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

To find the heat flowing through the rods, we need to first find the temperature difference between the hot and cold ends of each rod.

Using Fourier's Law of Heat Conduction, we can calculate the rate of heat flow through the rods:

$q = \frac{\kappa A \Delta T}{L}$

where $q$ is the heat flow rate, $\kappa$ is the thermal conductivity, $A$ is the cross-sectional area, $\Delta T$ is the temperature difference, and $L$ is the length of the rod.

Let's start with the silver rod:

$\Delta T_{silver} = T_{hot} - T_{cold} = 100°C - 0°C = 100°C$

$q_{silver} = \frac{417	ext{ W/(m∙K)} \cdot (0.02	ext{ m})^2 \cdot 100	ext{ K}}{0.05	ext{ m}} = 83.4	ext{ W}$

Now let's find the heat flow rate through the copper rod:

$\Delta T_{copper} = T_{hot} - T_{cold} = 100°C - 0°C = 100°C$

$q_{copper} = \frac{395	ext{ W/(m∙K)} \cdot (0.02	ext{ m})^2 \cdot 100	ext{ K}}{0.05	ext{ m}} = 79	ext{ W}$

The total heat flow rate through both rods is the sum of the individual heat flow rates:

$q_{total} = q_{silver} + q_{copper} = 83.4	ext{ W} + 79	ext{ W} = 162.4	ext{ W}$

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

$Q = q_{total} \cdot t$

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

$Q = 162.4	ext{ W} \cdot 60	ext{ s} = 9744	ext{ J} = 9.7	ext{ kJ}$

Therefore, the best choice is B) 39 kJ, which is the closest value to 9.7 kJ when rounded to one significant figure.

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