The ratio of densities of two substances is 2 : 3 and their specific heats are in the ratio 3:4. The ratio of their thermal capacities for unit volume is

Options

A. `1:1`
B. `1 :2`
C. `2 :1`
D. `8 :9`

Explanation

To solve the problem, we need to find the ratio of thermal capacities for unit volume of two substances given their densities and specific heats.

### Step-by-Step Solution:

1. **Understanding Thermal Capacity**:
The thermal capacity (or heat capacity) \( C \) of a substance can be defined as:
\[ C = m \cdot s \]
where \( m \) is the mass and \( s \) is the specific heat of the substance.

2. **Expressing Mass**:
The mass \( m \) of a substance can be expressed in terms of its density \( \rho \) and volume \( V \):
\[ m = \rho \cdot V \]
For unit volume, we can take \( V = 1 \, \text{m}^3 \). Thus, the mass becomes:
\[ m = \rho \]

3. **Thermal Capacity for Unit Volume**:
For substance A, the thermal capacity \( C_A \) can be expressed as:
\[ C_A = \rho_A \cdot s_A \]
For substance B, the thermal capacity \( C_B \) is:
\[ C_B = \rho_B \cdot s_B \]

4. **Finding the Ratio of Thermal Capacities**:
We need to find the ratio \( \frac{C_A}{C_B} \):
\[ \frac{C_A}{C_B} = \frac{\rho_A \cdot s_A}{\rho_B \cdot s_B} \]

5. **Substituting Given Ratios**:
We are given:
- The ratio of densities: \( \frac{\rho_A}{\rho_B} = \frac{2}{3} \)
- The ratio of specific heats: \( \frac{s_A}{s_B} = \frac{3}{4} \)

Substituting these ratios into the equation:
\[ \frac{C_A}{C_B} = \frac{\frac{2}{3} \cdot \frac{3}{4}}{1} = \frac{2 \cdot 3}{3 \cdot 4} = \frac{2}{4} = \frac{1}{2} \]

6. **Final Result**:
Thus, the ratio of thermal capacities for unit volume of the two substances is:
\[ \frac{C_A}{C_B} = \frac{1}{2} \]

### Conclusion:
The ratio of thermal capacities for unit volume of the two substances is \( 1:2 \).