Two metal rods, one silver and the other gold, are attached to each other end-to-end. The free end of the silver rod is immersed in a steam chamber at $100 ^ { \circ } \mathrm { C }$ and the free end of the gold rod in an ice water bath at $0 ^ { \circ } \mathrm { C }$ The rods are both 5.0 cm long and have a square cross-section that is 2.0 cm on a side. No heat is exchanged between the rods and their surroundings, except at the ends. What is the temperature at the point where the two rods are in contact with one another? The thermal conductivity of silver is 417 $\mathrm { W } / \mathrm { m } \cdot \mathrm { K } .$ and that of gold is 291 $\mathrm { W } / \mathrm { m } \cdot \mathrm { K } .$

Question

Two metal rods, one silver and the other gold, are attached to each other end-to-end. The free end of the silver rod is immersed in a steam chamber at $100 ^ { \circ } \mathrm { C }$ and the free end of the gold rod in an ice water bath at $0 ^ { \circ } \mathrm { C }$ The rods are both  5.0 cm long and have a square cross-section that is  2.0 cm  on a side. No heat is exchanged between the rods and their surroundings, except at the ends. What is the temperature at the point where the two rods are in contact with one another? The thermal conductivity of silver is 417 $\mathrm { W } / \mathrm { m } \cdot \mathrm { K } .$ and that of gold is 291 $\mathrm { W } / \mathrm { m } \cdot \mathrm { K } .$

Quizplus · Accepted Answer

The temperature at the point where the two rods are in contact can be found by setting the heat flow through each rod equal to each other, as they are in a steady state. The heat flow, $Q$, through a material is given by $Q = \frac{kA(T_{hot} - T_{cold})}{L}$, where $k$ is the thermal conductivity, $A$ is the cross-sectional area, $T_{hot}$ and $T_{cold}$ are the temperatures at the hot and cold ends, respectively, and $L$ is the length of the material. Since the heat flow through both rods is the same, we can set the equations equal to each other and solve for the temperature at the point of contact, $T$. Given the thermal conductivities and dimensions, the calculation shows that the temperature at the contact point leans closer to the silver rod's end due to its higher thermal conductivity, resulting in a temperature of $59 ^ { \circ } \mathrm { C }$.

Two Metal Rods, One Silver and the Other Gold, Are 100∘C100 ^ { \circ } \mathrm { C }100∘C

Two Metal Rods, One Silver and the Other Gold, Are $100 ^ { \circ } \mathrm { C }$