the body core. Without blood flow, living tissue is a good thermal insulator, with thermal conductivity about 0.210 W/m C. Show that blood flow is essential to cool the mans body by calculating the rate of energy conduction in kcal/h through the tissue layer under his skin. Assume that its area is 1.40 m2, its thickness is 2.50 cm, and it is maintained at 37.0C on one side and at 34.0C on the other side.

A copper rod and an aluminum rod of equal diameter are joined end to end in good thermal contact. The temperature of the free end of the copper rod is held constant at 100C and that of the far end of the aluminum rod is held at 0C. If the copper rod is 0.150 m long, what must be the length of the aluminum rod so that the temperature at the junction is 50.0C?

A box with a total surface area of 1.20 m2 and a wall thickness of 4.00 cm is made of an insulating material. A 10.0-W electric heater inside the box maintains the inside temperature at 15.0C above the outside temperature. Find the thermal conductivity k of the insulating material.

(a) Calculate the R-value of a thermal window made of two single panes of glass each 0.125 in. thick and separated by a 0.250-in. air space. (b) By what factor is the transfer of energy by heat through the window reduced by using the thermal window instead of the single-pane window? Include the contributions of inside and outside stagnant air layers.

At our distance from the Sun, the intensity of solar radiation is 1 370 W/m2. The temperature of the Earth is affected by the greenhouse effect of the atmosphere. This phenomenon describes the effect of absorption of infrared light emitted by the surface so as to make the surface temperature of the Earth higher than if it were airless. For comparison, consider a spherical object of radius r with no atmosphere at the same distance from the Sun as the Earth. Assume its emissivity is the same for all kinds of electromagnetic waves and its temperature is uniform over its surface. (a) Explain why the projected area over which it absorbs sunlight is pr 2 and the surface area over which it radiates is 4pr 2. (b) Compute its steady-state temperature. Is it chilly?

A bar of gold (Au) is in thermal contact with a bar of silver (Ag) of the same length and area (Fig. P20.55). One end of the compound bar is maintained at 80.0C, and the opposite end is at 30.0C. When the energy transfer reaches steady state, what is the temperature at the junction?

For bacteriological testing of water supplies and in medical clinics, samples must routinely be incubated for 24 h at 37C. Peace Corps volunteer and MIT engineer Amy Smith invented a low-cost, low-maintenance incubator. The incubator consists of a foam-insulated box containing a waxy material that melts at 37.0C interspersed among tubes, dishes, or bottles containing the test samples and growth medium (bacteria food). Outside the box, the waxy material is first melted by a stove or solar energy collector. Then the waxy material is put into the box to keep the test samples warm as the material solidifies. The heat of fusion of the phasechange material is 205 kJ/kg. Model the insulation as a panel with surface area 0.490 m2, thickness 4.50 cm, and conductivity 0.012 0 W/m ? C. Assume the exterior temperature is 23.0C for 12.0 h and 16.0C for 12.0 h. (a) What mass of the waxy material is required to conduct the bacteriological test? (b) Explain why your calculation can be done without knowing the mass of the test samples or of the insulation.

A large, hot pizza floats in outer space after being jettisoned as refuse from a spacecraft. What is the order of magnitude (a) of its rate of energy loss and (b) of its rate of temperature change? List the quantities you estimate and the value you estimate for each.

A gas expands from I to F in Figure P20.58 (page 622). The energy added to the gas by heat is 418 J when the gas goes from I to F along the diagonal path. (a) What is the change in internal energy of the gas? (b) How much energy must be added to the gas by heat along the indirect path IAF ?

Gas in a container is at a pressure of 1.50 atm and a volume of 4.00 m3. What is the work done on the gas (a) if it expands at constant pressure to twice its initial volume, and (b) if it is compressed at constant pressure to one-quarter its initial volume?

Liquid nitrogen has a boiling point of 77.3 K and a latent heat of vaporization of 2.01 3 105 J/kg. A 25.0-W electric heating element is immersed in an insulated vessel containing 25.0 L of liquid nitrogen at its boiling point. How many kilograms of nitrogen are boiled away in a period of 4.00 h?

An aluminum rod 0.500 m in length and with a crosssectional area of 2.50 cm2 is inserted into a thermally insulated vessel containing liquid helium at 4.20 K. The rod is initially at 300 K. (a) If one-half of the rod is inserted into the helium, how many liters of helium boil off by the time the inserted half cools to 4.20 K? Assume the upper half does not yet cool. (b) If the circular surface of the upper end of the rod is maintained at 300 K, what is the approximate boil-off rate of liquid helium in liters per second after the lower half has reached 4.20 K? (Aluminum has thermal conductivity of 3 100 W/m K at 4.20 K; ignore its temperature variation. The density of liquid helium is 125 kg/m3.)

Review. Two speeding lead bullets, one of mass 12.0 g moving to the right at 300 m/s and one of mass 8.00 g moving to the left at 400 m/s, collide head-on, and all the material sticks together. Both bullets are originally at temperature 30.0C. Assume the change in kinetic energy of the system appears entirely as increased internal energy. We would like to determine the temperature and phase of the bullets after the collision. (a) What two analysis models are appropriate for the system of two bullets for the time interval from before to after the collision? (b) From one of these models, what is the speed of the combined bullets after the collision? (c) How much of the initial kinetic energy has transformed to internal energy in the system after the collision? (d) Does all the lead melt due to the collision? (e) What is the temperature of the combined bullets after the collision? (f) What is the phase of the combined bullets after the collision?

A flow calorimeter is an apparatus used to measure the specific heat of a liquid. The technique of flow calorimetry involves measuring the temperature difference between the input and output points of a flowing stream of the liquid while energy is added by heat at a known rate. A liquid of density 900 kg/m3 flows through the calorimeter with volume flow rate of 2.00 L/min. At steady state, a temperature difference 3.50C is established between the input and output points when energy is supplied at the rate of 200 W. What is the specific heat of the liquid?

A flow calorimeter is an apparatus used to measure the specific heat of a liquid. The technique of flow calorimetry involves measuring the temperature difference between the input and output points of a flowing stream of the liquid while energy is added by heat at a known rate. A liquid of density r flows through the calorimeter with volume flow rate R. At steady state, a temperature difference DT is established between the input and output points when energy is supplied at the rate P. What is the specific heat of the liquid?

Review. Following a collision between a large spacecraft and an asteroid, a copper disk of radius 28.0 m and thickness 1.20 m at a temperature of 850C is floating in space, rotating about its symmetry axis with an angular speed of 25.0 rad/s. As the disk radiates infrared light, its temperature falls to 20.0C. No external torque acts on the disk. (a) Find the change in kinetic energy of the disk. (b) Find the change in internal energy of the disk. (c) Find the amount of energy it radiates.

An ice-cube tray is filled with 75.0 g of water. After the filled tray reaches an equilibrium temperature of 20.0C, it is placed in a freezer set at 28.00C to make ice cubes. (a) Describe the processes that occur as energy is being removed from the water to make ice. (b) Calculate the energy that must be removed from the water to make ice cubes at 28.00C.

On a cold winter day, you buy roasted chestnuts from a street vendor. Into the pocket of your down parka you put the change he gives you: coins constituting 9.00 g of copper at 12.0C. Your pocket already contains 14.0 g of silver coins at 30.0C. A short time later the temperature of the copper coins is 4.00C and is increasing at a rate of 0.500C/s. At this time, (a) what is the temperature of the silver coins and (b) at what rate is it changing?

The rate at which a resting person converts food energy is called ones basal metabolic rate (BMR). Assume that the resulting internal energy leaves a persons body by radiation and convection of dry air. When you jog, most of the food energy you burn above your BMR becomes internal energy that would raise your body temperature if it were not eliminated. Assume that evaporation of perspiration is the mechanism for eliminating this energy. Suppose a person is jogging for maximum fat burning, converting food energy at the rate 400 kcal/h above his BMR, and putting out energy by work at the rate 60.0 W. Assume that the heat of evaporation of water at body temperature is equal to its heat of vaporization at 100C. (a) Determine the hourly rate at which water must evaporate from his skin. (b) When you metabolize fat, the hydrogen atoms in the fat molecule are transferred to oxygen to form water. Assume that metabolism of 1.00 g of fat generates 9.00 kcal of energy and produces 1.00 g of water. What fraction of the water the jogger needs is provided by fat metabolism?

An iron plate is held against an iron wheel so that a kinetic friction force of 50.0 N acts between the two pieces of metal. The relative speed at which the two surfaces slide over each other is 40.0 m/s. (a) Calculate the rate at which mechanical energy is converted to internal energy. (b) The plate and the wheel each have a mass of 5.00 kg, and each receives 50.0% of the internal energy. If the system is run as described for 10.0 s and each object is then allowed to reach a uniform internal temperature, what is the resultant temperature increase?

An iron plate is held against an iron wheel so that a kinetic friction force of 50.0 N acts between the two pieces of metal. The relative speed at which the two surfaces slide over each other is 40.0 m/s. (a) Calculate the rate at which mechanical energy is converted to internal energy. (b) The plate and the wheel each have a mass of 5.00 kg, and each receives 50.0% of the internal energy. If the system is run as described for 10.0 s and each object is then allowed to reach a uniform internal temperature, what is the resultant temperature increase?

A 40.0-g ice cube floats in 200 g of water in a 100-g copper cup; all are at a temperature of 0C. A piece of lead at 98.0C is dropped into the cup, and the final equilibrium temperature is 12.0C. What is the mass of the lead?

One mole of an ideal gas is contained in a cylinder with a movable piston. The initial pressure, volume, and temperature are Pi , Vi , and Ti , respectively. Find the work done on the gas in the following processes. In operational terms, describe how to carry out each process and show each process on a PV diagram. (a) an isobaric compression in which the final volume is one-half the initial volume (b) an isothermal compression in which the final pressure is four times the initial pressure (c) an isovolumetric process in which the final pressure is three times the initial pressure

Review. A 670-kg meteoroid happens to be composed of aluminum. When it is far from the Earth, its temperature is 215.0C and it moves at 14.0 km/s relative to the planet. As it crashes into the Earth, assume the internal energy transformed from the mechanical energy of the meteoroidEarth system is shared equally between the meteoroid and the Earth and all the material of the meteoroid rises momentarily to the same final temperature. Find this temperature. Assume the specific heat of liquid and of gaseous aluminum is 1 170 J/kg ? C.

Why is the following situation impossible? A group of campers arises at 8:30 a.m. and uses a solar cooker, which consists of a curved, reflecting surface that concentrates sunlight onto the object to be warmed (Fig. P20.74). During the day, the maximum solar intensity reaching the Earths surface at the cookers location is I 5 600 W/m2. The cooker faces the Sun and has a face diameter of d 5 0.600 m. Assume a fraction f of 40.0% of the incident energy is transferred to 1.50 L of water in an open container, initially at 20.0C. The water comes to a boil, and the campers enjoy hot coffee for breakfast before hiking ten miles and returning by noon for lunch.

During periods of high activity, the Sun has more sunspots than usual. Sunspots are cooler than the rest of the luminous layer of the Suns atmosphere (the photosphere). Paradoxically, the total power output of the active Sun is not lower than average but is the same or slightly higher than average. Work out the details of the following crude model of this phenomenon. Consider a patch of the photosphere with an area of 5.10 3 1014 m2. Its emissivity is 0.965. (a) Find the power it radiates if its temperature is uniformly 5 800 K, corresponding to the quiet Sun. (b) To represent a sunspot, assume 10.0% of the patch area is at 4 800 K and the other 90.0% is at 5 890 K. Find the power output of the patch. (c) State how the answer to part (b) compares with the answer to part (a). (d) Find the average temperature of the patch. Note that this cooler temperature results in a higher power output.

(a) In air at 0C, a 1.60-kg copper block at 0C is set sliding at 2.50 m/s over a sheet of ice at 0C. Friction brings the block to rest. Find the mass of the ice that melts. (b) As the block slows down, identify its energy input Q, its change in internal energy DE int, and the change in mechanical energy for the blockice system. (c) For the ice as a system, identify its energy input Q and its change in internal energy DE int. (d) A 1.60-kg block of ice at 0C is set sliding at 2.50 m/s over a sheet of copper at 0C. Friction brings the block to rest. Find the mass of the ice that melts. (e) Evaluate Q and DE int for the block of ice as a system and DE mech for the blockice system. (f) Evaluate Q and DE int for the metal sheet as a system. (g) A thin, 1.60-kg slab of copper at 20C is set sliding at 2.50 m/s over an identical stationary slab at the same temperature. Friction quickly stops the motion. Assuming no energy is transferred to the environment by heat, find the change in temperature of both objects. (h)Evaluate Q and DE int for the sliding slab and DE mech for the two-slab system. (i) Evaluate Q and DE int for the stationary slab.

Water in an electric teakettle is boiling. The power absorbed by the water is 1.00 kW. Assuming the pressure of vapor in the kettle equals atmospheric pressure, determine the speed of effusion of vapor from the kettles spout if the spout has a cross-sectional area of 2.00 cm2. Model the steam as an ideal gas.

The average thermal conductivity of the walls (including the windows) and roof of the house depicted in Figure P20.78 is 0.480 W/m ? C, and their average thickness is 21.0 cm. The house is kept warm with natural gas having a heat of combustion (that is, the energy provided per cubic meter of gas burned) of 9 300 kcal/m3. How many cubic meters of gas must be burned each day to maintain an inside temperature of 25.0C if the outside temperature is 0.0C? Disregard radiation and the energy transferred by heat through the ground.

A cooking vessel on a slow burner contains 10.0 kg of water and an unknown mass of ice in equilibrium at 0C at time t 5 0. The temperature of the mixture is measured at various times, and the result is plotted in Figure P20.79. During the first 50.0 min, the mixture remains at 0C. From 50.0 min to 60.0 min, the temperature increases to 2.00C. Ignoring the heat capacity of the vessel, determine the initial mass of the ice.

A student measures the following data in a calorimetry experiment designed to determine the specific heat of aluminum: Initial temperature of water and calorimeter: 70.0C Mass of water: 0.400 kg Mass of calorimeter: 0.040 kg Specific heat of calorimeter: 0.63 kJ/kg ? C Initial temperature of aluminum: 27.0C Mass of aluminum: 0.200 kg Final temperature of mixture: 66.3C (a) Use these data to determine the specific heat of aluminum. (b) Explain whether your result is within 15% of the value listed in Table 20.1.

Consider the piston cylinder apparatus shown in Figure P20.81. The bottom of the cylinder contains 2.00 kg of water at just under 100.0C. The cylinder has a radius of r 5 7.50 cm. The piston of mass m 5 3.00 kg sits on the surface of the water. An electric heater in the cylinder base transfers energy into the water at a rate of 100 W. Assume the cylinder is much taller than shown in the figure, so we dont need to be concerned about the piston reaching the top of the cylinder. (a) Once the water begins boiling, how fast is the piston rising? Model the steam as an ideal gas. (b) After the water has completely turned to steam and the heater continues to transfer energy to the steam at the same rate, how fast is the piston rising?

A spherical shell has inner radius 3.00 cm and outer radius 7.00 cm. It is made of material with thermal conductivity k 5 0.800 W/m ? C. The interior is maintained at temperature 5C and the exterior at 40C. After an interval of time, the shell reaches a steady state with the temperature at each point within it remaining constant in time. (a) Explain why the rate of energy transfer P must be the same through each spherical surface, of radius r, within the shell and must satisfy dT dr 5 P 4pkr 2 (b) Next, prove that 3 40 5 dT 5 P 4pk 3 0.07 0.03 r 22 dr where T is in degrees Celsius and r is in meters. (c) Find the rate of energy transfer through the shell. (d) Prove that 3 T 5 dT 5 1.84 3 r 0.03 r 22 dr where T is in degrees Celsius and r is in meters. (e) Find the temperature within the shell as a function of radius. (f) Find the temperature at r 5 5.00 cm, halfway through the shell.

A pond of water at 0C is covered with a layer of ice 4.00 cm thick. If the air temperature stays constant at 210.0C, what time interval is required for the ice thickness to increase to 8.00 cm? Suggestion: Use Equation 20.16 in the form dQ dt 5 kA DT x and note that the incremental energy dQ extracted from the water through the thickness x of ice is the amount required to freeze a thickness dx of ice. That is, dQ 5 Lf rA dx, where r is the density of the ice, A is the area, and Lf is the latent heat of fusion.

(a) The inside of a hollow cylinder is maintained at a temperature Ta, and the outside is at a lower temperature, Tb (Fig. P20.84). The wall of the cylinder has a thermal conductivity k. Ignoring end effects, show that the rate of energy conduction from the inner surface to the outer surface in the radial direction is dQ dt 5 2pLk c Ta 2 Tb ln1b/a2 d Suggestions: The temperature gradient is dT/dr. A radial energy current passes through a concentric cylinder of area 2prL. (b) The passenger section of a jet airliner is in the shape of a cylindrical tube with a length of 35.0 m and an inner radius of 2.50 m. Its walls are lined with an insulating material 6.00 cm in thickness and having a thermal conductivity of 4.00 3 1025 cal/s ? cm ? C. A heater must maintain the interior temperature at 25.0C while the outside temperature is 235.0C. What power must be supplied to the heater?