To find the difference in the amount of heat given out by 4 kg of steam and 4 kg of water when both are cooled from 100°C to 80°C, we need to calculate the heat released by each.

Heat Given Out by Water

The heat given out by water as it cools can be calculated using the formula:

\(Q = mc\Delta T\)

Where:
-  m = 4 kg (mass of water)
-  c = 4200  (specific heat capacity of water)
- \( \Delta\) T = 100 - 80 = 20K (change in temperature)

Calculating the heat given out by water:

\(Q_{\text{water}} = 4 \times 4200 \times 20 = 336,000 \, \text{J}\)

 Heat Given Out by Steam

For steam, we need to consider both the cooling of the steam to water and the cooling of the water from 100°C to 80°C. The total heat given out by the steam is the sum of the heat released during condensation and the heat released during cooling.

Heat released during condensation: \(Q_{\text{condensation}} = mL\)

Where:
- \( L = 2,260,000 \, \text{J/kg} \) (specific latent heat of steam)

Calculating the heat released during condensation:

\(Q_{\text{condensation}} = 4 \, \text{kg} \times 2,260,000 \, \text{J/kg} = 9,040,000 \, \text{J}\)

Heat released by water as it cools from 100°C to 80°C:

This is the same calculation we did for water above:

\(Q_{\text{cooling}} = 4 \times 4200  \times 20  = 336,000 \, \text{J}\)

Total heat given out by steam:

\(Q_{\text{steam}} = Q_{\text{condensation}} + Q_{\text{cooling}} = 9,040,000 \, \text{J} + 336,000 \, \text{J} = 9,376,000 \, \text{J}\)

Difference in Heat Given Out

Now, we find the difference in heat given out by steam and water:

\(\text{Difference} = Q_{\text{steam}} - Q_{\text{water}} = 9,376,000 \, \text{J} - 336,000 \, \text{J} = 9,040,000 \, \text{J}\)

The difference in the amount of heat given out by 4 kg of steam and 4 kg of water when both are cooled from 100°C to 80°C is 9,040,000 J.