Fifty grams of ice at `0^(@)C` is places in a thermos bottle containing one hundred grams of water at `6^(@)C`. How many grams of ice will melt? The heat of fusion of water is 333 kJ `//` kg and the specific heat of water is 4190J `//` (kg K ).

Options

A. 7.5g
B. 2.0g
C. 8.3g
D. 17g

Answer

To solve the problem of how many grams of ice will melt when 50 grams of ice at 0°C is placed in 100 grams of water at 6°C, we can follow these steps:

### Step 1: Calculate the heat lost by the water
The heat lost by the water when it cools down from 6°C to 0°C can be calculated using the formula:
\[ Q = m \cdot c \cdot \Delta T \]
where:
- \( Q \) = heat lost (in Joules)
- \( m \) = mass of the water (in kg)
- \( c \) = specific heat of water (in J/kg·K)
- \( \Delta T \) = change in temperature (in K)

Given:
- Mass of water, \( m = 100 \, \text{g} = 0.1 \, \text{kg} \)
- Specific heat of water, \( c = 4190 \, \text{J/(kg·K)} \)
- Initial temperature, \( T_i = 6°C \)
- Final temperature, \( T_f = 0°C \)
- Change in temperature, \( \Delta T = T_i - T_f = 6 - 0 = 6 \, \text{K} \)

Now substituting the values:
\[ Q = 0.1 \, \text{kg} \cdot 4190 \, \text{J/(kg·K)} \cdot 6 \, \text{K} \]
\[ Q = 0.1 \cdot 4190 \cdot 6 = 2514 \, \text{J} \]

### Step 2: Calculate the heat required to melt the ice
The heat required to melt the ice can be calculated using the formula:
\[ Q = m \cdot L_f \]
where:
- \( L_f \) = latent heat of fusion of ice (in J/kg)

Given:
- Latent heat of fusion of ice, \( L_f = 333 \, \text{kJ/kg} = 333000 \, \text{J/kg} \)
- Mass of ice melted, \( m \) (unknown)

Thus, the heat required to melt \( m \) grams of ice is:
\[ Q = m \cdot 333000 \, \text{J/kg} \]

### Step 3: Set the heat lost equal to the heat gained
According to the principle of calorimetry, the heat lost by the water will be equal to the heat gained by the ice:
\[ 2514 \, \text{J} = m \cdot 333000 \, \text{J/kg} \]

### Step 4: Solve for \( m \)
Rearranging the equation to solve for \( m \):
\[ m = \frac{2514 \, \text{J}}{333000 \, \text{J/kg}} \]
\[ m = \frac{2514}{333000} \]
\[ m \approx 0.00755 \, \text{kg} \]

### Step 5: Convert to grams
To convert kilograms to grams:
\[ m \approx 0.00755 \, \text{kg} \times 1000 \, \text{g/kg} \]
\[ m \approx 7.55 \, \text{g} \]

### Conclusion
Thus, approximately **7.55 grams of ice will melt**.