Imagine that 25.0 of ice at -3.0C is placed in a container holding water vapor at temperature of 120.0C . After a certain amount of time, the container has expanded slightly, but holds only water vapor at 110.0C. what is the final volume of the container? you may assume the water vapor is an ideal gas, and no heat is lost to the surroundings. some other information is useful: deltaH fus(H2O) = 6.01KJ/mol
delta H vap(H2O) = 40.66kJ/mol
Cp,m H2O solid =38.02J/mol K
Cp,m H2O liquid = 75.35J/mol K
Cp,m H2O gas = 33.60J/mol K
pressure= 1.000 atm
To find the final volume of the container, we need to calculate the amount of water vapor that is produced from the ice, and then use the ideal gas law to find the final volume.
Step 1: Calculate the moles of water in the ice.
Given the mass of ice (25.0 g) and the molar mass of water (18.015 g/mol), we can calculate the moles of water in the ice:
moles of H2O = mass of ice / molar mass of water
moles of H2O = 25.0 g / 18.015 g/mol = 1.387 mol
Step 2: Calculate the heat released during the phase change from ice to water at 0°C.
The heat released can be calculated using the equation:
q = ΔHfus * moles of H2O
q = 6.01 kJ/mol * 1.387 mol = 8.329 kJ
Step 3: Calculate the heat required to raise the temperature of the water from 0°C to 110°C.
The heat required can be calculated using the equation:
q = Cp,m * moles of H2O * ΔT
ΔT = final temperature - initial temperature = 110°C - 0°C = 110 K
q = Cp,m H2O solid * moles of H2O * ΔT
q = 38.02 J/mol K * 1.387 mol * 110 K = 5674 J = 5.674 kJ
Step 4: Calculate the heat required to vaporize the water at 100°C.
The heat required can be calculated using the equation:
q = ΔHvap * moles of H2O
q = 40.66 kJ/mol * 1.387 mol = 56.395 kJ
Step 5: Calculate the total heat change.
The total heat change is the sum of the heat released during the phase change and the heat required to raise the temperature and vaporize the water:
q total = q phase change + q temperature change + q vaporization
q total = 8.329 kJ + 5.674 kJ + 56.395 kJ = 70.398 kJ
Step 6: Calculate the final volume of the water vapor.
Using the ideal gas law, we can calculate the final volume of the water vapor at 110°C and 1.000 atm:
PV = nRT
V = (nRT) / P
P = 1.000 atm
R = 0.0821 L∙atm / mol∙K (ideal gas constant)
T = temperature in Kelvin = 110°C + 273.15 = 383.15 K
n = moles of water vapor = moles of H2O (after vaporization) = 1.387 mol
V = (1.387 mol * 0.0821 L∙atm/mol∙K * 383.15 K) / 1.000 atm
V = 43.98 L
Therefore, the final volume of the container is 43.98 L.
To solve this problem, we'll use the principles of thermodynamics and the ideal gas law. We'll determine the amount of heat transferred to the ice to change its temperature and melt it, and then use that to calculate the final volume of the container.
Step 1: Calculate the heat transferred to the ice to raise its temperature to 0°C:
The heat transferred can be calculated using the formula:
q = m * Cp * ΔT
where q is the heat transferred, m is the mass, Cp is the specific heat capacity, and ΔT is the change in temperature.
Given:
Mass of ice (m) = 25.0 g
Change in temperature (ΔT) = 0°C - (-3.0°C) = 3.0°C
Specific heat capacity of ice (Cp,m) = 38.02 J/mol K
Convert mass to moles:
Molar mass of H2O = 18.015 g/mol
Number of moles (n) of ice = mass/molar mass = 25.0 g/18.015 g/mol
Calculate the heat transferred:
q1 = (n * Cp,m * ΔT)
q1 = (25.0 g/18.015 g/mol) * (38.02 J/mol K) * (3.0°C)
q1 = 76.715 J
Step 2: Calculate the heat transferred to melt the ice:
The heat transferred can be calculated using the formula:
q = n * ΔHfus
where q is the heat transferred, n is the number of moles, and ΔHfus is the enthalpy of fusion.
Given:
ΔHfus (H2O) = 6.01 kJ/mol
Convert ΔHfus to joules:
1 kJ = 1000 J
ΔHfus (H2O) = 6.01 kJ/mol * 1000 J/kJ = 6010 J/mol
Calculate the heat transferred:
q2 = (n * ΔHfus)
q2 = (25.0 g/18.015 g/mol) * (6010 J/mol)
q2 = 8353.523 J
Step 3: Calculate the heat transferred to raise the temperature of water vapor to 110.0°C:
The heat transferred can be calculated using the formula:
q = m * Cp * ΔT
where q is the heat transferred, m is the mass, Cp is the specific heat capacity, and ΔT is the change in temperature.
Given:
Change in temperature (ΔT) = 110.0°C - 100.0°C = 10.0°C
Specific heat capacity of water vapor (Cp,m) = 33.60 J/mol K
Convert mass to moles:
Molar mass of H2O = 18.015 g/mol
Number of moles (n) of water vapor = mass/molar mass
Calculate the heat transferred:
q3 = (n * Cp,m * ΔT)
Step 4: Calculate the total heat transferred to the system:
The total heat transferred is the sum of the individual heat transfers:
q_total = q1 + q2 + q3
Step 5: Use the ideal gas law to calculate the final volume of the container:
The ideal gas law equation is:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant (0.0821 L·atm/mol·K), and T is the temperature in Kelvin.
Given:
Pressure (P) = 1.000 atm
Final temperature (T) = 110.0°C = 383.15 K (converted to Kelvin)
Rearrange the ideal gas law equation to solve for volume (V):
V = (nRT) / P
Calculate the final volume:
V = (n * R * T) / P
Now, we'll substitute the values and solve for the final volume.