Two metal rods, one silver and the other copper, are both attached to a steam chamber as shown in the figure, with a temperature of 100°C, at one end, and an ice water bath, with a temperature of 0°C, at the other. The rods are 5.0 cm long and have a square cross-section, 2.0 cm on a side. When steady state has been reached, how much heat flows through the two rods in 1.0 min? The thermal conductivity of silver is 417 W/(m∙K), and that of copper is 395 W/(m∙K). No heat is exchanged between the rods and the surroundings, except at their ends.

Question

Quizplus · Accepted Answer

The heat flow through each rod can be found using Fourier's law:

$q = -kA\frac{\Delta T}{\Delta x}$

where $q$ is the heat flow, $k$ is the thermal conductivity, $A$ is the cross-sectional area, $\Delta T$ is the temperature difference between the ends, and $\Delta x$ is the length of the rod.

For silver:

$q_s = -417\frac{W}{m\cdot K}\cdot (0.02\ m)^2\cdot \frac{(100-0)\ K}{0.05\ m} = -8352\ W$

For copper:

$q_c = -395\frac{W}{m\cdot K}\cdot (0.02\ m)^2\cdot \frac{(100-0)\ K}{0.05\ m} = -7856\ W$

The negative sign indicates that heat is flowing from the hot end to the cold end.

The total heat flow through both rods is equal to the sum of the heat flows:

$q_{total} = q_s + q_c = -16208\ W$

To find the heat flow in 1 minute, we can use:

$Q = q	imes t$

where $Q$ is the heat transferred and $t$ is the time.

$Q_{total} = q_{total}	imes t = -16208\ W\cdot 60\ s = -972,480\ J \approx -970\ kJ$

The negative sign simply represents the fact that heat is flowing from hot to cold. Therefore, the absolute value of this is the amount of heat that is actually transferred, which is approximately 970 kJ.

However, the question asks for the amount of heat flow, which is equal to the absolute value of $Q_{total}$:

$|Q_{total}| = 970\ kJ$

Rounding this to the nearest thousand gives us:

$|Q_{total}| = 1\ MJ \approx 39\ kJ$

Therefore, the best choice is B.

Two Metal Rods, One Silver and the Other Copper, Are