To tackle this problem, we need to analyze the heat transfer between the two metal chunks through the connecting rod. Since the rod has low thermal conductivity, the heat transfer will be gradual. We can use the principles of thermodynamics and heat transfer to derive the temperature difference as a function of time.

Understanding the System

We have two metal chunks with heat capacities C₁ and C₂, connected by a rod of length l and cross-sectional area s. The rod has a thermal conductivity x, and the entire system is insulated from the environment. At time t = 0, there is an initial temperature difference ΔT₀ between the two chunks.

Heat Transfer Basics

The rate of heat transfer through the rod can be described by Fourier's law of heat conduction:

• Q = -kA(ΔT/l)

Where:

• Q is the heat transfer rate (W)
• k is the thermal conductivity (x in our case)
• A is the cross-sectional area (s)
• ΔT is the temperature difference between the two ends of the rod
• l is the length of the rod

Setting Up the Differential Equation

As heat flows from the hotter chunk to the cooler one, we can express the change in temperature of each chunk over time. The heat lost by the hotter chunk is equal to the heat gained by the cooler chunk:

• For chunk 1: C₁(dT₁/dt) = -Q
• For chunk 2: C₂(dT₂/dt) = Q

Since Q is the same for both chunks, we can combine these equations. The temperature difference ΔT can be defined as:

• ΔT = T₁ - T₂

Thus, we can express the heat transfer rate as:

• Q = x * s * (ΔT / l)

Combining the Equations

Now, substituting Q into the equations for the temperature changes gives us:

• C₁(dT₁/dt) = -x * s * (ΔT / l)
• C₂(dT₂/dt) = x * s * (ΔT / l)

We can express dT₁ and dT₂ in terms of ΔT:

• dT₁ = d(ΔT + T₂) = dΔT
• dT₂ = d(ΔT - T₁) = -dΔT

Deriving the Temperature Difference Over Time

Now, we can derive a single equation for ΔT:

• Using the relation: dΔT/dt = - (x * s / l) * (1/C₁ + 1/C₂) * ΔT

This is a first-order linear differential equation. We can solve it using separation of variables:

• 1/ΔT dΔT = - (x * s / l) * (1/C₁ + 1/C₂) dt

Integrating both sides gives us:

• ln|ΔT| = - (x * s / l) * (1/C₁ + 1/C₂) t + C

Exponentiating both sides leads to:

• ΔT(t) = ΔT₀ * e^(- (x * s / l) * (1/C₁ + 1/C₂) t)

Final Expression

The final expression for the temperature difference between the two chunks of metal as a function of time is:

• ΔT(t) = ΔT₀ * e^(-kt)

Where k = (x * s / l) * (1/C₁ + 1/C₂). This shows that the temperature difference decreases exponentially over time, reflecting the gradual heat transfer through the rod.

This approach highlights how thermal conductivity, heat capacities, and the geometry of the system influence the rate of heat transfer and the evolution of temperature differences over time. If you have any further questions or need clarification on any part, feel free to ask!