In a perfectly insulating container, what is the equilibrium state (temperature and phases) if one mixes 475 g of water vapor at 140 degrees Celsius with 25 g of ice at 0 degrees Celsius?
To determine the equilibrium state of a system when mixing water vapor and ice in a perfectly insulating container, we need to consider the energy exchange that occurs during the process. In this case, the container is perfectly insulating, meaning there is no heat transfer between the system and its surroundings.
To find the equilibrium state, we need to calculate the final temperature and phases of the system after the mixing process. This can be done by applying the principles of energy conservation.
Step 1: Calculate the energy of the water vapor.
The specific heat capacity (Cv) of water vapor is known to be 1.996 J/g°C. We can calculate the energy using the formula:
Energy (water vapor) = mass (water vapor) × specific heat capacity (water vapor) × temperature change (water vapor)
Energy (water vapor) = 475 g × 1.996 J/g°C × (140°C - Tf)
Here, Tf is the final temperature of the system.
Step 2: Calculate the energy of the ice.
The heat of fusion (ΔHfus) of ice is the amount of energy required to convert it from a solid to a liquid at its melting point. The heat of fusion of ice is approximately 334 J/g. To calculate the energy of the ice, we need to consider two parts:
1. The energy required to warm the ice from its initial temperature (0°C) to its melting point.
Energy (ice warming) = mass (ice) × specific heat capacity (ice) × temperature change (ice)
Energy (ice warming) = 25 g × 2.09 J/g°C × (Tf - 0°C)
2. The energy required to melt the ice at its melting point (0°C).
Energy (ice melting) = mass (ice) × heat of fusion (ice)
Energy (ice melting) = 25 g × 334 J/g
Step 3: Apply energy conservation.
Since the system is isolated and there is no heat exchange, the energy lost by the water vapor will be gained by the ice. Therefore, we can equate the energies in Step 1 and Step 2 to find the final temperature (Tf) of the system:
475 g × 1.996 J/g°C × (140°C - Tf) = 25 g × 2.09 J/g°C × (Tf - 0°C) + 25 g × 334 J/g
Solving this equation will give us the final temperature (Tf) of the system.
After obtaining the final temperature, we can determine the phases of the components based on their respective melting and boiling points. At temperatures below 0°C, water is in the solid phase (ice). Between 0°C and 100°C, water is in the liquid phase. Above 100°C, water exists as a gas (water vapor).
By evaluating the equilibrium state considering the energy exchange and the melting/freezing point of water, you can determine the final temperature and phases of the system.