Two identical bodies are made of a material for which the heat capacity increases with temperature. One of these is at $100^{\circ} C$ , while the other one is at $0^{\circ} C$. If the two bodies are brought into contact, then, assuming no heat loss, the final common temperature is -
- $50^{\circ} C$
- more than $50^{\circ} C$
- less than $50^{\circ} C$ but greater than $0^{\circ} C$
- $0^{\circ} C$
The Correct Option is B
Solution and Explanation
Heat Capacity and Temperature Relationship
As the temperature increases, the heat capacity also increases. Hence, the heat capacity of the second substance is greater than the first. Let the heat capacities be \( c_1 \) and \( c_2 \) for the first and second substances, respectively, and let the common temperature be \( T \). The equation for heat transfer is:
\(mc_1 (T - 0) = mc_2 (100 - T)\)
We can simplify this equation to:
\(\frac{T}{100 - T} = \frac{c_2}{c_1} > 1\)
This implies:
\(c_2 > c_1\)
Since \( \frac{T}{100 - T} > 1 \), it follows that:
\(T > 50\)
Conclusion:
The temperature \( T \) must be greater than 50 for the heat capacity of the second substance to be greater than the first.
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Concepts Used:
Specific Heat Capacity
Specific heat of a solid or liquid is the amount of heat that raises the temperature of a unit mass of the solid through 1°C.
Molar Specific Heat:
The Molar specific heat of a solid or liquid of a material is the heat that you provide to raise the temperature of one mole of solid or liquid through 1K or 1°C.
Specific Heat at Constant Pressure or Volume:
The volume of solid remains constant when heated through a small range of temperature. This is known as specific heat at a constant volume. It is denoted as CV.
The pressure of solid remains constant when heated through a small range of temperature. This is known as specific heat at constant pressure which can be denoted as CP.