Video: Combined Variation and Its Applications

Video Transcript

When one inflates a balloon, the pressure of gas inside the balloon varies directly with the quantity of gas injected inside. The unit for this is moles and the pressure varies inversely with the volume of the balloon, which increases when the balloon expands. Assume that the temperature of the gas inside the balloon is constant. Write an equation for the pressure of gas inside the balloon 𝑃 in terms of the quantity of gas inside the balloon 𝑛 and the volume of the balloon 𝑉. Let 𝑘 be constant. Given that the pressure inside the balloon is 1.1 bar with 0.089 moles of air inside the balloon of volume two decimeters cubed, find the pressure inside the balloon when there are 0.24 moles of air and its volume is four decimeters cubed. Round your answer to one decimal place.

𝑃 will stand for pressure; 𝑛 will stand for the quantity of gas; and 𝑉 with stand for the volume of the balloon. We’re told that the pressure of gas inside the balloon varies directly with the quantity of gas injected inside, and the pressure varies inversely with the volume of the balloon. When variables such as 𝑦 and 𝑥 vary directly with each other, we have an equation 𝑦 equals 𝑘𝑥 where 𝑘 is a constant, whereas it they would very inversely we would have the equation 𝑦 equals 𝑘 divided by 𝑥.

So notice we’re talking about the pressure, so we can replace 𝑦 with 𝑃. And we know that 𝑃 varies directly with 𝑛, so we would have 𝑃 equals 𝑘 times 𝑛. And we know that 𝑃 varies inversely with 𝑉. So we would have 𝑃 equals 𝑘 divided by 𝑉. So if we’re trying to create one equation, we need to put these two equations together. Creating 𝑃 equals 𝑘 times 𝑛 divided by 𝑉.

So we’re told that the pressure is 1.1 bar, and the quantity 𝑛 is 0.089 moles, and the volume is two cubic decimeters. So we plug in 1.1 for 𝑃, 0.089 for 𝑛, and two for 𝑉. So to solve for 𝑘, we need to multiply both sides of the equation by two. So we have that 2.2 equals 𝑘 times 0.089. So to solve for 𝑘, we need to divide both sides of the equation by 0.089. And we find that 𝑘 is equal to 24.7191011.

So now we can use this to write our equation. So instead of having 𝑃 equals 𝑘 times 𝑛 divided by 𝑉, we now have 𝑃 equals 24.7191011 𝑛 divided by 𝑉. So we’re asked to find the pressure inside the balloon when there are 0.24 moles of air and its volume is four decimeters cubed. After plugging in our numbers, we need to multiply the two numbers on the top, the numerator, together.

And it’s good not to round just yet. This prevents rounding errors. So we have 5.93258427 divided by four, resulting in 1.48314607. But we’re told to round to one decimal place, which is the four. So we will either keep the four a four or round it up to a five. So we look at the number to the right, the eight. And since eight is five or larger, we will round the four up to a five, resulting in 1.5. So the pressure of the balloon would be 1.5 bar.