The amount of heat required to raise the temperature of 75 kg of ice at `0^oC` to water at `10^oC` is (latent heat of fusion of ice is 80 cal/g, specific heat of water is 1 cal/`g^oC`)

Options

A. 750 kcal
B. `6.75xx10^3kcal`
C. `60xx10^3kcal`
D. 6.75 kcal

Answer

To solve the problem of calculating the amount of heat required to raise the temperature of 75 kg of ice at 0 °C to water at 10 °C, we will follow these steps:

### Step 1: Convert the mass of ice from kg to grams
The mass of ice is given as 75 kg. Since 1 kg = 1000 g, we convert:
\[ \text{Mass of ice} = 75 \, \text{kg} \times 1000 \, \text{g/kg} = 75000 \, \text{g} \]

### Step 2: Calculate the heat required for melting the ice
The latent heat of fusion of ice is given as 80 cal/g. The heat required to melt the ice (Q1) is calculated using the formula:
\[ Q_1 = \text{mass} \times \text{latent heat of fusion} \]
Substituting the values:
\[ Q_1 = 75000 \, \text{g} \times 80 \, \text{cal/g} = 6000000 \, \text{cal} \]

### Step 3: Calculate the heat required to raise the temperature of water
After the ice has melted, we need to raise the temperature of the resulting water from 0 °C to 10 °C. The specific heat of water is given as 1 cal/g°C. The heat required for this (Q2) is calculated using the formula:
\[ Q_2 = \text{mass} \times \text{specific heat} \times \Delta T \]
where \(\Delta T\) is the change in temperature (10 °C - 0 °C = 10 °C).
Substituting the values:
\[ Q_2 = 75000 \, \text{g} \times 1 \, \text{cal/g°C} \times 10 \, \text{°C} = 750000 \, \text{cal} \]

### Step 4: Calculate the total heat required
The total heat required (Q_total) is the sum of the heat required for melting the ice and the heat required to raise the temperature of the water:
\[ Q_{\text{total}} = Q_1 + Q_2 \]
Substituting the values:
\[ Q_{\text{total}} = 6000000 \, \text{cal} + 750000 \, \text{cal} = 6750000 \, \text{cal} \]

### Step 5: Convert the total heat from calories to kilocalories
Since 1 kcal = 1000 cal, we convert:
\[ Q_{\text{total}} = \frac{6750000 \, \text{cal}}{1000} = 6750 \, \text{kcal} \]

### Final Answer
The amount of heat required to raise the temperature of 75 kg of ice at 0 °C to water at 10 °C is **6750 kcal**.

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