Two liquids have the densities in the ratio of 1:2 and specific heats in the ratio of 2:1. The ratio of thermal capacity of equal volume of those liquids is

Options

A. `1:1`
B. `1:4`
C. `4:1`
D. `2:1`

Answer

To find the ratio of thermal capacities of equal volumes of two liquids with given densities and specific heats, we can follow these steps:

### Step 1: Understand the given ratios
We have:
- Densities of the two liquids in the ratio of 1:2
- Specific heats of the two liquids in the ratio of 2:1

Let:
- Density of liquid 1 = ρ₁
- Density of liquid 2 = ρ₂
- Specific heat of liquid 1 = c₁
- Specific heat of liquid 2 = c₂

From the problem:
- ρ₁ / ρ₂ = 1 / 2
- c₁ / c₂ = 2 / 1

### Step 2: Express the masses in terms of volume
Since we are considering equal volumes (V) of both liquids, we can express the masses (m) of the liquids as:
- Mass of liquid 1 (m₁) = ρ₁ * V
- Mass of liquid 2 (m₂) = ρ₂ * V

### Step 3: Calculate the masses using the density ratios
Using the density ratio:
- If we let ρ₁ = 1x (for simplicity), then ρ₂ = 2x.
- Therefore, the masses become:
- m₁ = ρ₁ * V = (1x) * V = xV
- m₂ = ρ₂ * V = (2x) * V = 2xV

### Step 4: Calculate the thermal capacities
The thermal capacity (C) is given by the formula:
\[ C = m \cdot c \]

Thus, the thermal capacities of the two liquids are:
- Thermal capacity of liquid 1 (C₁) = m₁ * c₁ = (xV) * (2) = 2xV
- Thermal capacity of liquid 2 (C₂) = m₂ * c₂ = (2xV) * (1) = 2xV

### Step 5: Find the ratio of thermal capacities
Now, we can find the ratio of thermal capacities of the two liquids:
\[ \frac{C₁}{C₂} = \frac{2xV}{2xV} = 1 \]

### Conclusion
The ratio of thermal capacities of equal volumes of the two liquids is:
\[ \text{Ratio} = 1:1 \]

### Final Answer
The correct answer is Option 1: 1:1.

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