The rates of fall temperature of two identical solid spheres of different materials are equal at a certain temperature if:

Options

A. Their specific heat capacities are equal
B. Their heat capacities are equal
C. Their specific heat capacities are proportional to their densities
D. Their specific heat capacities are inversely proportional to their densities

Explanation

To solve the question regarding the rates of fall of temperature of two identical solid spheres of different materials being equal at a certain temperature, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two identical solid spheres made of different materials. We need to find the condition under which the rates of fall of their temperatures are equal at a certain temperature.

2. **Rate of Cooling**:
The rate of cooling of an object can be described by Newton's law of cooling, which states that the rate of change of temperature (dT/dt) is proportional to the temperature difference between the object and its surroundings. However, in this case, we need to consider the heat capacities of the spheres.

3. **Heat Capacity**:
The heat capacity (H) of a body is defined as the product of its mass (m) and its specific heat capacity (S). For two spheres, we have:
\[ H_1 = m_1 \cdot S_1 \quad \text{and} \quad H_2 = m_2 \cdot S_2 \]

4. **Equal Rates of Cooling**:
For the rates of fall of temperature to be equal, we can set up the equation:
\[ \frac{dT_1}{dt} = \frac{dT_2}{dt} \]
This implies:
\[ \frac{1}{H_1} = \frac{1}{H_2} \]
Therefore, we can conclude:
\[ H_1 = H_2 \]
This means that the heat capacities of both spheres must be equal.

5. **Specific Heat Capacity and Density**:
If we consider the relationship between specific heat capacity and density, we can express the specific heat capacity (S) in terms of density (ρ) and volume (V):
\[ S = \frac{H}{m} = \frac{H}{\rho \cdot V} \]
If the specific heat capacities are inversely proportional to their densities, we can write:
\[ S_1 \propto \frac{1}{\rho_1} \quad \text{and} \quad S_2 \propto \frac{1}{\rho_2} \]
This leads to the conclusion that:
\[ \frac{S_1}{S_2} = \frac{\rho_2}{\rho_1} \]

6. **Conclusion**:
Therefore, the condition for the rates of fall of temperature of the two identical solid spheres to be equal at a certain temperature is that their heat capacities must be equal, which can also imply that their specific heat capacities are inversely proportional to their densities.

### Final Answer:
The rates of fall of temperature of two identical solid spheres of different materials are equal at a certain temperature if their heat capacities are equal, which can also imply that their specific heat capacities are inversely proportional to their densities.