A 10-cm-diameter, 40-cm-tall gas cylinder, sealed at the top by a frictionless 50 kg piston, is surrounded by a bath of 20°C water. Then 50 kg of sand is slowly poured onto the top of the piston, where it stays. Afterward, what is the height of the piston?

Video transcript

Hello, fellow physicists today, we're going to solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use. In order to solve this problem, a container designed to hold argon gas for welding applications has a base of 1.0, multiplied by 10 to the power of negative two m squared. The container is fastened with a movable and impermeable piston which is initially at a height of 0.50 m. The piston has a mass of 12 kg and is free to move up and down without any friction. The argon gas in the container is maintained at a constant temperature of 25 degrees Celsius using a thermostat, 10 kg of iron powder is carefully deposited onto the top of the piston where it remains, calculate the final height of the piston. So our end goal is to calculate the final height of the piston. So we're given some multiple choice answers. They're all in the same units as centimeters. Let's read them up to see what our final answer might be. A is 13 B is 28 C is 35 D is 46. So first off, let us assume that argon is an ideal gas. Therefore, we could recall and use the ideal gas law equation which states that pressure multiplied by the volume is equal to the number of moles multiplied by the universal gas constant multiplied by the temperature. Also, since the piston is impermeable, it means that it's airtight and leakproof. So the number of moles will remain constant. Let's make a note of that. So number of moles equals a constant value. Ok. So applying the ideal gas law to consider depositing the iron powder for the initial and final conditions. For this container, we can write the ideal gas light equation as follows. So we can use it to write the following that the initial pressure is equal to the initial volume, which is equal to the final pressure multiplied by the final volume, which is equal to the number of moles multiplied by the universal gas constant, multiplied by the temperature equals some constant value. There's a constant I should say. So we can then write, take it even further to write that the final pressure multiplied by, I'm sorry, the initial pressure multiplied by the cross sectional area, the container multiplied by the initial height of the piston is equal to the final pressure multiplied by the cross sectional area, the container multiplied by the final height of the piston. And we'll call this equation one. So Now, we need to recall the equation to solve for the initial pressure. Let's call it the equation two. So let's recall that. So how we find initial pressure? So initial pressure is equal to the atmospheric pressure. I'm gonna denote it as P subscript zero plus the mass of the piston, which I'm gonna denote it as mass M subscript P multiplied by gravity divided by the cross sectional area of the container. OK. So let's make a note that for a, the cross sectional area, in this case, we're gonna assume that it's the areas for like the area of a square. So let's just remember that it's just the area of a square is just the side squared which in this case, the side that's given to us is 1.0 multiplied by 10 of the power of negative two m squared. OK. Which will make sense in a second. OK. So now we need to write the equation for the final pressure. So P F is equal to the atmospheric pressure multiplied by the mass of the piston plus the mass of the iron, which I'm going to known as M subscript. I mean this is when the iron is deposited onto the top of the piston multiplied by gravity divided by the cross sectional area. OK. OK. So by combining equations, 12 and three, we can write the following, let's call it equation four. So let's combine equations 12 and three So the atmospheric pressure multiply. So I mean I'm adding so the atmospheric pressure plus the mass of the piston multiplied by gravity divided by the cross sectional area multiplied by the cross sectional area multiplied by the initial height of the piston equals the atmospheric pressure plus the mass of the piston plus the mass of iron multiplied by gravity divided by cross sectional area multiplied by the cross sectional area multiplied by the final height of the piston. So we need to rearrange equation four to solve for the final height of the piston. So when we do that, when we rearrange using algebra, the final height of the piston is equal to the atmospheric pressure multiplied by the cross sectional area plus the mass of the piston multiplied by gravity divided by the atmospheric pressure multiplied by the cross sectional area plus the mass of the piston plus the mass of iron multiplied by gravity all multiplied by the initial height of the piston. So at this stage, we need to plug in all of our known variables to solve for the final height of the piston. So let's do that. OK. So the atmospheric pressure, the numerical value for that is 1. multiplied by 10 to the fifth power pascals multiplied by the area which we determined and to be the area of the square. So it's 1.0 multiplied by 10 to the power of negative two meters squared. Squared plus the mass of the piston which is 12 kg multiplied by gravity which is 9.81 m per second squared, all divided by the atmospheric pressure again, which is 1.13 multiplied by 10 to the fifth. Power past scales multiplied by the cross sectional area which is 1.0, multiplied by 10 to the power of negative two m squared squared plus the mass of the piston plus the mass of iron, which was 10 kg multiplied by gravity 9.81 m per second squared. All multiplied by the initial height of the piston which was 0.5 m. So we're running out of room here. So I'll bring it down here. So it's 0.50 m. OK? So when you plug this all into a calculator and then don't forget to multiply it by the initial height of the piston, your final height of the piston should be when you plug that all into a calculator, 0.46 m. And that is our final answer. So that means our final answer has to be D centimeters. Oh yeah, let's make a quick note here. So we got 0.46 m, but we can quickly use dimensional analysis to convert meters to centimeters. So when there is 100 centimeters in one m, and that will give us the 46 centimeters, which is our final answer there. We go. So D is our final answer. So thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.