A 34.05 gram liquid sample of an unstable compound is deposited in a metal piston assembly that has a cylinder with a diameter of 20.0 cm. After the sample is added, the piston is placed in contact with the liquid and the piston vent is closed. This process removes all residual gas from the piston. The piston assembly is placed in 15.00 Kg of water at 30.00 oC. The barometric pressure is 778 mm Hg (760 mm Hg= 1 atm). As the compound spontaneously decomposes the piston moves up and the temperature of the water increases to 34.52oC. The piston rises 59.1 cm during the reaction. The contents of the cylinder at the end of the complete decomposition consists of 0.450 mol of carbon dioxide gas,0 .375 mol of liquid water, 0.0375 mol of oxygen gas and an unknown amount of gas from an unidentified element. Other useful information:
∆Hfo CO2 = -393.5 Kj/mol
∆Hfo H2O = -286 kj/mol
Enthalpy of decomposition for unknown = -1893Kj/mol
Heat capacity of water = 4.184 J/goC
R = 0.08206 L x atm / mol x K or 8.3145 J/mol x K
Vapor pressure of H2O at 34.5oC = 41 mm Hg
Assume that the heat capacity of the piston assembly is negligible and the mass of the piston is negligible.
To find the unknown amount of gas from the unidentified element, we need to use the ideal gas law equation and the given information.
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, and T is the temperature.
First, let's calculate the initial volume of the gas before the reaction using the diameter of the cylinder:
Given:
- Diameter of the cylinder = 20.0 cm
- Radius of the cylinder = diameter/2 = 10.0 cm = 0.1 m
- Height of the piston rise = 59.1 cm = 0.591 m
The initial volume of the gas inside the cylinder can be calculated using the formula for the volume of a cylinder:
V_initial = π * r^2 * h
V_initial = 3.14 * (0.1)^2 * 0.591
V_initial ≈ 0.0185 m^3
Now, let's calculate the molar amount of CO2 formed during the reaction, which is given as 0.450 mol.
Using the ideal gas law equation, we can calculate the final pressure of carbon dioxide:
P_CO2 = n_CO2 * R * T / V_initial
P_CO2 = 0.450 mol * (0.08206 L x atm / mol x K) * (34.52 + 273.15) K / 0.0185 m^3
P_CO2 ≈ 112.8 atm
Since the total pressure inside the cylinder is the sum of the partial pressures of all the gases:
Total Pressure = P_CO2 + P_H2O + P_O2 + P_unknown
We have the following information:
- Barometric pressure = 778 mm Hg
- Vapor pressure of H2O at 34.5°C = 41 mm Hg
Using the conversion 760 mmHg = 1 atm, we can convert the barometric pressure to atm:
Barometric pressure = 778 mm Hg / 760 mm Hg/atm ≈ 1.024 atm
Now, we can calculate the partial pressures of water vapor and oxygen:
P_H2O = Vapor pressure of H2O at 34.5°C = 41 mm Hg / 760 mm Hg/atm ≈ 0.054 atm
P_O2 = total pressure - Partial pressure of CO2 - Partial pressure of H2O
P_O2 = (1.024 atm) - (112.8 atm) - (0.054 atm) ≈ -111.830 atm
The partial pressure of oxygen comes out to be negative, which indicates that the calculated total pressure exceeds the actual total pressure due to some unidentified gas.
To find the partial pressure of the unidentified gas, we need to subtract the partial pressures of CO2, water vapor, and oxygen from the total pressure:
P_unknown = Total pressure - (P_CO2 + P_H2O + P_O2)
P_unknown = (1.024 atm) - (112.8 atm + 0.054 atm + (-111.830 atm))
P_unknown ≈ 0 atm
The partial pressure of the unidentified gas comes out to be approximately 0 atm, indicating that there is no additional gas from the unidentified element present in the contents of the cylinder at the end of the reaction.
Therefore, the unknown amount of gas from the unidentified element is negligible.