19 g of water at `30^@C` and 5 g of ice at `-20^@C` are mixed together in a calorimeter. What is the final temperature of the mixture? Given specific heat of ice `=0.5calg^(-1) (.^(@)C)^(-1)` and latent heat of fusion of ice `=80calg^(-1)`

Options

A. `0^@C`
B. `-5^@C`
C. `5^@C`
D. `10^@C`

Explanation

To solve the problem of finding the final temperature when 19 g of water at 30°C is mixed with 5 g of ice at -20°C, we will follow these steps:

### Step 1: Calculate the heat required to melt the ice
The heat required to raise the temperature of the ice from -20°C to 0°C and then to melt it can be calculated using the formula:

\[ Q_1 = m \cdot c \cdot \Delta T + m \cdot L_f \]

Where:
- \( m \) = mass of ice = 5 g
- \( c \) = specific heat of ice = 0.5 cal/g°C
- \( \Delta T \) = change in temperature = 0°C - (-20°C) = 20°C
- \( L_f \) = latent heat of fusion of ice = 80 cal/g

Calculating \( Q_1 \):

\[ Q_1 = 5 \cdot 0.5 \cdot 20 + 5 \cdot 80 \]
\[ Q_1 = 5 \cdot 10 + 400 = 50 + 400 = 450 \text{ cal} \]

### Step 2: Calculate the heat given by the water
The heat lost by the water as it cools from 30°C to the final temperature \( T \) can be calculated as:

\[ Q_2 = m \cdot c \cdot (T_{initial} - T_{final}) \]

Where:
- \( m \) = mass of water = 19 g
- \( c \) = specific heat of water = 1 cal/g°C
- \( T_{initial} \) = 30°C
- \( T_{final} \) = \( T \)

Calculating \( Q_2 \):

\[ Q_2 = 19 \cdot 1 \cdot (30 - T) = 570 - 19T \text{ cal} \]

### Step 3: Set up the heat balance equation
Since no heat is lost to the surroundings, the heat gained by the ice must equal the heat lost by the water:

\[ Q_2 = Q_1 \]

Substituting the values we found:

\[ 570 - 19T = 450 \]

### Step 4: Solve for \( T \)
Rearranging the equation:

\[ 570 - 450 = 19T \]
\[ 120 = 19T \]
\[ T = \frac{120}{19} \approx 6.32°C \]

### Step 5: Check if all ice melts
We need to check if all the ice melts at this temperature. The heat required to melt the ice (450 cal) is less than the heat given by the water (570 cal), confirming that all the ice will melt.

### Final Answer
The final temperature of the mixture is approximately **6.32°C**.