A bar of gold is in thermal contact with a bar of silver of the same length and area. One end of the compound bar is maintained at 80.0°C while the opposite end is at 30.0°C. When the energy transfer
reaches steady state, what is the temperature at the junction?
To determine the temperature at the junction of the gold and silver bars when the energy transfer reaches steady state, we can apply the principle of thermal equilibrium.
In steady state, the heat transfer rate from the hotter end of the compound bar to the cooler end will be equal. This means that the rate of heat transfer from the bar of gold and the rate of heat transfer from the bar of silver will be the same.
We can use the formula for heat transfer, given by the equation:
Q = k * A * (T2 - T1) / L
Where:
Q is the rate of heat transfer
k is the thermal conductivity of the material
A is the cross-sectional area of the bar
T2 is the temperature at one end of the bar
T1 is the temperature at the other end of the bar
L is the length of the bar
Since the bars have the same length and area, we can simplify the equation:
Qgold = Qsilver
k_gold * A_gold * (T_junction - T_gold_hot) = k_silver * A_silver * (T_silver_cold - T_junction)
Where:
T_junction is the temperature at the junction of the two bars
T_gold_hot is the temperature of the hot end of the gold bar (80.0°C)
T_silver_cold is the temperature of the cold end of the silver bar (30.0°C)
To solve for T_junction, we can rearrange the equation:
(T_junction - T_gold_hot) / (T_silver_cold - T_junction) = (k_silver * A_silver) / (k_gold * A_gold)
Now, we can substitute the known values:
(T_junction - 80.0°C) / (30.0°C - T_junction) = (k_silver * A_silver) / (k_gold * A_gold)
Since we don't know the specific values of the thermal conductivities and cross-sectional areas, we cannot find the exact temperature at the junction. However, we can solve the equation numerically by substituting approximate values for the thermal conductivities and cross-sectional areas if they are given.
Alternatively, if the specific values of the thermal conductivities and cross-sectional areas are provided, substituting those values into the equation will give the temperature at the junction.