Two chunks of metal with heat capacities `C_(1)` and `C_(2)`, are interconnected by a rod length `l` and cross-sectional area `S` and fairly low heat conductivity `K`. The whole system is thermally insulated from the environment. At a moment `t = 0` the temperature difference betwene the two chunks of metal equals `(DeltaT)_(0)`. Assuming the heat capacity of the rod to be negligible, find the temperature difference between the chucks as a function of time.

Options

A. `T_(0) "exp" ((-KA(C_(1) + C_(2))t)/(C_(1)C_(2)))`
B. `T_(0) "exp" ((-KA(C_(1) + C_(2)))/(C_(1)C_(2)))`
C. `T_(0) "exp" ((KA(C_(1) + C_(2))t)/(C_(1)C_(2)))`
D. `T_(0) "exp" ((KA(C_(1) + C_(2))t^(2))/(C_(1)C_(2)))`

Explanation

To solve the problem of finding the temperature difference between two chunks of metal as a function of time, we can follow these steps:

### Step 1: Define the Variables
Let:
- \( C_1 \) = Heat capacity of the first chunk of metal
- \( C_2 \) = Heat capacity of the second chunk of metal
- \( l \) = Length of the rod connecting the two chunks
- \( S \) = Cross-sectional area of the rod
- \( K \) = Thermal conductivity of the rod
- \( \Delta T_0 \) = Initial temperature difference between the two chunks at \( t = 0 \)
- \( T_1 \) and \( T_2 \) = Temperatures of the first and second chunks of metal, respectively

### Step 2: Establish the Heat Transfer Equation
At any moment, the heat transfer between the two chunks can be described by the equation:
\[ C_1 \frac{dT_1}{dt} = -C_2 \frac{dT_2}{dt} \]
This indicates that the heat lost by one chunk is equal to the heat gained by the other.

### Step 3: Relate Temperature Change to Heat Conductivity
The rate of heat transfer through the rod can be expressed using Fourier's law of heat conduction:
\[ Q = \frac{K \cdot S}{l} (T_1 - T_2) \]
Where \( Q \) is the heat transferred per unit time.

### Step 4: Express the Temperature Change
Using the definitions of heat transfer, we can write:
\[ C_1 \frac{dT_1}{dt} = -\frac{K \cdot S}{l} (T_1 - T_2) \]
\[ C_2 \frac{dT_2}{dt} = -\frac{K \cdot S}{l} (T_2 - T_1) \]

### Step 5: Define the Temperature Difference
Let \( \Delta T = T_1 - T_2 \). Then, we can express the changes in temperature as:
\[ \frac{d\Delta T}{dt} = \frac{dT_1}{dt} - \frac{dT_2}{dt} \]

### Step 6: Substitute and Rearrange
Substituting the expressions for \( \frac{dT_1}{dt} \) and \( \frac{dT_2}{dt} \) into the equation for \( \frac{d\Delta T}{dt} \):
\[ \frac{d\Delta T}{dt} = -\frac{K \cdot S}{l} \left( \frac{1}{C_1} + \frac{1}{C_2} \right) \Delta T \]

### Step 7: Solve the Differential Equation
This is a first-order linear differential equation. We can separate variables and integrate:
\[ \frac{d\Delta T}{\Delta T} = -\frac{K \cdot S}{l} \left( \frac{1}{C_1} + \frac{1}{C_2} \right) dt \]
Integrating both sides gives:
\[ \ln|\Delta T| = -\frac{K \cdot S}{l} \left( \frac{1}{C_1} + \frac{1}{C_2} \right) t + C \]
Where \( C \) is the integration constant.

### Step 8: Exponentiate to Solve for \( \Delta T \)
Exponentiating both sides results in:
\[ \Delta T = e^{C} e^{-\frac{K \cdot S}{l} \left( \frac{1}{C_1} + \frac{1}{C_2} \right) t} \]
At \( t = 0 \), \( \Delta T = \Delta T_0 \), hence:
\[ e^{C} = \Delta T_0 \]
Thus, the final solution is:
\[ \Delta T(t) = \Delta T_0 e^{-\frac{K \cdot S}{l} \left( \frac{1}{C_1} + \frac{1}{C_2} \right) t} \]

### Final Result
The temperature difference between the two chunks of metal as a function of time is:
\[ \Delta T(t) = \Delta T_0 e^{-\frac{K \cdot S}{l} \left( \frac{1}{C_1} + \frac{1}{C_2} \right) t} \]