Two cannisters are connected by a closed tap. In the 8.4 L cannister, the pressure of Helium is 18.6 kPa and the pressure of Argon is {p2} kPa. In the 2.6 L cannister, which is at a pressure of 109.4 kPa, the same two gases are present and the partial pressure of the Argon is 35.2 kPa. Determine the partial pressure of the helium when the tap is opened.
Question
Two cannisters are connected by a closed tap. In the 8.4 L cannister, the pressure of Helium is 18.6 kPa and the pressure of Argon is {p2} kPa. In the 2.6 L cannister, which is at a pressure of 109.4 kPa, the same two gases are present and the partial pressure of the Argon is 35.2 kPa. Determine the partial pressure of the helium when the tap is opened.
Solution 1
To solve this problem, we need to use the law of partial pressures, also known as Dalton's Law. This law states that the total pressure exerted by a mixture of gases is equal to the sum of the pressures that each gas would exert if it were alone.
First, we need to find the total pressure in each canister before the tap is opened.
In the 8.4 L canister, the total pressure is the sum of the pressure of Helium and Argon. We know the pressure of Helium is 18.6 kPa, but we don't know the pressure of Argon (p2). So, the total pressure in the 8.4 L canister is 18.6 + p2.
In the 2.6 L canister, the total pressure is given as 109.4 kPa. This is the sum of the pressures of Helium and Argon. We know the pressure of Argon is 35.2 kPa, so the pressure of Helium in the 2.6 L canister is 109.4 - 35.2 = 74.2 kPa.
When the tap is opened, the gases mix and the total pressure and volume become the sum of the initial pressures and volumes. The total volume is 8.4 L + 2.6 L = 11 L.
The total pressure is (18.6 + p2) + 109.4 = 128 + p2.
The partial pressure of a gas is given by its mole fraction times the total pressure. The mole fraction is given by the pressure of the gas divided by the total pressure.
So, the partial pressure of Helium when the tap is opened is (18.6/128 + 74.2/128) * (128 + p2) = 0.724 * (128 + p2).
To find the value of p2, we need more information about the system, such as the temperature or the number of moles of each gas.
Solution 2
To solve this problem, we need to use the law of partial pressures, also known as Dalton's Law. This law states that the total pressure exerted by a mixture of gases is equal to the sum of the pressures that each gas would exert if it were alone.
First, we need to find the total pressure in each canister before the tap is opened.
In the 8.4 L canister, the total pressure is the sum of the pressure of Helium and Argon. We know the pressure of Helium is 18.6 kPa, but we don't know the pressure of Argon (p2). So, the total pressure in the 8.4 L canister is 18.6 + p2.
In the 2.6 L canister, the total pressure is given as 109.4 kPa. This is the sum of the pressures of Helium and Argon. We know the pressure of Argon is 35.2 kPa, so the pressure of Helium in the 2.6 L canister is 109.4 - 35.2 = 74.2 kPa.
When the tap is opened, the gases mix and the total pressure and volume become the sum of the initial pressures and volumes. The pressure of each gas is proportional to its volume (Boyle's Law), so the final pressure of each gas is its initial pressure times the ratio of the final volume to the initial volume.
The final volume is 8.4 L + 2.6 L = 11 L.
The final pressure of Helium is its initial pressure in the 8.4 L canister (18.6 kPa) times the ratio of the final volume to the initial volume (11 L / 8.4 L) plus its initial pressure in the 2.6 L canister (74.2 kPa) times the ratio of the final volume to the initial volume (11 L / 2.6 L).
So, the final pressure of Helium is 18.6 * (11 / 8.4) + 74.2 * (11 / 2.6) = 24.3 kPa + 312.5 kPa = 336.8 kPa.
Therefore, the partial pressure of the helium when the tap is opened is 336.8 kPa.
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