5.00 liter steel cylinder is filled with water vapor at 100 °C and 650 mm Hg. The problem I am having is whether the 5.00L significant in the problem and if so where?
a. Determine the density of this gas sample.
b. Determine the mass of water that condenses to a liquid when the closed
system is cooled to 20 °C. The vapor pressure of water at 20 °C is 18 mm
Hg.
6.
PM = dRT
P is pressure in atm.
M is molar mass H2O
d is density of gas sample.
R is 0.08206
T is 373K.
To solve this problem, we need to consider the given information and use the appropriate equations and relationships.
a. To determine the density of the gas sample, we need to use the ideal gas law: PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature. Rearranging the equation, we have:
n/V = P/RT
Given:
P = 650 mm Hg
T = 100 °C = (100 + 273.15) K (converting Celsius to Kelvin)
R = 0.0821 L atm/(mol K) (ideal gas constant)
Now we can calculate the number of moles (n) using the given pressure and temperature:
n/V = (650 mm Hg) / (0.0821 L atm/(mol K) * (100 + 273.15) K)
n/V = 0.0256 mol/L
Since the units cancel out, the volume of 5.00 L is not significant to calculate the density.
b. To determine the mass of water that condenses to a liquid, we need to use the ideal gas law and the relationship between pressure and mole fraction.
First, we find the number of moles of water vapor initially present in the cylinder using the given conditions:
n_initial = (P_initial * V) / (R * T_initial)
n_initial = (650 mm Hg * 5.00 L) / (0.0821 L atm/(mol K) * (100 + 273.15) K)
Next, we need to find the number of moles of water vapor at the final temperature (20 °C) using the relationship between pressure and mole fraction:
P_final = P_total - P_water_vapor
P_water_vapor = P_total - P_final
Given:
P_total = 650 mm Hg
P_final = 18 mm Hg
Substituting these values, we get:
P_water_vapor = 650 mm Hg - 18 mm Hg
P_water_vapor = 632 mm Hg
Now, using the new pressure and the ideal gas law, we can calculate the number of moles at the final temperature:
n_final = (P_water_vapor * V) / (R * T_final)
n_final = (632 mm Hg * 5.00 L) / (0.0821 L atm/(mol K) * (20 + 273.15) K)
Finally, to determine the mass of water that condenses, we subtract the number of moles at the final temperature from the number of moles initially present:
mass_condensed = (n_initial - n_final) * molar mass of water
Please note that in these calculations, the volume of the cylinder (5.00 L) is significant as it directly affects the number of moles present in the system.