The temperature of a hot liquid in a container of negligible heat capacity falls at the rate of `3K//min` due to heat emission to the surroundings, just before it begins to solidify. The temperature then remains constant for `30min`, by the time the liquid has all solidfied. Find the ratio of specific heat capacity of liquid to specific latent heat of fusion.

Answer

To solve the problem, we need to find the ratio of the specific heat capacity of the liquid (S) to the specific latent heat of fusion (L). Let's break down the solution step by step.

### Step 1: Understand the problem
The temperature of the liquid falls at a rate of \(3 \, \text{K/min}\) until it reaches the point of solidification, after which it remains constant for \(30 \, \text{min}\) while it solidifies.

### Step 2: Calculate the total temperature drop before solidification
The temperature drop in \(30 \, \text{min}\) can be calculated as follows:
\[ \Delta T = \text{Rate of temperature drop} \times \text{Time} = 3 \, \text{K/min} \times 30 \, \text{min} = 90 \, \text{K} \]

### Step 3: Calculate the heat released during cooling
The heat released during the cooling of the liquid can be expressed using the formula:
\[ Q_R = m \cdot S \cdot \Delta T \]
Substituting the values we have:
\[ Q_R = m \cdot S \cdot 90 \]

### Step 4: Calculate the heat required for solidification
When the liquid begins to solidify, the heat required for this process is given by:
\[ Q_S = m \cdot L \]

### Step 5: Set the heat released equal to the heat required
Since the heat released during cooling equals the heat required for solidification, we can set the two equations equal to each other:
\[ Q_R = Q_S \]
Substituting the expressions we derived:
\[ m \cdot S \cdot 90 = m \cdot L \]

### Step 6: Simplify the equation
We can cancel \(m\) from both sides (assuming \(m \neq 0\)):
\[ S \cdot 90 = L \]

### Step 7: Find the ratio of specific heat capacity to specific latent heat of fusion
Rearranging the equation gives us:
\[ \frac{S}{L} = \frac{1}{90} \]

### Final Answer
Thus, the ratio of the specific heat capacity of the liquid to the specific latent heat of fusion is:
\[ \frac{S}{L} = \frac{1}{90} \]

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