When a man does exercise on a treadmill machine, work doen by him is supplied in the form of heat of an ice bath. Find how much he has to do to convert a piece of 5 gm of ice at `-3^(@)C` to steam at `100^(@)C`.Given that the specific heat of ice is 0.5 cal/`gm^(@)C`, latent heat of fusion of ice is 80 cal/gm and laten heat of vaporization of water is 540 cal/gm.

Answer

To solve the problem of converting 5 grams of ice at -3°C to steam at 100°C, we need to calculate the total heat required for each step of the process. The steps involved are:

1. Heating the ice from -3°C to 0°C.
2. Melting the ice at 0°C to water.
3. Heating the water from 0°C to 100°C.
4. Converting the water at 100°C to steam.

Let's go through each step in detail:

### Step 1: Heating the Ice from -3°C to 0°C
To calculate the heat required to raise the temperature of the ice, we use the formula:
\[ Q_1 = m \cdot c \cdot \Delta T \]
where:
- \( m = 5 \, \text{g} \) (mass of ice)
- \( c = 0.5 \, \text{cal/g°C} \) (specific heat of ice)
- \( \Delta T = 0 - (-3) = 3 \, \text{°C} \) (change in temperature)

Calculating \( Q_1 \):
\[ Q_1 = 5 \, \text{g} \cdot 0.5 \, \text{cal/g°C} \cdot 3 \, \text{°C} = 7.5 \, \text{cal} \]

### Step 2: Melting the Ice at 0°C
Next, we calculate the heat required to melt the ice:
\[ Q_2 = m \cdot L_f \]
where:
- \( L_f = 80 \, \text{cal/g} \) (latent heat of fusion)

Calculating \( Q_2 \):
\[ Q_2 = 5 \, \text{g} \cdot 80 \, \text{cal/g} = 400 \, \text{cal} \]

### Step 3: Heating the Water from 0°C to 100°C
Now, we calculate the heat required to raise the temperature of the water:
\[ Q_3 = m \cdot c_w \cdot \Delta T \]
where:
- \( c_w = 1 \, \text{cal/g°C} \) (specific heat of water)
- \( \Delta T = 100 - 0 = 100 \, \text{°C} \)

Calculating \( Q_3 \):
\[ Q_3 = 5 \, \text{g} \cdot 1 \, \text{cal/g°C} \cdot 100 \, \text{°C} = 500 \, \text{cal} \]

### Step 4: Converting Water at 100°C to Steam
Finally, we calculate the heat required to convert the water to steam:
\[ Q_4 = m \cdot L_v \]
where:
- \( L_v = 540 \, \text{cal/g} \) (latent heat of vaporization)

Calculating \( Q_4 \):
\[ Q_4 = 5 \, \text{g} \cdot 540 \, \text{cal/g} = 2700 \, \text{cal} \]

### Total Heat Required
Now, we sum all the heat quantities calculated:
\[ Q_{\text{total}} = Q_1 + Q_2 + Q_3 + Q_4 \]
\[ Q_{\text{total}} = 7.5 \, \text{cal} + 400 \, \text{cal} + 500 \, \text{cal} + 2700 \, \text{cal} \]
\[ Q_{\text{total}} = 3607.5 \, \text{cal} \]

### Conclusion
The total work done by the man on the treadmill, which is converted to heat, must be equal to the total heat required:
\[ \text{Work done} = Q_{\text{total}} = 3607.5 \, \text{cal} \]