Question

A 2.00 m long lead bar and a 2.00 m long copper bar with the same diameter are connected together in series to make one 4.00 m long bar. The lead end is in thermal contact with a cold reservoir at 2∘C and the copper end is in thermal contact with a hot reservoir at 106∘C. What is the temperature at the junction of the lead and copper bars? The thermal conductivity of lead is 34.7 W/(m∘C) and for copper is 397 W/(m∘C).

Answer 1

The temperature at the junction of the lead and copper bars in thermal contact with a cold reservoir at 2°C and a hot reservoir at 106°C, is calculated using Fourier's Law and found to be 97.36°C.

Step-by-step explanation:

To determine the temperature at the junction of the lead and copper bars, we use the fact that at steady state, the heat flow through each bar is the same, because there is no heat build-up at any point in the system. The rate of heat transfer Q through a material is given by Fourier's Law:

Q = (k × A × ΔT) / L,

where k is the thermal conductivity, A is the cross-sectional area of the material, ΔT is the temperature difference across the material, and L is its length.

Let's denote the temperature at the junction as Tj. Since the heat flow through each bar is equal, we can set up the following equations:

• For the lead bar: Q = (34.7 W/(m°C) × A × (Tj - 2°C)) / 2.00 m
• For the copper bar: Q = (397 W/(m°C) × A × (106°C - Tj)) / 2.00 m

Since the cross-sectional areas and the lengths for both bars are the same and the heat transfer Q is equivalent, we can simplify and find Tj:

1. 34.7*(Tj - 2) = 397*(106 - Tj)
2. 34.7*Tj - 69.4 = 397*106 - 397*Tj
3. 34.7*Tj + 397*Tj = 397*106 + 69.4
4. (34.7 + 397) Tj = 42031.4
5. Tj = 42031.4 / 431.7
6. Tj = 97.36°C

So, the temperature at the junction of the lead and copper bars is 97.36°C.