Video Transcript
A balloon is inflated by adding
more air to it, increasing its volume, as shown in the diagram. The pressure of the air in the
balloon is 101 kilopascals before it is inflated and is 121 kilopascals after it is
inflated. The temperature of the air in the
balloon does not change when it is inflated. The air can be assumed to behave as
an ideal gas. Find the mass of the balloon when
it is inflated as a percent of the mass of the balloon before it is inflated. Give your answer to the nearest
percent.
In this question, we have a balloon
that has been inflated so that its volume increases from 0.012 cubic meters to 0.033
cubic meters. We need to find the balloon’s final
mass as a percentage of its initial mass. This is quite an unusual
question. Usually when we think about ideal
gasses, we have a fixed amount of gas, and we’re thinking about changes to its
pressure, volume, or temperature. However, in this question, the
amount of gas we’re thinking about changes. As the balloon is inflated, the
amount of gas it contains increases, and so does its mass.
Clearing some space on screen,
let’s start by looking at the ideal gas law for a fixed amount of gas. This is often written in the form
𝑃 times 𝑉 equals 𝑘 times 𝑇, where 𝑃 is the pressure of the gas, 𝑉 is the
volume, and 𝑇 is the temperature. Here, 𝑘 is a constant of
proportionality. If we divide both sides of the
equation by 𝑇, we can make 𝑘 the subject, 𝑘 equals 𝑃 times 𝑉 divided by 𝑇. When we’re looking at a fixed
amount of gas, 𝑘 has a constant value. But it turns out that 𝑘 is
actually proportional to the total number of particles in the gas, 𝑁. So, if we’re not looking at a fixed
amount of gas, 𝑘 isn’t a constant number. Using the fact that 𝑘 is
proportional to the number of gas particles, 𝑁, we can rewrite the ideal gas
equation in a more helpful way. 𝑃 times 𝑉 divided by 𝑇 is
proportional to 𝑁.
So, how do we link this to the mass
of the gas? Well, the total mass of a volume of
gas, which we’ll call 𝑚 sub gas, is equal to the number of particles in the gas,
𝑁, multiplied by the mass of each individual gas particle. If we divide both sides of this
equation by the mass of an individual gas particle, we find that the number of
particles, 𝑁, is equal to the total mass of the gas divided by the mass of each
particle. Now let’s substitute this back into
our expression for the ideal gas law. This gives us 𝑃 times 𝑉 divided
by 𝑇 is proportional to the mass of the gas divided by the mass of the single gas
particle. Because the mass of the particle
has a constant value, we can absorb it into the proportionality relation. This gives us 𝑃 times 𝑉 divided
by 𝑇 is proportional to the mass of the gas.
With this mass relation known, we
can now begin to answer this question. We need to find the mass of the
balloon after it’s been inflated as a percentage of its initial mass. Clearing space to work, we’ll call
the balloon’s initial mass 𝑚 one and its final mass 𝑚 two. As a percentage of the initial
mass, the final mass is equal to 𝑚 two divided by 𝑚 one multiplied by 100. Using our expression for the mass
of a gas, we can rewrite this in terms of the pressure, volume, and temperature of
the air in the balloon. As a percentage of 𝑚 one, 𝑚 two
is equal to 𝑃 two times 𝑉 two divided by 𝑇 two all divided by 𝑃 one times 𝑉 one
divided by 𝑇 one all multiplied by 100.
We’re told that before the balloon
is inflated, the pressure of the air in the balloon is 101 kilopascals and its
volume is 0.012 cubic meters. We aren’t told the initial
temperature of the air in the balloon, so we’ll just call this 𝑇. After the balloon is inflated, the
pressure of the air in the balloon is 121 kilopascals and its volume is 0.033 cubic
meters. We’re told that the temperature of
the balloon doesn’t change, so 𝑇 two is also just 𝑇.
Now we can substitute these values
into our equation. We see that these two 𝑇 terms
cancel. So, as a percentage of 𝑚 one, 𝑚
two is equal to 121 kilopascals times 0.033 cubic meters divided by 101 kilopascals
times 0.012 cubic meters. Working through the calculation, we
get a value of 329.455 percent. When we round this to the nearest
percent, we get 329 percent.
So, we have found that the mass of
the inflated balloon is equal to 329 percent of its original mass. The answer to this question is then
329 percent.
