A piece of ice (heat capacity=`2100Jkg^-1^@C^-1` and latent heat `=3.36xx10^5Jkg^-1`) of mass m grams is at `-5^@C` at atmospheric pressure. It is given 420J of heat so that the ice starts melting. Finally when the ice-water mixture is in equilibrium, it is found that 1gm of ice has melted. Assuming there is no other heat exchange in the process, the value of m is

Explanation

To solve the problem, we will break it down into steps, calculating the heat absorbed by the ice and the heat required for melting it.

### Step 1: Understand the problem
We have a piece of ice at -5°C, and we are given 420 J of heat. We need to find the mass of the ice (m grams) when it is known that 1 gram of ice has melted.

### Step 2: Convert units
Since the mass is given in grams, we will convert it to kilograms for calculations.
- \( m \) grams = \( \frac{m}{1000} \) kg

### Step 3: Calculate the heat required to raise the temperature of ice from -5°C to 0°C
The formula to calculate the heat (Q1) required to raise the temperature of ice is:
\[ Q_1 = m \cdot C \cdot \Delta T \]
Where:
- \( C = 2100 \, \text{J/kg°C} \) (heat capacity of ice)
- \( \Delta T = 0 - (-5) = 5 \, \text{°C} \)

Substituting the values:
\[ Q_1 = \frac{m}{1000} \cdot 2100 \cdot 5 \]
\[ Q_1 = \frac{m \cdot 10500}{1000} \]
\[ Q_1 = 10.5m \, \text{J} \]

### Step 4: Calculate the heat required to melt 1 gram of ice
The latent heat (Q2) required to melt ice is given by:
\[ Q_2 = m_{ice} \cdot L \]
Where:
- \( L = 3.36 \times 10^5 \, \text{J/kg} \)
- \( m_{ice} = 1 \, \text{g} = 0.001 \, \text{kg} \)

Substituting the values:
\[ Q_2 = 0.001 \cdot 3.36 \times 10^5 \]
\[ Q_2 = 336 \, \text{J} \]

### Step 5: Set up the energy balance equation
The total heat supplied (420 J) is used to raise the temperature of the ice and to melt 1 gram of ice:
\[ Q_1 + Q_2 = 420 \]
Substituting the values we found:
\[ 10.5m + 336 = 420 \]

### Step 6: Solve for m
Rearranging the equation:
\[ 10.5m = 420 - 336 \]
\[ 10.5m = 84 \]
\[ m = \frac{84}{10.5} \]
\[ m = 8 \, \text{grams} \]

### Final Answer
The mass of the ice is \( m = 8 \, \text{grams} \).