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

Options

A. The masses of the liquids are equal
B. Equal masses of the liquids at the same temperature are taken
C. Different volumes of the liquids at the same
D. Equal volumes of the liquids at the same temperature are taken

Explanation

To solve the problem regarding the rates of cooling of two different liquids in identical calorimeters, we can follow these steps:

### Step 1: Understand the Concept of Cooling
The rate of cooling of a liquid is influenced by various factors, including its mass, specific heat capacity, and the temperature difference between the liquid and its surroundings. According to Newton's Law of Cooling, the rate of heat loss of a body is directly proportional to the temperature difference between the body and its surroundings.

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

### Step 3: Relate Heat Transfer to Mass and Specific Heat
The heat transferred can also be expressed in terms of mass (m), specific heat (s), and change in temperature (dT):
\[ dq = m \cdot s \cdot dT \]
Thus, we can relate the rate of heat transfer to the mass and specific heat:
\[ \frac{dq}{dt} = m \cdot s \cdot \frac{dT}{dt} \]

### Step 4: Combine the Equations
Setting the two expressions for dq/dt equal gives:
\[ m \cdot s \cdot \frac{dT}{dt} = \sigma A (T^4 - T_0^4) \]
From this, we can solve for the rate of cooling (dT/dt):
\[ \frac{dT}{dt} = \frac{\sigma A (T^4 - T_0^4)}{m \cdot s} \]

### Step 5: Analyze the Conditions for Equal Rates of Cooling
For the rates of cooling of the two liquids to be the same, the right-hand side of the equation must be equal for both liquids. This means:
- The surface area (A) and the temperature difference (T^4 - T_0^4) must be the same.
- The product \( m \cdot s \) must also be the same for both liquids.

### Step 6: Evaluate the Given Options
Now, let's evaluate the options provided in the question:
1. **The masses of the liquids are equal**: This alone does not ensure equal rates of cooling because specific heat can vary.
2. **Equal masses of liquid at the same temperature are taken**: This does not guarantee equal rates of cooling since specific heat may differ.
3. **Different volumes of the liquid at the same temperature**: This is incorrect as different volumes will affect the mass and thus the rate of cooling.
4. **Equal volume of liquid at the same temperature are taken**: This is the most appropriate condition as it ensures that the mass and specific heat can be controlled to yield the same rate of cooling.

### Conclusion
The correct answer is:
**D: Equal volume of liquid at the same temperature are taken.**