A 24-cm-diameter vertical cylinder is sealed at the top by a fric...

Question

A 24-cm-diameter vertical cylinder is sealed at the top by a frictionless 20 kg piston. The piston is 84 cm above the bottom when the gas temperature is 303°C. The air above the piston is at 1.00 atm pressure.

b. What will the height of the piston be if the temperature is lowered to 15°C?

Accepted Answer

Hello, fellow physicists today, we're gonna 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. So a massless piston is used to cover a 320 millimeter diameter container. When a 25 kg metal block is placed on the piston. The height of the gas in the container is millimeters and temperature equals 285 degrees Celsius. The outer side of the piston is in contact with air at 1.0 atmosphere pressure. If the gas is cooled to five degrees Celsius, find the new height of the gas. Awesome. So our end goal is to find the new height of the gas. So we're given some multiple choice answers and they're all in the same units of millimeters. So let's read off our multiple choice answers and see what our final answer might be. So A is 362 B is C is 259 and D is 9.1. Awesome. So let's recall that a piston's weight exerts a pressure. And the equation for that is pressure is equal to force divided by area. OK. And that the, and also we need to recall and remember that the pressure exerted by the piston on the gas is as follows. So the total pressure is equal to the pressure of the atmosphere plus the forest divided by the area. OK. So note that the piston will maintain a constant pressure. Thus, we are dealing with a constant pressure process. So we can write our first important in equation here, which is that the initial volume divided by the initial temperature is equal to the final volume divided by the final temperature. So this is derived from the ideal gas law which as we should remember and recall, the ideal gas law is pressure multiplied by the volume is equal to the number of moles multiplied by the universal gas constant multiplied by temperature. OK. So now we need to recall and consider that volume. We'll call this equation to that volume is equal to the area multiplied by the height and that a the area is constant. That's a very important that we remember and recall that the area is constant. So now when we rewrite equation one, considering equation two, so then we have to take the area multiplied by the height for V. So we're taking this and we plug it in for the V value. So let's do that. OK. So for equation three, when we take the value for V which is area times height or area multiplied by height, and we plug it in to equation one, we should get the following three. Equation three. So the area multiplied by the initial height divided by the initial temperature is equal to area multiplied by the final height divided by the final temperature. So determine the final height. We need to rearrange equation three to solve for the final height. So when we rearrange equation three to solve for the final height, we should get that the final height is equal to the initial height multiplied by the final temperature divided by the initial temperature. So at this stage, we need to convert our temperature values both our initial and final temperature from degrees Celsius to Kelvin in order to properly cancel out units and solve for the numerical value for the final height. So let's do that. OK. So we need to recall really quick how to do a U the unit conversion, how to convert Celsius to Kelvin. It's really easy. So let's do that for both the initial and the final temperature. So let's start with the initial temperature. So the initial temperature was given to us in the prom as 285 degrees Celsius. So we take 285 and all we have to do to convert it to Kelvin is add 273. So we add that together, we should get 558 Kelvin. And then doing the same for the final temperature which our final temperature was cool to five degrees Celsius. So it's five plus 2, 73 equals 278 Kelvin. So now we can finally plug in our numerical values to sol for the final height. So let's do that. Let's plug in our numerical values back into equation four. So the height, the initial height was 520 millimeters multiplied by the final temperature which was 278 Kelvin divided by 558 Calvin, which was our initial temperature. And when you plug that all into a calculator, you should get 259 millimeters. Awesome. So that means that our final answer is C millimeters. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video. Bye.

A 24-cm-diameter vertical cylinder is sealed at the top by a frictionless 20 kg piston. The piston is 84 cm above the bottom when the gas temperature is 303°C. The air above the piston is at 1.00 atm pressure. b. What will the height of the piston be if the temperature is lowered to 15°C?

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