A helium balloon rises because of a buoyant force. By what per...

Question

A helium balloon rises because of a buoyant force. By what percentage does the buoyant force on a helium balloon change if the temperature of the helium is increased from 15°C to 25°C while the temperature of the surrounding air is unchanged? Assume that the pressure of the helium remains constant.

Accepted Answer

Hey, everyone in this problem, a hot air balloon is floating in the air with some riders. The initial temperature of the air inside the balloon is at 20 °C afterward, the temperature of the air inside the balloon is increased to 30 °C. We're asked to determine by what percentage the buoyant force on the balloon changes. If the temperature of the surrounding air and the pressure of the air inside the balloon remains unchanged. We're given four answer choices, all in percentages. Option A 3.4 option B 4.2 option C 5.1 and option D 6.5. So what we're asked to find here is this percent change in the buoyant force is a call for a percent change. And in this case, in the buoyant force, which we'll call FB, that's gonna be equal to the final point force. So we'll call that FBF minus the initial buoyant force. FV, not divided by that initial buoyant force. FB. Not, we're gonna take all of this and we're gonna multiply by 100 to change it from a decimal into a person. Ok. So that's where we're gonna calculate this percent change in that buoyant force. Now, we'll see in this equation that we have FB not FBF. So what is the buoyant force? How can we calculate it? We'll recall that the buoyant force FB can be written as the density. OK. And that is the density of the surrounding fluid. So in this case, the density of the surrounding fluid is gonna be the density of air. We have this hot air balloon with air surrounding it. So we have the density of air multiplied by the volume multiplied by the acceleration due to gravity. So we can write our buoyant force initially FB not as row air, the density of air multiplied by the initial volume being not multiplied by the acceleration due to gravity G. And similarly, we can write the final buoyant force. FBF as ro air VF multiplied by G. OK. So now we have an expression for these two buoyant forces initially and after the temperature has increased. OK. So let's go ahead, substitute those into our percent change formula. So our percent change in that point force is now going to be equal to ro air VFG minus ro air V, not G all divided by row air VRG multiplied by 100. Now what we notice is that we can divide the entire equation by the density of air row air, we can divide the entire equation by the acceleration due to gravity G. And what we're left with is that the percent change in the buoyant force can be written as VF minus V not divided by V not multiplied by 100. So in this case, a percent change in the buoyant force is equivalent to the percent change in the volume. All right. So that's kind of neat. Now, let's keep going. Ok, what about these volumes? We aren't given the volumes in the problem? So how can we figure out these volumes or figure out an expression for these volumes in order to simplify the problem more? And what we're gonna do is think about the ideal gas law and we're dealing with air, we can assume that that's an ideal gas. If we take a look at our ideal gas law recall that we can write PV is equal to N RT where P is the pressure V is the volume and is the number of moles R is the ideal gas constant and T is the temperature. In this case, we want an expression for the volume. So we can write that the volume V is equal to N RT divided by P. Now, similarly to what we did for the buoyant force above, we're gonna write out this expression for the volume VNO and VF. So we can write that V knot is equal to N R tea not divided by peanut. And we can also write that VF is equal to N RT F divided by PF. OK. And the number of moles are just that constant though, stay the same, even when the temperature changes. OK, those are constant, they're not going to change throughout this process. So we use the same N and the same R for both equations. So now we have these expressions for the volume and we're gonna go back to our percent change equation and we're gonna go ahead and we're gonna substitute that was it. So again, we're looking at the percent change in the buoyant force FB we found that it was equivalent to the percent change in the volume. In this case. Now, we have this expression for volume. So we can write our equation as N RT F divided by PF minus N RT, not divided by P NT, all divided by N RT, not divided by peanut. We take all of this and we multiply it by 100. Now, this expression looks really messy. We might be at the point where we're thinking, why are we doing this? We're making it more complicated, but we're gonna see that things simplify it very nicely here. OK. So just like before we have some common terms that we can divide by. So the N, the number of moles is gonna divide it. Same with R the ideal gas concept right now. The other thing we have to remember is that the pressure PN is equal to the pressure PF we have the same pressure, we have this constant pressure. We're told that in the problem. So if that's the case, then these PN and PF terms will all cancel each other out as well. What that leaves us with is that the percent change in the point force is going to be equal to TF minus tina divided by tino all multiplied by 100. OK. So what we've shown is that the percent change in the buoyant force in this case is equivalent to the percent change in the volume, which is also equivalent to the percent change in the temperature. Now, the temperature is something we're told about in this problem, we know the initial temperature, we know the final temperature. So we've been able to get this problem in terms of some values that we know we can substitute those in now and actually solve. Now, what we're gonna wanna do is have these temperatures in Kelvin. OK. So if we go back up to the problem, we were given our temperatures in degrees Celsius. So we have 20 °C and 30 °C. In order to convert to Calvin, we're going to add 273 to each of them. That means our initial temperature will be 20 plus 273 which gives us 293 Kelvin, our final temperature, 30 °C plus 273 gives us 303 Kelvin. So those are the values we're gonna use in our equation. We can substitute them in and we have that our percent change in that buoyant force is going to be 303 Calvin minus 293. Kelvin divided by 293 Calvin multiplied by 100. If we work this out on our calculator, we are gonna find that, that percent change is about 3.413 per se. Right? And that is that final answer we were looking for that is the percent change in the buoyant force when we have this rise in temperature in our hot air balloon. If we go to our answer choices, we're going round to two significant digits comparing what we found. And we can see that the correct answer here is going to be option a 3.4%. Thanks everyone for watching. I hope this video helped see you in the next one.

Key Concepts

Buoyant Force

Ideal Gas Law

Density and Temperature Relationship

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