Problems & Exercises

Show that $1\text{cal/g}\cdot \text{ºC}=1\text{kcal/kg}\cdot \text{ºC}$.

The same heat transfer into identical masses of different substances produces different temperature changes. Calculate the final temperature when 1.00 kcal of heat transfers into 1.00 kg of the following, originally at $\text{20}\text{.}\mathrm{0º}\text{C}$: (a) water; (b) concrete; (c) steel; and (d) mercury.

A 0.250-kg block of a pure material is heated from $\text{20}\text{.}\mathrm{0º}\text{C}$ to $\text{65}\text{.}\mathrm{0º}\text{C}$ by the addition of 4.35 kJ of energy. Calculate its specific heat and identify the substance of which it is most likely composed.

(a) The number of kilocalories in food is determined by calorimetry techniques in which the food is burned and the amount of heat transfer is measured. How many kilocalories per gram are there in a 5.00-g peanut if the energy from burning it is transferred to 0.500 kg of water held in a 0.100-kg aluminum cup, causing a $\text{54}\text{.}\mathrm{9º}\text{C}$ temperature increase? (b) Compare your answer to labeling information found on a package of peanuts and comment on whether the values are consistent.

Even when shut down after a period of normal use, a large commercial nuclear reactor transfers thermal energy at the rate of 150 MW by the radioactive decay of fission products. This heat transfer causes a rapid increase in temperature if the cooling system fails (1 watt = 1 joule/second or 1 W = 1 J/s and 1 MW = 1 megawatt). (a) Calculate the rate of temperature increase in degrees Celsius per second $(\text{ºC/s})$ if the mass of the reactor core is $1\text{.}\text{60}\times {\text{10}}^5\text{kg}$ and it has an average specific heat of $\text{0.3349 kJ/kgº}\cdot \text{C}$. (b) How long would it take to obtain a temperature increase of $\text{2000º}\text{C}$, which could cause some metals holding the radioactive materials to melt? (The initial rate of temperature increase would be greater than that calculated here because the heat transfer is concentrated in a smaller mass. Later, however, the temperature increase would slow down because the $5\times {\text{10}}^5\text{-kg}$ steel containment vessel would also begin to heat up.)

A bag containing $\mathrm{0º}\text{C}$ ice is much more effective in absorbing energy than one containing the same amount of $\mathrm{0º}\text{C}$ water.

1. How much heat transfer is necessary to raise the temperature of 0.800 kg of water from $\mathrm{0º}\text{C}$ to $\text{30}\text{.}\mathrm{0º}\text{C?}$
2. How much heat transfer is required to first melt 0.800 kg of $\mathrm{0º}\text{C}$ ice and then raise its temperature?
3. Explain how your answer supports the contention that the ice is more effective.

The formation of condensation on a glass of ice water causes the ice to melt faster than it would otherwise. If 8.00 g of condensation forms on a glass containing both water and 200 g of ice, how many grams of the ice will melt as a result? Assume no other heat transfer occurs.

On a certain dry sunny day, a swimming pool’s temperature would rise by $1\text{.}\text{50º}\text{C}$ if not for evaporation. What fraction of the water must evaporate to carry away precisely enough energy to keep the temperature constant?

In 1986, a gargantuan iceberg broke away from the Ross Ice Shelf in Antarctica. It was approximately a rectangle 160 km long, 40.0 km wide, and 250 m thick.

(a) What is the mass of this iceberg, given that the density of ice is $\text{917}{\text{ kg/m}}^3$?

(b) How much heat transfer (in joules) is needed to melt it?

(c) How many years would it take sunlight alone to melt ice this thick, if the ice absorbs an average of $\text{100 }{\text{W/m}}^2$, 12.00 h per day?

The energy released from condensation in thunderstorms can be very large. Calculate the energy released into the atmosphere for a small storm of radius 1 km, assuming that 1.0 cm of rain is precipitated uniformly over this area.

A 0.250-kg aluminum bowl holding 0.800 kg of soup at $\text{25}\text{.}\mathrm{0º}\text{C}$ is placed in a freezer. What is the final temperature if 377 kJ of energy is transferred from the bowl and soup, assuming the soup’s thermal properties are the same as that of water? Explicitly show how you follow the steps in Problem-Solving Strategies for the Effects of Heat Transfer.

If you pour 0.0100 kg of $\text{20}\text{.}\mathrm{0º}\text{C}$ water onto a 1.20-kg block of ice (which is initially at $-\text{15}\text{.}\mathrm{0º}\text{C}$), what is the final temperature? You may assume that the water cools so rapidly that effects of the surroundings are negligible.

What would be the final temperature of the pan and water in Calculating the Final Temperature When Heat Is Transferred Between Two Bodies: Pouring Cold Water in a Hot Pan if 0.260 kg of water was placed in the pan and 0.0100 kg of the water evaporated immediately, leaving the remainder to come to a common temperature with the pan?

Some gun fanciers make their own bullets, which involves melting and casting the lead slugs. How much heat transfer is needed to raise the temperature and melt 0.500 kg of lead, starting from $\text{25}\text{.}\mathrm{0º}\text{C}$?

The rate of heat conduction out of a window on a winter day is rapid enough to chill the air next to it. To see just how rapidly the windows transfer heat by conduction, calculate the rate of conduction in watts through a $3\text{.}{\text{00-m}}^2$ window that is $0\text{.635 cm}$ thick (1/4 in) if the temperatures of the inner and outer surfaces are $5\text{.00ºC}$ and $-\text{10}\text{.}\mathrm{0º}\text{C}$, respectively. This rapid rate will not be maintained—the inner surface will cool, and even result in frost formation.

Suppose you stand with one foot on ceramic flooring and one foot on a wool carpet, making contact over an area of $\text{80}\text{.}0{\text{cm}}^2$ with each foot. Both the ceramic and the carpet are 2.00 cm thick and are $\text{10.}\mathrm{0º}\text{C}$ on their bottom sides. At what rate must heat transfer occur from each foot to keep the top of the ceramic and carpet at $\text{33}\text{.}\mathrm{0º}\text{C}$? For ceramic, use the thermal conductivity for glass.

(a) A firewalker runs across a bed of hot coals without sustaining burns. Calculate the heat transferred by conduction into the sole of one foot of a firewalker given that the bottom of the foot is a 3.00-mm-thick callus with a conductivity at the low end of the range for wood and its density is $\text{300}{\text{ kg/m}}^3$. The area of contact is $\text{25}\text{.}0{\text{ cm}}^2$, the temperature of the coals is $\text{700º}\text{C}$, and the time in contact is 1.00 s.

(b) What temperature increase is produced in the $\text{25}\text{.}0{\text{ cm}}^3$ of tissue affected?

(c) What effect do you think this will have on the tissue, keeping in mind that a callus is made of dead cells?

A walrus transfers energy by conduction through its blubber at the rate of 150 W when immersed in $-1\text{.00ºC}$ water. The walrus’s internal core temperature is $\text{37.}\mathrm{0º}\text{C}$, and it has a surface area of $2\text{.00}{\text{m}}^2$. What is the average thickness of its blubber, which has the conductivity of fatty tissues without blood?

Figure 14.33 Walrus on ice. (credit: Captain Budd Christman, NOAA Corps)

Suppose a person is covered head to foot by wool clothing with average thickness of 2.00 cm and is transferring energy by conduction through the clothing at the rate of 50.0 W. What is the temperature difference across the clothing, given the surface area is $1\text{.}\text{40}{\text{ m}}^2$?

One easy way to reduce heating (and cooling) costs is to add extra insulation in the attic of a house. Suppose the house already had 15 cm of fiberglass insulation in the attic and in all the exterior surfaces. If you added an extra 8.0 cm of fiberglass to the attic, then by what percentage would the heating cost of the house drop? Take the single story house to be of dimensions 10 m by 15 m by 3.0 m. Ignore air infiltration and heat loss through windows and doors.

Many decisions are made on the basis of the payback period: the time it will take through savings to equal the capital cost of an investment. Acceptable payback times depend upon the business or philosophy one has. (For some industries, a payback period is as small as two years.) Suppose you wish to install the extra insulation in Exercise 14.41. If energy cost $1.00 per million joules and the insulation was $4.00 per square meter, then calculate the simple payback time. Take the average $\text{Δ}T$ for the 120 day heating season to be $\text{15.}\mathrm{0º}\text{C}$.

At what temperature does still air cause the same chill factor as $-\mathrm{5º}\text{C}$ air moving at 15 m/s?

(a) How many kilograms of water must evaporate from a 60.0-kg woman to lower her body temperature by $0\text{.}\text{750ºC}$?

(b) Is this a reasonable amount of water to evaporate in the form of perspiration, assuming the relative humidity of the surrounding air is low?

One winter day, the climate control system of a large university classroom building malfunctions. As a result, $\text{500 }{\text{m}}^3$ of excess cold air is brought in each minute. At what rate in kilowatts must heat transfer occur to warm this air by $\text{10}\text{.}\text{0ºC}$ (that is, to bring the air to room temperature)?

During heavy exercise, the body pumps 2.00 L of blood per minute to the surface, where it is cooled by $2\text{.}\text{00ºC}$. What is the rate of heat transfer from this forced convection alone, assuming blood has the same specific heat as water and its density is $\text{1050}{\text{ kg/m}}^3$?

A glass coffee pot has a circular bottom with a 9.00-cm diameter in contact with a heating element that keeps the coffee warm with a continuous heat transfer rate of 50.0 W

(a) What is the temperature of the bottom of the pot, if it is 3.00 mm thick and the inside temperature is $\text{60}\text{.}\text{0ºC}$?

(b) If the temperature of the coffee remains constant and all of the heat transfer is removed by evaporation, how many grams per minute evaporate? Take the heat of vaporization to be 2340 kJ/kg.

(a) Cherry-red embers in a fireplace are at $\text{850º}\text{C}$ and have an exposed area of $0\text{.}\text{200}{\text{ m}}^2$ and an emissivity of 0.980. The surrounding room has a temperature of $\text{18}\text{.}\mathrm{0º}\text{C}$. If 50% of the radiant energy enters the room, what is the net rate of radiant heat transfer in kilowatts? (b) Does your answer support the contention that most of the heat transfer into a room by a fireplace comes from infrared radiation?

(a) Calculate the rate of heat transfer by radiation from a car radiator at $110°\text{C}$ into a $\mathrm{50.0º}\text{C}$ environment, if the radiator has an emissivity of 0.750 and a $1.20{\mathrm{-m}}^2$ surface area. (b) Is this a significant fraction of the heat transfer by an automobile engine? To answer this, assume a horsepower of $200\text{hp}\left(1.5\text{kW}\right)$ and the efficiency of automobile engines as 25%.

Suppose you walk into a sauna that has an ambient temperature of $\text{50}\text{.0ºC}$. (a) Calculate the rate of heat transfer to you by radiation given your skin temperature is $\text{37}\text{.0ºC}$, the emissivity of skin is 0.98, and the surface area of your body is $1\text{.50}{\text{m}}^2$. (b) If all other forms of heat transfer are balanced (the net heat transfer is zero), at what rate will your body temperature increase if your mass is 75.0 kg?

The Sun radiates like a perfect black body with an emissivity of exactly 1. (a) Calculate the surface temperature of the Sun, given that it is a sphere with a $7\text{.}\text{00}\times {\text{10}}^8\text{-m}$ radius that radiates $3\text{.}\text{80}\times {\text{10}}^{\text{26}}\text{ W}$ into 3-K space. (b) How much power does the Sun radiate per square meter of its surface? (c) How much power in watts per square meter is that value at the distance of Earth, $1\text{.}\text{50}\times {\text{10}}^{\text{11}}\text{ m}$ away? (This number is called the solar constant.)

Calculate the temperature the entire sky would have to be in order to transfer energy by radiation at $\text{1000}{\text{W/m}}^2$ —about the rate at which the Sun radiates when it is directly overhead on a clear day. This value is the effective temperature of the sky, a kind of average that takes account of the fact that the Sun occupies only a small part of the sky but is much hotter than the rest. Assume that the body receiving the energy has a temperature of $\text{27}\text{.}\text{0ºC}$.

Integrated Concepts

One $\text{30}\text{.}\text{0ºC}$ day the relative humidity is $\text{75}\text{.}\text{0\%}$, and that evening the temperature drops to $\text{20}\text{.}\text{0ºC}$, well below the dew point. (a) How many grams of water condense from each cubic meter of air? (b) How much heat transfer occurs by this condensation? (c) What temperature increase could this cause in dry air?

Integrated Concepts

Frozen waste from airplane toilets has sometimes been accidentally ejected at high altitude. Ordinarily it breaks up and disperses over a large area, but sometimes it holds together and strikes the ground. Calculate the mass of $\text{0ºC}$ ice that can be melted by the conversion of kinetic and gravitational potential energy when a $\text{20}\text{.0 kg}$ piece of frozen waste is released at 12.0 km altitude while moving at 250 m/s and strikes the ground at 100 m/s (since less than 20.0 kg melts, a significant mess results).

Integrated Concepts

(a) Suppose you start a workout on a Stairmaster, producing power at the same rate as climbing 116 stairs per minute. Assuming your mass is 76.0 kg and your efficiency is $\text{20}\text{.}\text{0\%}$, how long will it take for your body temperature to rise $1\text{.}\text{00ºC}$ if all other forms of heat transfer in and out of your body are balanced? (b) Is this consistent with your experience in getting warm while exercising?

Integrated Concepts

In certain large geographic regions, the underlying rock is hot. Wells can be drilled and water circulated through the rock for heat transfer for the generation of electricity. (a) Calculate the heat transfer that can be extracted by cooling $1\text{.00}{\text{ km}}^3$ of granite by $\text{100ºC}$. (b) How long will this take if heat is transferred at a rate of 300 MW, assuming no heat transfers back into the 1.00 km of rock by its surroundings?

Integrated Concepts

(a) What is the temperature increase of water falling 55.0 m over Niagara Falls? (b) What fraction must evaporate to keep the temperature constant?

Unreasonable Results

An Arctic inventor surrounded by ice thinks it would be much less mechanically complex to cool a car engine by melting ice on it than by having a water-cooled system with a radiator, water pump, antifreeze, and so on. (a) If $\text{80}\text{.}\text{0\%}$ of the energy in 1.00 gal of gasoline is converted into “waste heat” in a car engine, how many kilograms of $\text{0ºC}$ ice could it melt? (b) Is this a reasonable amount of ice to carry around to cool the engine for 1.00 gal of gasoline consumption? (c) What premises or assumptions are unreasonable?

Unreasonable Results

A meteorite 1.20 cm in diameter is so hot immediately after penetrating the atmosphere that it radiates 20.0 kW of power. (a) What is its temperature, if the surroundings are at $\text{20.0ºC}$ and it has an emissivity of 0.800? (b) What is unreasonable about this result? (c) Which premise or assumption is responsible?

Construct Your Own Problem

Consider a new model of commercial airplane having its brakes tested as a part of the initial flight permission procedure. The airplane is brought to takeoff speed and then stopped with the brakes alone. Construct a problem in which you calculate the temperature increase of the brakes during this process. You may assume most of the kinetic energy of the airplane is converted to thermal energy in the brakes and surrounding materials, and that little escapes. Note that the brakes are expected to become so hot in this procedure that they ignite and, in order to pass the test, the airplane must be able to withstand the fire for some time without a general conflagration.

Construct Your Own Problem

Consider a person outdoors on a cold night. Construct a problem in which you calculate the rate of heat transfer from the person by all three heat transfer methods. Make the initial circumstances such that at rest the person will have a net heat transfer and then decide how much physical activity of a chosen type is necessary to balance the rate of heat transfer. Among the things to consider are the size of the person, type of clothing, initial metabolic rate, sky conditions, amount of water evaporated, and volume of air breathed. Of course, there are many other factors to consider and your instructor may wish to guide you in the assumptions made as well as the detail of analysis and method of presenting your results.