What mass (in grams) of steam at 100°C must be mixed with 446 g of ice at its melting point, in a thermally insulated container, to produce liquid water at 65.0°C? The specific heat of water is 4186 J/kg·K. The latent heat of fusion is 333 kJ/kg, and the latent heat of vaporization is 2256 kJ/kg.
To solve this problem, we need to understand the energy changes happening during the phase transitions and heating processes.
First, we need to determine the energy required to heat the ice from its melting point to the final temperature (65.0°C). Then, we need to calculate the energy required to melt the ice completely, and finally, the energy released when the steam condenses to liquid water at 65.0°C.
Let's break down the solution into separate parts:
1. Heating the ice:
To determine the energy required to heat the ice, we can use the specific heat capacity formula:
Energy = mass * specific heat capacity * temperature change
The mass of the ice is given as 446 grams, and the specific heat capacity of water is 4186 J/kg·K. However, we need to convert the mass to kilograms since the specific heat capacity is given in terms of kilograms:
mass_ice = 446 g = 0.446 kg
The temperature change is from the melting point of ice (0°C) to the final temperature (65.0°C):
ΔT_ice = 65.0°C - 0°C = 65.0°C
Using the specific heat capacity formula, we can calculate the energy required to heat the ice:
Energy_ice = mass_ice * specific heat capacity * ΔT_ice
2. Melting the ice:
To determine the energy required to melt the ice completely, we can use the latent heat of fusion formula:
Energy = mass * latent heat of fusion
Since we are melting all the ice, the mass of the ice remains the same:
mass_ice = 0.446 kg
Using the formula, we can calculate the energy required to melt the ice:
Energy_melt = mass_ice * latent heat of fusion
3. Condensing the steam:
To determine the energy released when the steam condenses to liquid water, we can use the latent heat of vaporization formula:
Energy = mass * latent heat of vaporization
Let's assume the mass of steam required is m grams. So, the mass of liquid water produced will also be m grams.
mass_water = m g = m/1000 kg (converting to kilograms)
Using the formula, we can calculate the energy released when the steam condenses:
Energy_condense = mass_water * latent heat of vaporization
Now, the total energy required to heat the ice, melt the ice, and condense the steam is equal to the energy released when the steam condenses:
Energy_total = Energy_ice + Energy_melt = Energy_condense
By substituting the relevant equations and solving for m (mass of steam required), we can find the answer.