An apparatus consists of a 4.0 dm3 flask containing nitrogen gas at 25 oC and 803 kPa. It is joined by a valve to a 10.0 dm3 flask containing argon gas 25 oC and 47.2 kPa. The valve is opened and the gases mix.
i. What is the partial pressure of each gas after mixing?
ii. What is the total pressure of the gas mixture?
i. Use PV = nRT and solve for n = # mols in the 4 dm3 flask and # mols in the 10 dm3 flask. Add mols in flask 1 to mols in flask 2 for total mols. Add volume flask 1 to volume flask 2 for total volume. Then use PV = nRT plugging in total Volume for V, the proper conditions for R and T, Use n for N2 and solve for pN2. Do the same for pAr.
ii. Ptotal = pN2 + pAr
To solve this problem, we can use the ideal gas law equation, which states that PV = nRT, where:
- P is the pressure of the gas
- V is the volume of the gas
- n is the number of moles of the gas
- R is the ideal gas constant (0.0821 L·atm/(mol·K))
- T is the temperature of the gas in Kelvin
Since we are given volumes and pressures, we can rearrange the ideal gas law equation to solve for the number of moles of each gas. Then, we can use the mole fractions to determine the partial pressure and total pressure after mixing.
i. To find the partial pressure of each gas after mixing, we need to determine the number of moles of each gas.
For nitrogen gas:
Given volume = 4.0 dm^3
Given temperature = 25°C = 298 K
Given pressure = 803 kPa
Using the ideal gas law, we can rearrange it to solve for the number of moles (n):
n = PV / RT
For nitrogen gas:
n = (803 kPa * 4.0 dm^3) / (0.0821 L·atm/(mol·K) * 298 K) = 107.5 moles (approx.)
For argon gas:
Given volume = 10.0 dm^3
Given temperature = 25°C = 298 K
Given pressure = 47.2 kPa
For argon gas:
n = (47.2 kPa * 10.0 dm^3) / (0.0821 L·atm/(mol·K) * 298 K) = 23.0 moles (approx.)
Now, we can calculate the mole fraction for each gas, which is the number of moles of the gas divided by the total number of moles.
For nitrogen gas:
Mole fraction of nitrogen = (107.5 moles) / (107.5 moles + 23.0 moles) ≈ 0.823
For argon gas:
Mole fraction of argon = (23.0 moles) / (107.5 moles + 23.0 moles) ≈ 0.177
To find the partial pressure of each gas after mixing, we need to multiply the mole fraction with the total pressure of the gas mixture.
For nitrogen gas:
Partial pressure of nitrogen = Mole fraction of nitrogen * Total pressure
Partial pressure of nitrogen = 0.823 * Total pressure
For argon gas:
Partial pressure of argon = Mole fraction of argon * Total pressure
Partial pressure of argon = 0.177 * Total pressure
ii. To find the total pressure of the gas mixture, we can use Dalton's Law of Partial Pressures, which states that the total pressure is the sum of the partial pressures.
Total pressure = Partial pressure of nitrogen + Partial pressure of argon
Substituting the values we calculated:
Total pressure = (0.823 * Total pressure) + (0.177 * Total pressure)
Simplifying the equation:
Total pressure = 0.823 Total pressure + 0.177 Total pressure
Total pressure - 0.823 Total pressure - 0.177 Total pressure = 0
0.177 Total pressure - 0.823 Total pressure = 0
-0.646 Total pressure = 0
Total pressure = 0 / (-0.646) (Since there is no negative pressure)
Total pressure = 0
Therefore, the total pressure of the gas mixture after mixing is 0 kPa (approx.).
To find the partial pressure of each gas after mixing, we can use Dalton's Law of Partial Pressures. According to Dalton's Law, the total pressure of a mixture of gases is equal to the sum of the partial pressures of each individual gas.
i. Partial Pressure of Nitrogen (Pn):
The initial pressure of nitrogen gas (Pn_initial) is given as 803 kPa. Since the total pressure of the mixture remains constant, the final pressure of nitrogen gas (Pn_final) will be the same. Hence, the partial pressure of nitrogen after mixing is 803 kPa.
ii. Partial Pressure of Argon (Pa):
The initial pressure of argon gas (Pa_initial) is given as 47.2 kPa. Similar to nitrogen gas, the final pressure of argon gas (Pa_final) will be the same as its initial pressure. Therefore, the partial pressure of argon after mixing is 47.2 kPa.
iii. Total Pressure of the Gas Mixture (Pt):
To find the total pressure of the gas mixture, we sum up the partial pressures of nitrogen and argon.
Pt = Pn + Pa
= 803 kPa + 47.2 kPa
= 850.2 kPa
Therefore:
i. The partial pressure of nitrogen after mixing is 803 kPa.
ii. The partial pressure of argon after mixing is 47.2 kPa.
iii. The total pressure of the gas mixture is 850.2 kPa.