A clown at a birthday party has brought along a helium cylinder, with which he intends to fill balloons. When full, each balloon contains 0.00280 m3 of helium at an absolute pressure of 1.20 x 105 Pa. The cylinder contains helium at an absolute pressure of 2.90 x 107 Pa and has a volume of 0.00280 m3. The temperature of the helium in the tank and in the balloons is the same and remains constant. What is the maximum number of people who will get a balloon?
I keep getting 242, which is incorrect. My lecturer said something about not being able to use all the gas in the tank due to pressure, which is where I am getting confused.
To solve this question, we need to consider the ideal gas law, which states:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles of gas
R = gas constant
T = temperature in Kelvin
In this problem, we are given the volume and pressure for both the cylinder and the balloon, and we know that the temperature remains constant.
Let's start by finding the number of moles of helium in the cylinder:
PV = nRT
We have:
P = 2.90 x 10^7 Pa (pressure in the cylinder)
V = 0.00280 m^3 (volume of the cylinder)
R = 8.314 J/(mol K) (gas constant)
T = constant temperature
Rearranging the equation, we can solve for n:
n = PV / RT
n = (2.90 x 10^7 Pa) * (0.00280 m^3) / (8.314 J/(mol K) * T)
Now, let's find the maximum number of moles that can be used to fill the balloons. We know the volume and pressure for each balloon, so we can use the ideal gas law again:
PV = nRT
P = 1.20 x 10^5 Pa (pressure in the balloon)
V = 0.00280 m^3 (volume of one balloon)
R = 8.314 J/(mol K) (gas constant)
T = constant temperature
Rearranging the equation, we can solve for n:
n = PV / RT
Now, to find the maximum number of people who will get a balloon, we divide the number of moles of helium in the cylinder by the number of moles needed for each balloon:
Maximum number of people = n_cylinder / n_balloon
Substituting the values we calculated earlier:
Maximum number of people = [(2.90 x 10^7 Pa) * (0.00280 m^3) / (8.314 J/(mol K) * T)] / [(1.20 x 10^5 Pa) * (0.00280 m^3) / (8.314 J/(mol K) * T)]
Simplifying, we can cancel out the T and V terms:
Maximum number of people = (2.90 x 10^7 Pa) / (1.20 x 10^5 Pa)
Maximum number of people = 241.66
Now, we need to consider the limitation mentioned by your lecturer. As the cylinder's pressure is much higher (2.90 x 10^7 Pa) than that of the balloons (1.20 x 10^5 Pa), we cannot completely use all the gas in the cylinder. This is due to the pressure difference, which creates a back pressure preventing the balloon from being completely filled.
Therefore, to account for this limitation, we need to round down the maximum number of people who can get a balloon. In this case, the maximum number of people who will get a balloon is 241.
So, the correct answer is 241 people.