What mass (in grams) of steam at 100°C must be mixed with 462 g of ice at its melting point, in a thermally insulated container, to produce liquid water at 40.0°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 consider the energy changes that occur during the phase transitions and heating processes. Here's how we can approach the problem step by step:
1. Calculate the heat required to raise the temperature of the ice from its melting point (-0.0°C) to the final temperature (40.0°C).
Q1 = mass of ice * specific heat of ice * change in temperature
The specific heat of ice is 2090 J/kg·K and the change in temperature is 40.0°C - (-0.0°C) = 40.0°C.
So, Q1 = 462 g * 2090 J/kg·K * 40.0°C
2. Calculate the heat required to melt the ice at its melting point (-0.0°C).
Q2 = mass of ice * latent heat of fusion
The latent heat of fusion is 333 kJ/kg.
To convert kJ to J, we multiply by 1000:
Q2 = 462 g * 333 kJ/kg * 1000 J/kJ
3. Calculate the heat required to raise the temperature of the steam from 100°C to the final temperature (40.0°C).
Q3 = mass of steam * specific heat of water * change in temperature
The specific heat of water is 4186 J/kg·K, and the change in temperature is 40.0°C - 100°C = -60.0°C.
So, Q3 = mass of steam * 4186 J/kg·K * -60.0°C
4. Calculate the heat required to condense the steam at 100°C.
Q4 = mass of steam * latent heat of vaporization
The latent heat of vaporization is 2256 kJ/kg.
To convert kJ to J, we multiply by 1000:
Q4 = mass of steam * 2256 kJ/kg * 1000 J/kJ
5. Since the entire process is thermally insulated, the total heat gained by the ice and steam must be equal.
Q1 + Q2 = Q3 + Q4
Substitute the calculated values into the equation and solve for the mass of steam.
By following these steps, you should be able to determine the mass of steam required to produce the desired outcome.