The rates of cooling of two different liquids put in exactly similar calorimeters and kept in identical surroundings are the same if

Options

A. Equal masses of the liquids at the same temperature are taken
B. Equal volumes of the liquids at the same temperature are taken
C. Both 1 and 2
D. Insufficient information

Answer

To solve the question regarding the rates of cooling of two different liquids in identical calorimeters, we will analyze the conditions under which the rates of cooling are the same.

### Step-by-Step Solution:

1. **Understanding the Cooling Process**:
The rate of cooling of a liquid can be described by Newton's Law of Cooling, which states that the rate of change of temperature of an object is proportional to the difference in temperature between the object and its surroundings.

2. **Applying Stefan's Law**:
According to Stefan's law, the heat transfer rate (dq/dt) can be expressed as:
\[ \frac{dq}{dt} = \sigma A (T^4 - T_0^4) \]
where:
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( A \) is the surface area,
- \( T \) is the temperature of the liquid,
- \( T_0 \) is the ambient temperature.

3. **Relating Heat to Mass and Specific Heat**:
The heat lost by the liquid can be expressed as:
\[ dq = m \cdot s \cdot dT \]
where:
- \( m \) is the mass of the liquid,
- \( s \) is the specific heat capacity,
- \( dT \) is the change in temperature.

4. **Setting Up the Rate of Cooling Equation**:
By combining the two equations, we can express the rate of cooling as:
\[ \frac{dT}{dt} \propto \frac{(T^4 - T_0^4)}{m \cdot s} \]

5. **Considering Equal Masses and Volumes**:
If we take equal masses of the liquids, we can denote them as \( m_1 \) and \( m_2 \). The condition for the rates of cooling to be the same requires:
\[ \frac{(T_1^4 - T_0^4)}{m_1 \cdot s_1} = \frac{(T_2^4 - T_0^4)}{m_2 \cdot s_2} \]
If \( m_1 = m_2 \), the equation simplifies to:
\[ \frac{(T_1^4 - T_0^4)}{s_1} = \frac{(T_2^4 - T_0^4)}{s_2} \]

6. **Considering Equal Volumes**:
If we take equal volumes of the liquids, we can express the mass in terms of density:
\[ m_1 = \rho_1 \cdot V \quad \text{and} \quad m_2 = \rho_2 \cdot V \]
The rates of cooling can also be analyzed under this condition.

7. **Conclusion**:
After analyzing both cases (equal mass and equal volume), we find that the rates of cooling will be the same if we take equal volumes of the liquids at the same temperature. Therefore, the correct answer is:
**Equal volumes of the liquids at the same temperature are taken.**

### Final Answer:
**The rates of cooling of two different liquids put in exactly similar calorimeters and kept in identical surroundings are the same if equal volumes of the liquids at the same temperature are taken.**