To solve this problem, we can use the Ideal Gas Law equation, which states PV = nRT, where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature (assuming temperature is constant).
Given:
Initial pressure (P1) = 2×10^2 atm
Initial volume (V1) = 13.5 L
Final pressure (P2) = 1 atm
Final volume (V2) = 2 L
We need to find the number of moles of hydrogen gas in the tank (n1) and the number of moles in each balloon (n2), and then calculate how many balloons can be inflated.
Step 1: Calculate the initial number of moles in the tank.
Using the Ideal Gas Law equation (PV = nRT), we rearrange it to find the number of moles (n).
n1 = (P1 * V1) / (R * T)
Since the temperature is constant in this problem, we can assume R and T will cancel out. The value of R is 0.0821 (atm L / mol K).
n1 = (2×10^2 atm * 13.5 L) / (0.0821 atm L / mol K)
n1 ≈ 275.30 mol
Therefore, the initial number of moles of hydrogen gas in the tank is approximately 275.30 mol.
Step 2: Calculate the final number of moles in each balloon.
Since the pressure in each balloon is 1 atm and the volume is 2 L, we can use the Ideal Gas Law equation again to find the number of moles (n2) in each balloon.
n2 = (P2 * V2) / (R * T)
n2 = (1 atm * 2 L) / (0.0821 atm L / mol K)
n2 ≈ 0.024 mol
Therefore, the final number of moles of hydrogen gas in each balloon is approximately 0.024 mol.
Step 3: Calculate the number of balloons that can be inflated.
To find the number of balloons that can be inflated, divide the initial number of moles in the tank (n1) by the final number of moles in each balloon (n2).
Number of balloons = n1 / n2
Number of balloons ≈ 275.30 mol / 0.024 mol
Number of balloons ≈ 11,470
Therefore, approximately 11,470 balloons can be inflated with the given conditions.
Note: It's important to consider that the Ideal Gas Law assumes ideal gas behavior, and in practice, real gases may deviate slightly from ideal behavior.