### Additional Problems

In 1701, the Danish astronomer Ole Rømer proposed a temperature scale with two fixed points, freezing water at 7.5 degrees, and boiling water at 60.0 degrees. What is the boiling point of oxygen, 90.2 K, on the Rømer scale?

What is the percent error of thinking the melting point of tungsten is $3695\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ instead of the correct value of 3695 K?

An engineer wants to design a structure in which the difference in length between a steel beam and an aluminum beam remains at 0.500 m regardless of temperature, for ordinary temperatures. What must the lengths of the beams be?

How much stress is created in a steel beam if its temperature changes from $\mathrm{\u201315}\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ to $40\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ but it cannot expand? For steel, the Young’s modulus $Y=210\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{9}\phantom{\rule{0.2em}{0ex}}{\text{N/m}}^{2}$ from Stress, Strain, and Elastic Modulus. (Ignore the change in area resulting from the expansion.)

A brass rod $\left(Y=90\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{9}\phantom{\rule{0.2em}{0ex}}{\text{N/m}}^{2}\right),$ with a diameter of 0.800 cm and a length of 1.20 m when the temperature is $25\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$, is fixed at both ends. At what temperature is the force in it at 36,000 N?

A mercury thermometer still in use for meteorology has a bulb with a volume of $0.780\phantom{\rule{0.2em}{0ex}}{\text{cm}}^{3}$ and a tube for the mercury to expand into of inside diameter 0.130 mm. (a) Neglecting the thermal expansion of the glass, what is the spacing between marks $1\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ apart? (b) If the thermometer is made of ordinary glass (not a good idea), what is the spacing?

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 $(1\phantom{\rule{0.2em}{0ex}}\text{watt}=1\phantom{\rule{0.2em}{0ex}}\text{joule/second}$ or $1\phantom{\rule{0.2em}{0ex}}\text{W}=1\phantom{\rule{0.2em}{0ex}}\text{J/s}$ and

$1\phantom{\rule{0.2em}{0ex}}\text{MW}=1\phantom{\rule{0.2em}{0ex}}\text{megawatt}).$ (a) Calculate the rate of temperature increase in degrees Celsius per second $\left(\text{\xb0}\text{C/s}\right)$ if the mass of the reactor core is $1.60\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{5}\phantom{\rule{0.2em}{0ex}}\text{kg}$ and it has an average specific heat of $0.3349\phantom{\rule{0.2em}{0ex}}\text{kJ/kg}\xb7\text{\xb0}\text{C}$. (b) How long would it take to obtain a temperature increase of $2000\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$, 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 500,000-kg steel containment vessel would also begin to heat up.)

You leave a pastry in the refrigerator on a plate and ask your roommate to take it out before you get home so you can eat it at room temperature, the way you like it. Instead, your roommate plays video games for hours. When you return, you notice that the pastry is still cold, but the game console has become hot. Annoyed, and knowing that the pastry will not be good if it is microwaved, you warm up the pastry by unplugging the console and putting it in a clean trash bag (which acts as a perfect calorimeter) with the pastry on the plate. After a while, you find that the equilibrium temperature is a nice, warm $38.3\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$. You know that the game console has a mass of 2.1 kg. Approximate it as having a uniform initial temperature of $45\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$. The pastry has a mass of 0.16 kg and a specific heat of $3.0\phantom{\rule{0.2em}{0ex}}\text{k}\phantom{\rule{0.2em}{0ex}}\text{J/}(\text{kg}\xb7\text{\xbaC}),$ and is at a uniform initial temperature of $4.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$. The plate is at the same temperature and has a mass of 0.24 kg and a specific heat of $0.90\phantom{\rule{0.2em}{0ex}}\text{J/}(\text{kg}\xb7\text{\xbaC})$. What is the specific heat of the console?

Two solid spheres, *A* and *B*, made of the same material, are at temperatures of $0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ and $100\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$, respectively. The spheres are placed in thermal contact in an ideal calorimeter, and they reach an equilibrium temperature of $20\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$. Which is the bigger sphere? What is the ratio of their diameters?

In some countries, liquid nitrogen is used on dairy trucks instead of mechanical refrigerators. A 3.00-hour delivery trip requires 200 L of liquid nitrogen, which has a density of $808\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3}.$ (a) Calculate the heat transfer necessary to evaporate this amount of liquid nitrogen and raise its temperature to $3.00\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$. (Use ${c}_{\text{P}}$ and assume it is constant over the temperature range.) This value is the amount of cooling the liquid nitrogen supplies. (b) What is this heat transfer rate in kilowatt-hours? (c) Compare the amount of cooling obtained from melting an identical mass of $0\text{-}\text{\xb0}\text{C}$ ice with that from evaporating the liquid nitrogen.

Some gun fanciers make their own bullets, which involves melting lead and casting it into lead slugs. How much heat transfer is needed to raise the temperature and melt 0.500 kg of lead, starting from $25.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$?

A 0.800-kg iron cylinder at a temperature of $1.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ is dropped into an insulated chest of 1.00 kg of ice at its melting point. What is the final temperature, and how much ice has melted?

Repeat the preceding problem with 2.00 kg of ice instead of 1.00 kg.

Repeat the preceding problem with 0.500 kg of ice, assuming that the ice is initially in a copper container of mass 1.50 kg in equilibrium with the ice.

A 30.0-g ice cube at its melting point is dropped into an aluminum calorimeter of mass 100.0 g in equilibrium at $24.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ with 300.0 g of an unknown liquid. The final temperature is $4.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$. What is the heat capacity of the liquid?

(a) Calculate the rate of heat conduction through a double-paned window that has a $1.50{\text{-m}}^{2}$ area and is made of two panes of 0.800-cm-thick glass separated by a 1.00-cm air gap. The inside surface temperature is $15.0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C},$ while that on the outside is $\mathrm{-10.0}\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}.$ (*Hint:* There are identical temperature drops across the two glass panes. First find these and then the temperature drop across the air gap. This problem ignores the increased heat transfer in the air gap due to convection.) (b) Calculate the rate of heat conduction through a 1.60-cm-thick window of the same area and with the same temperatures. Compare your answer with that for part (a).

(a) An exterior wall of a house is 3 m tall and 10 m wide. It consists of a layer of drywall with an *R* factor of 0.56, a layer 3.5 inches thick filled with fiberglass batts, and a layer of insulated siding with an *R* factor of 2.6. The wall is built so well that there are no leaks of air through it. When the inside of the wall is at $22\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$ and the outside is at $\mathrm{-2}\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$, what is the rate of heat flow through the wall? (b) More realistically, the 3.5-inch space also contains 2-by-4 studs—wooden boards 1.5 inches by 3.5 inches oriented so that 3.5-inch dimension extends from the drywall to the siding. They are “on 16-inch centers,” that is, the centers of the studs are 16 inches apart. What is the heat current in this situation? Don’t worry about one stud more or less.

For the human body, what is the rate of heat transfer by conduction through the body’s tissue with the following conditions: the tissue thickness is 3.00 cm, the difference in temperature is $2.00\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$, and the skin area is $1.50\phantom{\rule{0.2em}{0ex}}{\text{m}}^{2}$. How does this compare with the average heat transfer rate to the body resulting from an energy intake of about 2400 kcal per day? (No exercise is included.)

You have a Dewar flask (a laboratory vacuum flask) that has an open top and straight sides, as shown below. You fill it with water and put it into the freezer. It is effectively a perfect insulator, blocking all heat transfer, except on the top. After a time, ice forms on the surface of the water. The liquid water and the bottom surface of the ice, in contact with the liquid water, are at $0\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C}$. The top surface of the ice is at the same temperature as the air in the freezer, $\mathrm{-18}\phantom{\rule{0.2em}{0ex}}\text{\xb0}\text{C.}$ Set the rate of heat flow through the ice equal to the rate of loss of heat of fusion as the water freezes. When the ice layer is 0.700 cm thick, find the rate in m/s at which the ice is thickening.

An infrared heater for a sauna has a surface area of $0.050\phantom{\rule{0.2em}{0ex}}{\text{m}}^{2}$ and an emissivity of 0.84. What temperature must it run at if the required power is 360 W? Neglect the temperature of the environment.

(a) Determine the power of radiation from the Sun by noting that the intensity of the radiation at the distance of Earth is $1370\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}$. *Hint:* That intensity will be found everywhere on a spherical surface with radius equal to that of Earth’s orbit. (b) Assuming that the Sun’s temperature is 5780 K and that its emissivity is 1, find its radius.