20. Heat and Temperature

300g of water at 40°C is in an insulated cup. What mass of copper at 500°C must be added to heat the water to 60°C?

The blood plays an important role in removing heat from the body by bringing this energy directly to the surface where it can radiate away. Nevertheless, this heat must still travel through the skin before it can radiate away. Assume that the blood is brought to the bottom layer of skin at $37.0$°C and that the outer surface of the skin is at $30.0$°C. Skin varies in thickness from $0.50$ mm to a few millimeters on the palms and soles, so assume an average thickness of $0.75$ mm. A $165$-lb, $6$-ft-tall person has a surface area of about $2.0$ m² and loses heat at a net rate of $75$ W while resting. On the basis of our assumptions, what is the thermal conductivity of this person's skin?

Two rods, one made of brass and the other made of copper, are joined end to end. The length of the brass section is $0.300$ m and the length of the copper section is $0.800$ m. Each segment has cross-sectional area $0.00500$ m². The free end of the brass segment is in boiling water and the free end of the copper segment is in an ice–water mixture, in both cases under normal atmospheric pressure. The sides of the rods are insulated so there is no heat loss to the surroundings. What mass of ice is melted in $5.00$ min by the heat conducted by the composite rod?

The emissivity of tungsten is $0.350$. A tungsten sphere with radius $1.50$ cm is suspended within a large evacuated enclosure whose walls are at $290.0$ K. What power input is required to maintain the sphere at $3000.0$ K if heat conduction along the supports is ignored?

A spherical pot contains $0.75$ L of hot coffee (essentially water) at an initial temperature of $95$°C. The pot has an emissivity of $0.60$, and the surroundings are at $20.0$°C. Calculate the coffee's rate of heat loss by radiation.

An electric kitchen range has a total wall area of $1.40$ m² and is insulated with a layer of fiberglass $4.00$ cm thick. The inside surface of the fiberglass has a temperature of $175$°C, and its outside surface is at $35.0$°C. The fiberglass has a thermal conductivity of $0.040W/m\cdot K$. What is the heat current through the insulation, assuming it may be treated as a flat slab with an area of $1.40$ m² ?

Suppose that the rod in Fig. $17.24$a is made of copper, is $45.0$ cm long, and has a cross-sectional area of $1.25$ cm² . Let $TH=100.0$°C and $TC=0.0$°C. What is the final steady-state temperature gradient along the rod?

A carpenter builds an exterior house wall with a layer of wood $3.0$ cm thick on the outside and a layer of Styrofoam insulation $2.2$ cm thick on the inside wall surface. The wood has $k=0.080W/m\cdot K$ , and the Styrofoam has $k=0.027W/m\cdot K$. The interior surface temperature is $19.0$°C, and the exterior surface temperature is $-10.0$°C. What is the temperature at the plane where the wood meets the Styrofoam?

A carpenter builds an exterior house wall with a layer of wood $3.0$ cm thick on the outside and a layer of Styrofoam insulation $2.2$ cm thick on the inside wall surface. The wood has $k=0.080\,W/m\cdot K$ , and the Styrofoam has $k=0.027\,W/m\cdot K$. The interior surface temperature is $19.0$°C, and the exterior surface temperature is $-10.0$°C. What is the rate of heat flow per square meter through this wall?

Most stars are main-sequence stars, a group of stars for which size, mass, surface temperature, and radiated power are closely related. The sun, for instance, is a yellow main-sequence star with a surface temperature of 5800 K. For a main-sequence star whose mass M is more than twice that of the sun, the total radiated power, relative to the sun, is approximately P/P_sun=1.5(M/M_sun)^3.5. The star Regulus A is a bluish main-sequence star with mass 3.8M_sun and radius 3.1R_sun. What is the surface temperature of Regulus A?

Liquid helium, with a boiling point of 4.2 K, is used in ultralow-temperature experiments and also for cooling the superconducting magnets used in MRI imaging in medicine. Storing liquid helium so far below room temperature is a challenge because even a small 'heat leak' will boil the helium away. A standard helium dewar, shown in FIGURE P19.67, has an inner stainless-steel cylinder filled with liquid helium surrounded by an outer cylindrical shell filled with liquid nitrogen at –196°C. The space between is a vacuum. The small structural supports have very low thermal conductivity, so you can assume that radiation is the only heat transfer between the helium and its surroundings. Suppose the helium cylinder is 16 cm in diameter and 30 cm tall and that all walls have an emissivity of 0.25. The density of liquid helium is 125 kg/m³ and its heat of vaporization is 2.1×10⁴ J/kg. What is the mass of helium in the filled cylinder?

You are boiling pasta and absentmindedly grab a copper stirring spoon rather than your wooden spoon. The copper spoon has a 20 mm ×1.5 mm rectangular cross section, and the distance from the boiling water to your 35°C hand is 18 cm. How long does it take the spoon to transfer 25 J of energy to your hand?

The ends of a 20-cm-long, 2.0-cm-diameter rod are maintained at 0°C and 100°C by immersion in an ice-water bath and boiling water. Heat is conducted through the rod at 4.5×10⁴ J per hour. Of what material is the rod made?

A 2.0-cm-diameter metal sphere is glowing red, but a spectrum shows that its emission spectrum peaks at an infrared wavelength of 2.0 μm. How much power does the sphere radiate? Assume e=1 .