A fully sealed 50m^3 room at atmospheric pressure and temperature 20C contains two ballons, one containing helium and the other containing oxygen. The balloons each have a volume of 0.5m^3 and have gauge pressure 2m of water. The balloons are ruptured and helium and oxygen mix with the air in the room.

Determine mass and mol fractions of the gaseous species in the room, assuming that air contains 79% nitrogen and 21% oxygen by volume and atmospheric pressure is 100kPa.

Universal gas constant = 8.314
Atomic molar mass He = 4 g/mol"

To determine the mass and mole fractions of the gaseous species in the room after the balloons rupture, we can follow these steps:

1. Convert the gauge pressure of the balloons from meters of water to pascals:
- 1 meter of water = 9806.65 Pa
- Gauge pressure of 2 meters of water = 2 * 9806.65 Pa = 19613.3 Pa

2. Calculate the absolute pressure inside the balloons, taking atmospheric pressure into account:
- Absolute pressure inside the balloons = Atmospheric pressure + Gauge pressure
- Absolute pressure inside the balloons = 100,000 Pa + 19,613.3 Pa = 119,613.3 Pa

3. Use the ideal gas law to determine the number of moles of helium and oxygen in the balloons:
- n = PV / RT
- Assuming that the temperature inside the room is the same as the atmospheric temperature, which is 20°C = 293 K
- The volume (V) of each balloon is 0.5 m^3
- The gas constant (R) is 8.314 J/mol·K
- For helium:
- P_helium = Absolute pressure inside the balloons
- n_helium = P_helium * V_helium / R / T
- For oxygen:
- P_oxygen = Absolute pressure inside the balloons
- n_oxygen = P_oxygen * V_oxygen / R / T

4. Calculate the mass of helium and oxygen in the balloons using the molar mass:
- The atomic molar mass of helium (M_helium) is given as 4 g/mol
- The atomic molar mass of oxygen (M_oxygen) is given as 32 g/mol
- Mass_helium = n_helium * M_helium
- Mass_oxygen = n_oxygen * M_oxygen

5. Determine the initial number of moles and mass of nitrogen and oxygen in the room:
- Since air contains 79% nitrogen and 21% oxygen by volume, we can assume that the initial mole fractions are the same.
- To calculate the total moles of nitrogen and oxygen in the room before mixing, let's consider a volume of 50 m^3:
- Volume_of_nitrogen = 50 m^3 * 0.79
- Volume_of_oxygen = 50 m^3 * 0.21
- From the ideal gas law, we can calculate the moles of nitrogen and oxygen:
- n_nitrogen = P_atm * Volume_of_nitrogen / R / T
- n_oxygen_initial = P_atm * Volume_of_oxygen / R / T
- Then, calculate the mass of nitrogen and oxygen in the room:
- Mass_nitrogen_initial = n_nitrogen * M_nitrogen
- Mass_oxygen_initial = n_oxygen_initial * M_oxygen

6. Calculate the total mass of the gaseous species in the room after the balloons rupture:
- Total_mass_gases = Mass_nitrogen_initial + Mass_oxygen_initial + Mass_helium + Mass_oxygen

7. Determine the mole fractions of the gaseous species in the room after the balloons rupture:
- Mole_fraction_nitrogen = n_nitrogen / (n_nitrogen + n_oxygen_initial + n_helium + n_oxygen)
- Mole_fraction_oxygen = n_oxygen_initial / (n_nitrogen + n_oxygen_initial + n_helium + n_oxygen)
- Mole_fraction_helium = n_helium / (n_nitrogen + n_oxygen_initial + n_helium + n_oxygen)
- Mole_fraction_final_oxygen = n_oxygen / (n_nitrogen + n_oxygen_initial + n_helium + n_oxygen)

You can now use these steps to calculate the mass and mole fractions of the gaseous species in the room.

To determine the mass and molar fractions of the gaseous species in the room after the balloons rupture, we can use the ideal gas law and Dalton's law of partial pressures. Here's how you can approach this problem step by step:

Step 1: Convert the gauge pressure (2m of water) to Pascal.
Since 1 meter of water exerts a pressure of approximately 9810 Pa, the gauge pressure can be converted to Pascal by multiplying by this conversion factor:
2m of water * 9810 Pa/m of water = 19620 Pa

Step 2: Determine the total pressure in the room.
The total pressure in the room is the sum of the atmospheric pressure and the gauge pressure.
Total pressure = Atmospheric pressure + Gauge pressure = 100 kPa + 19620 Pa
Convert 100 kPa to Pascal: 100 kPa * 1000 Pa/kPa = 100000 Pa
Total pressure = 100000 Pa + 19620 Pa = 119620 Pa

Step 3: Calculate the number of moles of air in the room.
Use the ideal gas law: PV = nRT
Where:
P = pressure (in Pa)
V = volume (in m^3)
n = number of moles
R = universal gas constant (8.314 J/(mol·K))
T = temperature (in Kelvin)

Convert the volume of the room and the balloons to liters:
50 m^3 * 1000 L/m^3 = 50000 L
0.5 m^3 * 1000 L/m^3 = 500 L

Convert the temperature from Celsius to Kelvin: 20°C + 273.15 = 293.15 K

For air in the room:
P = 119620 Pa
V = 50000 L
R = 8.314 J/(mol·K)
T = 293.15 K

Solve for n:
n = (P * V) / (R * T)

Step 4: Calculate the number of moles of nitrogen and oxygen in the air.
Based on the given volume percentages, we can determine the number of moles of nitrogen and oxygen in the air using their molar masses and the total number of moles of air obtained in step 3.

Molar mass of nitrogen (N2) = 28 g/mol
Molar mass of oxygen (O2) = 32 g/mol

Moles of nitrogen = (79/100) * Total moles of air
Moles of oxygen = (21/100) * Total moles of air

Step 5: Calculate the number of moles of helium and oxygen released from the balloons.
Since the balloons have a gauge pressure of 2m of water, they are at atmospheric pressure. Therefore, the number of moles of helium and oxygen released is equal to the number of moles of air.

Step 6: Calculate the total moles of each species in the room.
Total moles of nitrogen = Moles of nitrogen in the air
Total moles of oxygen = Moles of oxygen in the air + Moles of oxygen released from the balloon
Total moles of helium = Moles of helium released from the balloon

Step 7: Calculate the mass and molar fractions for each species.
The mass fraction of a species is calculated by dividing its mass by the total mass of all species present in the room. The molar fraction is calculated by dividing the number of moles of a species by the total number of moles of all species present in the room.

Mass fraction of nitrogen = (Molar mass of nitrogen * Total moles of nitrogen) / (Molar mass of nitrogen * Total moles of nitrogen + Molar mass of oxygen * Total moles of oxygen + Molar mass of helium * Total moles of helium)
Molar fraction of nitrogen = Total moles of nitrogen / (Total moles of nitrogen + Total moles of oxygen + Total moles of helium)

Similarly, calculate the mass and molar fractions for oxygen and helium.

Note: The atomic molar mass of helium given (4 g/mol) is incorrect. The correct atomic molar mass of helium is 4.0026 g/mol.

Follow these steps with the corrected atomic molar mass of helium to obtain the mass and molar fractions of the gaseous species in the room.