A balloon 16 m in diameter is inflated with helium at 18^∘ C. (a) Calculate the mass of He in th

A balloon 16 m in diameter is inflated with helium at 18^∘...

Key Concepts

Volume Calculation for a Sphere

Calculating the volume of a sphere is essential when determining the amount of gas contained within a spherical balloon. This involves using the formula V = (4/3)?r³, where r is the radius of the sphere, which provides the spatial measurement needed for further gas law calculations.

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, relates the pressure (P), volume (V), number of moles (n), the ideal gas constant (R), and temperature (T) of a gas. This fundamental equation is used to determine the number of moles of helium present in the balloon, assuming the helium behaves ideally.

Molar Mass and Conversion of Moles to Mass

Once the number of moles of a gas is known using the ideal gas law, the molar mass (the mass per mole of a substance) is used to convert this quantity into mass. This concept is key in finding the mass of helium by multiplying the mole value by helium's molar mass.

Work Done in a Constant Pressure Process

Work done by or on a gas during expansion or compression at a constant pressure is given by W = P?V, where ?V is the change in volume. This concept is crucial when calculating the mechanical work done during the inflation process of the balloon under atmospheric pressure.

Transcript

00:01 So for this problem, we have two parts, and part i asked us to find the mass of helium and a balloon, assuming that the gas abides by the ideal behaviors of gases.

00:11 So the first step that we have to do is have to find the volume of the balloon, because that tells us the volume of the space that the helium takes up as well.

00:20 So even though balloons aren't perfectly spherical, that's the best way to model the volume of the balloon, and we know that that volume is four -thirds, are cubed.

00:33 And when you plug the radius in, which is 8 meters, as we know, you'll get that approximately the volume of the balloon is 2 ,000 and 100 meters.

00:47 However, this unit of cubic meters isn't very useful for us.

00:52 And the reason why is because that's not what the volume expression of the units that are used in the ideal gas flow.

01:00 So instead we need to convert this to liters.

01:03 Well, there is a conversion between cubic meters, and it's that there's a thousand liters in one cubic meter.

01:14 So we know that the volume, and more useful units, is 2 .1 times 10 to the 6 liters.

01:22 So that's kind of the first step of part a.

01:25 Now that we know the volume of the helium, we can then use the ideal gas law, because, as we know, the helium in the balloon is ideal.

01:34 As well as it pv equals n or t.

01:38 And so we still need to find a couple of these numbers.

01:41 So as we are told from the question, that the pressure is 98 .7 kilopascals.

01:53 We also know that the temperature is 18 degrees celsius.

02:00 But neither of these are the ideal units that we want.

02:02 We want the pressure to be in atm, and we want the temperature to be in kelvin.

02:07 But we can correct both of those things.

02:09 First, on pressure, we can simply multiply by a conversion factor that we have, more an atm.

02:27 And when you use this conversion factor between kilopascals and atm, we get the pressure is 0 .974 atm.

02:39 For temperature, all we need to do is to add 273, and that will give us the temperature of 291 kelvin.

02:50 So that gives us the numbers that we need to solve for the ideal gas equation.

02:58 So pv equals nrt, what we're trying to find here is n, the moles of helium.

03:05 Because we can then use the moles of helium and then multiply by the molar mass of helium, and that will give us the total mass of the helium in the balloon.

03:13 So we want to isolate for n.

03:16 So we know that we can write n as pv divided by our.....

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