A hollow sphere of glas whose external and internal radii are 11 cm and 9 cm respectively is completely filled with ice at `0^@C` and placed in a bath of boiling water. How long will it take for the ice to melt completely? Given that density of ice = `0.9 g//cm^3`, latent heat of fusion of ice `= 80 cal//g` and thermal conductivity of glass `=0.002 cal//cm-s^@C`.

Explanation

To solve the problem step by step, we will follow the outlined approach in the video transcript.

### Step 1: Calculate the Rate of Heat Flow (H)
We will use the formula for the rate of heat flow through a hollow sphere:

\[ H = \frac{4 \pi K R_1 R_2 \Delta T}{R_2 - R_1} \]

Where:
- \( K = 0.002 \, \text{cal/cm-s} \) (thermal conductivity of glass)
- \( R_1 = 11 \, \text{cm} \) (external radius)
- \( R_2 = 9 \, \text{cm} \) (internal radius)
- \( \Delta T = 100 - 0 = 100 \, \text{°C} \) (temperature difference)

Substituting the values into the formula:

\[ H = \frac{4 \pi (0.002) (11) (9) (100)}{9 - 11} \]

Calculating the denominator:

\[ H = \frac{4 \pi (0.002) (11) (9) (100)}{-2} \]

Calculating the numerator:

\[ H = \frac{4 \pi (0.002) (11) (9) (100)}{-2} = \frac{4 \pi (0.002) (9900)}{-2} \]

Calculating this gives:

\[ H \approx 124.4 \, \text{cal/s} \]

### Step 2: Relate Heat Flow to Mass Flow Rate
The heat required to melt the ice can be expressed as:

\[ H = L_f \frac{dm}{dt} \]

Where:
- \( L_f = 80 \, \text{cal/g} \) (latent heat of fusion)
- \( \frac{dm}{dt} \) is the mass flow rate of ice melting.

Rearranging gives:

\[ \frac{dm}{dt} = \frac{H}{L_f} = \frac{124.4}{80} \]

Calculating this gives:

\[ \frac{dm}{dt} \approx 1.555 \, \text{g/s} \]

### Step 3: Calculate the Total Mass of Ice
The total mass of ice can be calculated using the volume of the hollow sphere:

\[ \text{Volume} = \frac{4}{3} \pi (R_1^3 - R_2^3) \]

Calculating the volume:

\[ \text{Volume} = \frac{4}{3} \pi (11^3 - 9^3) = \frac{4}{3} \pi (1331 - 729) = \frac{4}{3} \pi (602) \]

Calculating the mass using the density of ice:

\[ \text{Mass} = \text{Density} \times \text{Volume} = 0.9 \, \text{g/cm}^3 \times \frac{4}{3} \pi (602) \approx 916 \, \text{g} \]

### Step 4: Calculate the Time to Melt the Ice
Using the total mass and the mass flow rate, we can find the time:

\[ t = \frac{m}{\frac{dm}{dt}} = \frac{916}{1.555} \]

Calculating this gives:

\[ t \approx 589 \, \text{seconds} \]

### Final Answer
The time taken for the ice to melt completely is approximately **589 seconds**.

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