A reaction inside a cylindrical container with a movable RAN piston causes the volume to change from 12.0 L to 18.0 L while the pressure outside the container remains constant at 0.975 atm. (The volume of a cylinder is V = pr2h, where h is the height; 1 L # atm = 101.325 J.) (b) The diameter of the piston is 17.0 cm. How far does the piston move?

Video transcript

Hi everyone. This problem reads. Consider a reaction taking place in a cylindrical container with a movable piston. After the reaction reached completion, the volume changed from 15 L to 10 L at a constant pressure of .843 atmospheres. If the diameter of the piston is 24 cm, how far will the piston move? And we're gay. Given some useful information, the volume of a cylinder is equal to pi r squared times H where H is the height. And we're also told that one leader atmosphere is equal to 101.325 jewels. All right, so let's go ahead and get started. The question that we want to answer is how far will the piston move? And essentially what we want to solve for here is height. Using our volume of cylinder equation. So let's go ahead and write that out. So our volume is going to equal pi R squared times H where R is radius and ages height. And because we want to know how far will the piston move, we need to isolate the height variable. So let's go ahead and divide both sides by pi r squared when we isolate volume. What we're going to get is excuse me. When we isolate height, What we're going to get is height is equal to volume over pi R squared. Alright, so taking a look at the problem, let's see what information we know. So volume is going to equal our change in volume. We're told the volume change from 15 liters 02, 10 liters. So our change and volume is going to be our final volume minus our initial volume. So we have 10 liters minus 15 liters. That's our change in volume. So this change in volume is negative five liters. Okay, and this is going to equal five liters. Alright, So we want this in units of centimeters. So we want our volume to go from leaders two cubic centimeters. All right, So we want to go from Leaders two cubic centimeters. So we have our five l and we are going to go first from Leaders two mL. Okay, so in one millimeter there is 10 to the negative three liters. Our units for leaders cancel. And now we're in milliliters. Alright, now we're gonna go from milliliters two cubic centimeters and one millimeter and one millimeter There's one cc. So our units for milliliters cancel and now we have cubic centimeters. So let's go ahead and do this calculation. And when we do it, we're going to get our five liters is going to be equal 5000 cubic centimeters. So now we have our volume in cubic centimeters. All right. And so we can write here, our change in volume is equal to 5000 cubic centimeters. So we'll put an arrow here put an arrow. Alright. And in the problem, what else are we given? Were given the diameter of the piston and in our equation that we re arrange an isolated height. We see the variable r where r is radius, so radius is going to equal the diameter divided by two. So we're told the diameter of the piston is 24 centimeters. So we're going to take 24 centimeters and divided by two and we're going to get 12 centimeters as our radius. So radius is equal to radius is equal to 12 centimeters. All right. So we have the volume and we have the radius. So now we have all the information that we need to solve for height. Let's go ahead and plug these values in. So height is equal to volume. So we have 5000 cubic centimeters divided by pi R squared. So we're going to take pi Times our radius which is 12 centimeters squared. Okay, So when we simplify this, we're going to get cubic centimeters over 452. centimeters squared. Excuse me. There's an error here. I did not mean to put a cube there. So that should just be 12 centimeters. All right, So that means our denominator then becomes centimeters squared. So we have three and two. So we're going to take three minus two, which is going to leave us with one. Alright, so our cubic centimeters will cancel out and for our centimeters at the bottom will be left with one. All right, So let's go ahead and do the calculation. So we're going to take The 5000 and divided by 452.3893, and we're going to get our height is equal to 11. cm, and this is going to be our final answer. Okay, so how far will the piston move? Our answer is 11.1 cm. That is it for this problem? I hope this was helpful.