Heat and the First Law of Thermodynamics

5-1

Calculate the specific heat of a metal from the following data: a container made of the metal has a mass of 3.6 kg and contains 14 kg of water. A 1.8 kg piece of the metal initially at a temperature of $180 \rm ^o C$ is dropped into the water. The container and water initially have a temperature of $16.0 \rm ^o C$ and the final temperature of the entire system is $18.0 \rm ^o C$.

5-2

In a solar water heater, energy from the sun is gathered by rooftop collectors, which circulate through tubes in the collector. The solar radiation enters the collector through a transparent cover and warms the water in the tubes; this water is pumped into a holding tank. Assuming that the efficiency of the overall system is 20% (that is, 80% of the incident solar energy is lost from the system), what collector area is necessary to take water from 200-L tank and raise its temperature from 20 to $40 \rm ^o C$ in 1.0 h. The intensity of incident sunlight is $700  \rm W/m^2$.

5-3

What mass of steam at 100 C must be mixed with 150 g of ice at 0 C, in a thermally insulated container to produce liquid water at $50 \rm ^o C$?

5-4

A thermodynamic system is taken from an initial state $A$ to another $B$ and back again to $A$, via state $C$, as shown by the path $ABCA$ in the $p-V$ diagram of Fig. 7a. (a) Complete the table in Fig. 7b by filling in appropriate $+$ or $-$ indications for the signs of the thermodynamic quantities associated with each process. (b) Calculate the numerical value of the work done by the system for the complete cycle $ABCA$.

Figure 7: Prob 5-4

5-5

When a system is taken from state $i$ to state $f$ along the path $iaf$ in Fig. 8, it is found that $Q=50 \rm cal$ and $W=20 \rm cal$. Along the path $ibf$, $Q=36 \rm cal$. (a) What is $W$ along the path $ibf$? (b) If $W=-13 \rm cal$ for the curved return path $fi$, what is $Q$ for this path? (c) Take $U_i=10 \rm cal$. What is $U_f$? (d) If $U_b=22 \rm cal$, what are the $Q$ values for process $ib$ and process $bf$?

Figure 8: Prob 5-5

5-6

A cylinder has a well fitted 2.0-kg metal piston whose cross sectional area is $2.0 \rm cm^2$ (Fig. 9). The cylinder contains water and steam at constant temperature. The piston is observed to fall slowly at a rate of $0.30 \rm cm/s$ because heat flows out the cylinder through the cylinder walls. As this happens, some steam condenses in the chamber. The density of the steam inside the chamber is $6.0\times 10^{-4} \rm g/cm^3$ and the atmospheric pressure is 1.0 atm. (a) Calculate the rate of condensation of the steam. (b) At what rate is heat leaving the chamber? (c) What is the rate of change of internal energy of the steam and water inside the chamber?

Figure 9: Prob 5-6

5-7

The average rate at which heat flows out through the surface of the earth in North America is $54 \rm mW/m^2$, and the average thermal conductivity of the near surface rocks is $2.5  \rm W/m\cdot K$. Assuming a surface temperature of $10 \rm ^o C$, what should be the temperature at a depth of 35 km (near the base of the crust)? Ignore the heat generated by the presence of radioactive elements.

5-8

(a) What is the rate of heat loss in watts per square meter through a glass window 3.0 mm thick if the outside temperature is -20 F and the inside temperature is +72 F? (b) A storm window is installed having the same thickness as the glass but with an air-gap of 7.5 cm between the two windows. What will be the corresponding rate of heat loss presuming that conduction is the only important heat loss mechanism?