A rigid tank contains 38:0 kg of O2 gas at a pressure of 9:70 atm. If the oxygen is replaced by helium, how many kilogram of helium will be needed to produce a pressure of 8:00 atm?
To find the number of kilograms of helium needed to produce a pressure of 8.00 atm, we can use the ideal gas law equation:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles
R = ideal gas constant
T = temperature
Since the volume and temperature are constant, we can simplify the equation to:
P1/n1 = P2/n2
Let's assume the number of moles of oxygen in the tank is n1 and the number of moles of helium needed is n2.
Given:
P1 = 9.70 atm (initial pressure of oxygen)
P2 = 8.00 atm (desired pressure)
n1 = 38.0 kg of oxygen / molar mass of oxygen
n2 = ? (number of moles of helium)
To calculate the number of moles of oxygen (n1), we need to determine the molar mass of oxygen. The molar mass of oxygen (O2) is approximately 32.00 g/mol.
n1 = 38.0 kg / 32.00 g/mol = 1187.5 mol
Now, we can rearrange the equation to solve for n2:
P1/n1 = P2/n2
n2 = (P2 * n1) / P1
Substituting the given values:
n2 = (8.00 atm * 1187.5 mol) / 9.70 atm
n2 ≈ 984.54 mol
Finally, we can calculate the mass of helium (m2) using the molar mass of helium, which is approximately 4.00 g/mol:
m2 = n2 * molar mass of helium
m2 = 984.54 mol * 4.00 g/mol
m2 ≈ 3938.16 g ≈ 3.94 kg
Therefore, approximately 3.94 kilograms of helium will be needed to produce a pressure of 8.00 atm.
To answer this question, we first need to understand the relationship between pressure, volume, and the amount of gas (in moles). This relationship is described by the Ideal Gas Law equation:
PV = nRT
Where:
P = Pressure
V = Volume
n = Number of moles of gas
R = Ideal gas constant
T = Temperature
In this case, the volume and temperature are constant because the tank is rigid. Therefore, we can rewrite the equation as:
P1 * V1 = n1 * R * T1
P2 * V2 = n2 * R * T1
Where:
P1 = Initial pressure
V1 = Initial volume
n1 = Initial number of moles of gas (oxygen)
P2 = Final pressure
V2 = Final volume (same as V1 because the tank is rigid)
n2 = Final number of moles of gas
T1 = Initial temperature (assumed constant)
Now, to find the number of moles of helium needed to achieve a final pressure of 8.00 atm, we can rearrange the equation as follows:
n2 = (P2 * V1) / (P1 * R * T1)
Next, we need to convert the number of moles of helium to kilograms. To do this, we will use the molar mass of helium, which is 4.0026 g/mol.
First, we calculate the molar mass of oxygen (32 g/mol), which allows us to find the number of moles of oxygen:
n1 = (m1 / M1)
Where:
m1 = Mass of oxygen (38.0 kg)
M1 = Molar mass of oxygen (32 g/mol)
Then, we can calculate the mass of helium:
m2 = n2 * M2
Where:
M2 = Molar mass of helium (4.0026 g/mol)
Putting it all together:
1. Calculate the number of moles of oxygen:
n1 = (m1 / M1)
2. Calculate the number of moles of helium:
n2 = (P2 * V1) / (P1 * R * T1)
3. Convert moles of helium to kilograms:
m2 = n2 * M2
Therefore, to find the mass (in kilograms) of helium needed to achieve a pressure of 8.00 atm, follow the steps above with the given values for P1, V1, P2, m1, R, T1, M1, and M2.