The density of a material A is 1500`kg//m^(3)` and that of another material B is 2000 `kg//m^(3)`. It is found that the heat capacity of 8 volumes of A is equal to heat capacity of 12 volumes of B. The ratio of specific heats of A and B will be

Options

A. `1:2`
B. `3:1`
C. `3:2`
D. `2:1`

Answer

To find the ratio of specific heats of materials A and B, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- Density of material A, \( \rho_A = 1500 \, \text{kg/m}^3 \)
- Density of material B, \( \rho_B = 2000 \, \text{kg/m}^3 \)
- Heat capacity of 8 volumes of A is equal to heat capacity of 12 volumes of B.

2. **Define Heat Capacity**:
- The heat capacity \( C \) of a material can be expressed as:
\[ C = m \cdot c \]
where \( m \) is the mass and \( c \) is the specific heat capacity.

3. **Express Mass in Terms of Density and Volume**:
- For material A:
\[ m_A = \rho_A \cdot V_A \]
where \( V_A = 8 \, \text{volumes} \).
- Therefore, the heat capacity of A is:
\[ C_A = m_A \cdot c_A = \rho_A \cdot V_A \cdot c_A = \rho_A \cdot 8 \cdot c_A \]

- For material B:
\[ m_B = \rho_B \cdot V_B \]
where \( V_B = 12 \, \text{volumes} \).
- Therefore, the heat capacity of B is:
\[ C_B = m_B \cdot c_B = \rho_B \cdot V_B \cdot c_B = \rho_B \cdot 12 \cdot c_B \]

4. **Set the Heat Capacities Equal**:
- According to the problem, the heat capacities are equal:
\[ C_A = C_B \]
Substituting the expressions for \( C_A \) and \( C_B \):
\[ \rho_A \cdot 8 \cdot c_A = \rho_B \cdot 12 \cdot c_B \]

5. **Rearranging the Equation**:
- Rearranging gives:
\[ \frac{c_A}{c_B} = \frac{\rho_B \cdot 12}{\rho_A \cdot 8} \]

6. **Substituting the Densities**:
- Now substituting the values of \( \rho_A \) and \( \rho_B \):
\[ \frac{c_A}{c_B} = \frac{2000 \cdot 12}{1500 \cdot 8} \]

7. **Calculating the Ratio**:
- Simplifying the right side:
\[ \frac{c_A}{c_B} = \frac{24000}{12000} = 2 \]

8. **Final Result**:
- Thus, the ratio of specific heats of A and B is:
\[ \frac{c_A}{c_B} = 2:1 \]

### Conclusion:
The ratio of specific heats of materials A and B is \( 2:1 \).