4 kg of ice at `-15^(@)C` are added to 5 kg of water at `15^(@)C` . The temperature of the mixture equals

Options

A. `-15^(@)C`
B. `0^(@)C`
C. ` 5^(@)C`
D. `15^(@)C`

Explanation

To solve the problem of finding the final temperature of the mixture when 4 kg of ice at -15°C is added to 5 kg of water at 15°C, we will follow these steps:

### Step 1: Calculate the heat required to raise the temperature of ice from -15°C to 0°C.

The formula to calculate the heat (Q) required is:
\[ Q = m \cdot c \cdot \Delta T \]
where:
- \( m \) = mass of ice = 4 kg = 4000 g (since we will use calories, we convert kg to grams)
- \( c \) = specific heat of ice = 0.5 cal/g°C
- \( \Delta T \) = change in temperature = 0°C - (-15°C) = 15°C

Calculating the heat:
\[ Q_{\text{ice}} = 4000 \, \text{g} \cdot 0.5 \, \text{cal/g°C} \cdot 15 \, \text{°C} \]
\[ Q_{\text{ice}} = 4000 \cdot 0.5 \cdot 15 = 30000 \, \text{cal} \]

### Step 2: Calculate the heat released by the water as it cools from 15°C to 0°C.

Using the same formula:
- \( m \) = mass of water = 5 kg = 5000 g
- \( c \) = specific heat of water = 1 cal/g°C
- \( \Delta T \) = change in temperature = 15°C - 0°C = 15°C

Calculating the heat:
\[ Q_{\text{water}} = 5000 \, \text{g} \cdot 1 \, \text{cal/g°C} \cdot 15 \, \text{°C} \]
\[ Q_{\text{water}} = 5000 \cdot 1 \cdot 15 = 75000 \, \text{cal} \]

### Step 3: Determine if the heat from the water is sufficient to melt the ice.

The latent heat of fusion for ice is approximately 80 cal/g. To find out how much ice can be melted with the heat released by the water, we first calculate the total heat available from the water.

The total heat available from the water is 75000 cal.

Now, we need to calculate how much ice can be melted with this heat:
Let \( m \) be the mass of ice melted.
\[ Q_{\text{melt}} = m \cdot L_f \]
where \( L_f \) = latent heat of fusion = 80 cal/g.

Setting the heat from the water equal to the heat required to melt the ice:
\[ 75000 \, \text{cal} = m \cdot 80 \, \text{cal/g} \]
\[ m = \frac{75000}{80} = 937.5 \, \text{g} \]

### Step 4: Determine the final state of the system.

The initial mass of ice is 4000 g. After melting, the mass of ice remaining is:
\[ \text{Remaining ice} = 4000 \, \text{g} - 937.5 \, \text{g} = 3062.5 \, \text{g} \]

### Step 5: Find the final temperature of the mixture.

After the ice has melted, we have:
- 937.5 g of water from melted ice at 0°C
- 5 kg (5000 g) of water at 0°C

The total mass of water in the mixture is:
\[ \text{Total water} = 5000 \, \text{g} + 937.5 \, \text{g} = 5937.5 \, \text{g} \]

Since all the water is now at 0°C (the ice has melted to water at 0°C), the final temperature of the mixture is:
\[ \text{Final Temperature} = 0°C \]

### Final Answer:
The final temperature of the mixture is **0°C**.

---