336 g of ice at `0^@C` is mixed with 336 g of water at `80^@C`. What is the final temperature of the mixture?

Options

A. `0^@C`
B. `40^@C`
C. `80^@C`
D. `85^@C`

Answer

To solve the problem of finding the final temperature when 336 g of ice at 0°C is mixed with 336 g of water at 80°C, we can follow these steps:

### Step 1: Understand the heat exchange process
When ice at 0°C is mixed with water at a higher temperature (80°C), the heat from the warm water will be used to melt the ice and then raise the temperature of the resulting water until thermal equilibrium is reached.

### Step 2: Calculate the heat required to melt the ice
The heat required to melt ice can be calculated using the formula:
\[ Q = m \cdot L_f \]
where:
- \( Q \) = heat absorbed (in joules)
- \( m \) = mass of ice (in grams)
- \( L_f \) = latent heat of fusion of ice (approximately 334 J/g)

For our case:
- \( m = 336 \, \text{g} \)
- \( L_f = 334 \, \text{J/g} \)

Calculating the heat required to melt the ice:
\[ Q_{\text{melt}} = 336 \, \text{g} \cdot 334 \, \text{J/g} = 112224 \, \text{J} \]

### Step 3: Calculate the heat lost by the water
The heat lost by the warm water as it cools down can be calculated using the formula:
\[ Q = m \cdot c \cdot \Delta T \]
where:
- \( c \) = specific heat capacity of water (approximately 4.18 J/g°C)
- \( \Delta T \) = change in temperature (final temperature - initial temperature)

Let \( T_f \) be the final temperature of the mixture. The initial temperature of the water is 80°C, so:
\[ \Delta T = 80 - T_f \]

For our case:
- \( m = 336 \, \text{g} \)
- \( c = 4.18 \, \text{J/g°C} \)

Calculating the heat lost by the water:
\[ Q_{\text{lost}} = 336 \, \text{g} \cdot 4.18 \, \text{J/g°C} \cdot (80 - T_f) \]

### Step 4: Set up the heat balance equation
At thermal equilibrium, the heat gained by the ice will be equal to the heat lost by the water:
\[ Q_{\text{melt}} + Q_{\text{gained}} = Q_{\text{lost}} \]

Since the melted ice will also increase in temperature from 0°C to \( T_f \):
\[ Q_{\text{gained}} = 336 \, \text{g} \cdot 4.18 \, \text{J/g°C} \cdot T_f \]

Setting up the equation:
\[ 112224 + 336 \cdot 4.18 \cdot T_f = 336 \cdot 4.18 \cdot (80 - T_f) \]

### Step 5: Solve the equation for \( T_f \)
Expanding and simplifying:
\[ 112224 + 1404.48 T_f = 26880 - 1404.48 T_f \]
Bringing all \( T_f \) terms to one side:
\[ 112224 + 1404.48 T_f + 1404.48 T_f = 26880 \]
\[ 112224 + 2808.96 T_f = 26880 \]
Now, isolate \( T_f \):
\[ 2808.96 T_f = 26880 - 112224 \]
\[ 2808.96 T_f = -85344 \]
\[ T_f = \frac{-85344}{2808.96} \]
\[ T_f \approx 30.4 \, °C \]

### Final Answer
The final temperature of the mixture is approximately **30.4°C**.

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