If a ball of `80` kg mass hits an ice cube and temperature of ball is `100 .^(@)C`, then how much ice converted into water ? (Specific heat of ball is `0.2 cal g^(-1)`, Latent heat of ice `= 80 cal g^(-1)`)

Options

A. a. 20 g
B. b. `200 g`
C. c. `2 xx 10^(3) kg`
D. d. `2 xx 10^(4)g`

Explanation

To solve the problem of how much ice is converted into water when a ball of mass 80 kg at a temperature of 100 °C hits it, we can follow these steps:

### Step 1: Understand the heat transfer
When the hot ball hits the ice, it will lose heat, and this heat will be used to melt the ice. The heat lost by the ball will be equal to the heat gained by the ice.

### Step 2: Write the heat transfer equations
The heat lost by the ball can be expressed as:
\[ Q_{\text{ball}} = m_b \cdot c_b \cdot \Delta T \]
where:
- \( m_b \) = mass of the ball in grams
- \( c_b \) = specific heat of the ball (0.2 cal/g°C)
- \( \Delta T \) = change in temperature of the ball (100°C to 0°C, so ΔT = 100°C)

The heat gained by the ice to melt it can be expressed as:
\[ Q_{\text{ice}} = m_i \cdot L \]
where:
- \( m_i \) = mass of the ice melted (in grams)
- \( L \) = latent heat of ice (80 cal/g)

### Step 3: Convert the mass of the ball to grams
The mass of the ball is given as 80 kg. To convert this to grams:
\[ m_b = 80 \, \text{kg} \times 1000 \, \text{g/kg} = 80000 \, \text{g} \]

### Step 4: Set the heat lost by the ball equal to the heat gained by the ice
From the conservation of energy:
\[ m_b \cdot c_b \cdot \Delta T = m_i \cdot L \]
Substituting the known values:
\[ 80000 \, \text{g} \cdot 0.2 \, \text{cal/g°C} \cdot 100 \, \text{°C} = m_i \cdot 80 \, \text{cal/g} \]

### Step 5: Calculate the left side of the equation
Calculating the left side:
\[ 80000 \cdot 0.2 \cdot 100 = 1600000 \, \text{cal} \]

### Step 6: Solve for the mass of ice melted
Now we can solve for \( m_i \):
\[ 1600000 = m_i \cdot 80 \]
\[ m_i = \frac{1600000}{80} = 20000 \, \text{g} \]

### Step 7: Final answer
Thus, the mass of ice converted into water is:
\[ m_i = 20000 \, \text{g} = 2 \times 10^4 \, \text{g} \]