© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Understand the Physical Situation
We have two identical bodies made of a material whose heat capacity increases with temperature. One body is at a higher temperature, 100°C, and the other is at a lower temperature, 0°C. They are placed in contact so that they exchange heat until they reach a common final temperature, with no heat loss to the surroundings.
Step 2: Represent the Heat Capacities
Let:
$C_{\text{h}}$ = effective heat capacity of the hotter body around its higher temperature range
$C_{\text{c}}$ = effective heat capacity of the colder body around its lower temperature range
$T_{\text{c}}$ = final common temperature of both bodies
Step 3: Apply the Principle of Calorimetry
According to the principle of calorimetry:
Heat lost by the hotter body = Heat gained by the colder body
This translates into the equation:
$C_{\text{h}} \, (100 - T_{\text{c}}) = C_{\text{c}} \, (T_{\text{c}} - 0)$
Since the lower body starts at 0°C, its heat gained is $C_{\text{c}} \, T_{\text{c}}$, and the hotter body’s heat lost is $C_{\text{h}} \, (100 - T_{\text{c}})$.
Step 4: Solve for the Final Temperature
Rearrange the above equation to find $T_{\text{c}}$:
$C_{\text{h}} \, (100 - T_{\text{c}}) = C_{\text{c}} \, T_{\text{c}}$
$C_{\text{h}} \times 100 - C_{\text{h}} \, T_{\text{c}} = C_{\text{c}} \, T_{\text{c}}$
$C_{\text{h}} \times 100 = T_{\text{c}} (C_{\text{h}} + C_{\text{c}})$
$T_{\text{c}} = \frac{C_{\text{h}}}{C_{\text{h}} + C_{\text{c}}} \times 100$
Step 5: Compare $C_{\text{h}}$ and $C_{\text{c}}$
Because the material's heat capacity increases with temperature, the body that starts at the higher temperature (100°C) effectively has a larger heat capacity than the one at 0°C. Therefore, $C_{\text{h}} > C_{\text{c}}$. Dividing both sides of the fraction inside the final temperature expression:
$T_{\text{c}} = \frac{100}{1 + \frac{C_{\text{c}}}{C_{\text{h}}}}$
Since $\frac{C_{\text{c}}}{C_{\text{h}}} < 1$, it follows that:
$1 + \frac{C_{\text{c}}}{C_{\text{h}}} < 2 \quad \Longrightarrow \quad T_{\text{c}} > \frac{100}{2} = 50\text{°C}$
Step 6: Final Conclusion
Hence, the final common temperature when these two bodies are brought into contact (with no heat loss to the surroundings) will be more than 50°C.
