You are a blacksmith and have been working with 13.5 kg of steel. When you are finished shaping it, the steel is at a temperature of 450°C. To cool it off, you drop it into a bucket containing 5.5 kg of water at 55°C. How much of this water is converted to steam? Assume the steel, the water, and the steam all have the same final temperature.

To solve this problem, we need to use the concept of conservation of energy. The heat lost by the steel will be equal to the heat gained by the water and the heat used to convert some of the water into steam.

First, let's calculate the heat lost by the steel. The specific heat capacity of steel is approximately 0.49 J/g°C.

The heat (Q) lost by the steel can be calculated using the formula Q = mcΔT, where m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.

For the steel: Q_steel = m_steel * c_steel * (T_initial_steel - T_final),

where: m_steel = 13.5 kg = 13500 g, c_steel = 0.49 J/g°C, T_initial_steel = 450°C, T_final is unknown at this point.

Next, let's calculate the heat gained by the water. The specific heat capacity of water is approximately 4.18 J/g°C.

For the water: Q_water = m_water * c_water * (T_final - T_initial_water),

where: m_water = 5.5 kg = 5500 g, c_water = 4.18 J/g°C, T_initial_water = 55°C, T_final is the same as for the steel.

Finally, let's calculate the heat used to convert the water into steam. The heat of vaporization of water is approximately 2260 J/g.

For the steam: Q_steam = m_steam * Hvap,

where: m_steam is the mass of the water converted into steam, Hvap = 2260 J/g.

According to the conservation of energy, the heat lost by the steel will be equal to the heat gained by the water and the heat used to convert the water into steam:

Q_steel = Q_water + Q_steam.

Substituting the expressions for Q_steel, Q_water, and Q_steam into this equation, we get:

m_steel * c_steel * (T_initial_steel - T_final) = m_water * c_water * (T_final - T_initial_water) + m_steam * Hvap.

This equation has two unknowns: T_final and m_steam. To solve it, we need to make an assumption about the final temperature. A reasonable assumption is that the final temperature is 100°C, the boiling point of water. This is because the water will start to boil and convert into steam when it reaches this temperature.

Substituting T_final = 100°C into the equation, we can solve it for m_steam. This will give us the mass of the water converted into steam.