Two vessels 1 and 2, made out of different materials are identical in all the geometrical aspects. In both the vessel the same quantity of ice gets melted in 20 min and 30 min respectively. The ratio of the thermal conductivity of the second one to that of the first is

Options

A. 1.5
B. 1
C. `2/3`
D. `4`

Answer

To solve the problem, we need to find the ratio of the thermal conductivity of the second vessel (K2) to that of the first vessel (K1). We know that both vessels are identical in geometrical aspects and that the same quantity of ice melts in each vessel over different times.

### Step-by-Step Solution:

1. **Understanding the Problem:**
- We have two vessels made of different materials but identical in shape and size.
- The time taken for the ice to melt in vessel 1 (T1) is 20 minutes.
- The time taken for the ice to melt in vessel 2 (T2) is 30 minutes.
- We need to find the ratio K2/K1, where K1 and K2 are the thermal conductivities of vessels 1 and 2, respectively.

2. **Heat Transfer and Melting Ice:**
- The heat transfer (Q) required to melt the ice can be expressed as:
\[ Q = m \cdot L \]
where \(m\) is the mass of the ice and \(L\) is the latent heat of fusion.

3. **Heat Transfer Rate:**
- The rate of heat transfer through conduction can be given by:
\[ Q = k \cdot A \cdot \frac{\Delta T}{L} \]
where \(k\) is the thermal conductivity, \(A\) is the cross-sectional area, \(\Delta T\) is the temperature difference, and \(L\) is the thickness of the material.

4. **Relating Heat Transfer to Time:**
- Since the quantity of ice melted is the same in both vessels, we can set the heat transfer equations equal to each other:
\[ k_1 \cdot A \cdot \frac{\Delta T}{L} \cdot T_1 = k_2 \cdot A \cdot \frac{\Delta T}{L} \cdot T_2 \]
- Here, \(A\), \(\Delta T\), and \(L\) are constant for both vessels, so they can be canceled out.

5. **Establishing the Ratio:**
- From the above equation, we can derive:
\[ k_1 \cdot T_1 = k_2 \cdot T_2 \]
- Rearranging gives:
\[ \frac{k_2}{k_1} = \frac{T_1}{T_2} \]

6. **Substituting the Values:**
- Substitute the values of T1 and T2:
\[ \frac{k_2}{k_1} = \frac{20}{30} = \frac{2}{3} \]

7. **Final Answer:**
- Therefore, the ratio of the thermal conductivity of the second vessel to that of the first vessel is:
\[ \frac{K_2}{K_1} = \frac{2}{3} \]