b. A gas cylinder has a piston at one end that is moving outward at speed vₚᵢₛₜₒₙ during an isobaric expansion of the gas. Find an expression for the rate at which vᵣₘₛ is changing in terms of vₚᵢₛₜₒₙ, the instantaneous value of vᵣₘₛ, and the instantaneous value L of the length of the cylinder.

Question

Accepted Answer

Hey, everyone. So this problem is dealing with the root means squared velocity. Let's see what it's asking us consider a cylindrical container of radius R filled with an ideal gas that has a movable piston. The gas undergoes an isobaric process. While the piston is moving at a constant speed and the gas expands, determine the expression for the rate of change of the root mean square speed in the gas. In terms of the, the instantaneous value of the root mean square speed V sub R MS and the instantaneous value of the cylinders height H our multiple choice answers are given here below. So the key to solving this problem is remembering two equations. So the equation for the root mean squared speed BR MS is equal to the square root of three K BT divided by M where KB is your Boltzmann's constant T is your temperature and M is your mass. And our ideal gas law PV uppercase V with volume is equal to NKB T. And so we can see here that this K BT our bolts and constant multiplied by our temperature is a common term. So if we isolate that we have K BT is equal to P uppercase V divided by N. And so we can replace, we can substitute that in, in our VR MS equation. So now we have three PV divided by nm, we have a cylindrical container. And so that volume of a cylinder is given by, we can recall pi R squared H and so plugging that in, we have VR MS as equal two, three P multiplied by pi R squared H divided by nm. And now we're asked for the expression for the rate of change. So we can recall that that means we're looking for the derivative in terms of time. So if we look at all of the terms that are constant here, three number constant P pressure is constant because it said it was an isobaric. Um The process I is constant, the radius of the cylinder is not changing, that's constant. Our height is changing as the piston moves and then the number of molecules of the gas and the mass of the gas is also not changing. So the only thing that is changing as the piston moves is the height. So just for ease of writing out our derivatives, I'm going to name this constant C to be equal to the square root of three P pi R squared divided by N multiplied by M because we can basically pull all of that out and say that's a constant. And that makes it easier to deal with in in derivative space. So now we'll take the derivative. So DDT of VR MS is equal to C are constant multiplied by H to the one half or the square root of H DH DT. And so when we take that derivative, we have DD TV R MS is equal to C divided by two multiplied by the squared of age. And so now how do we get this back into terms of VR MS and our uh piston speed? So we have this C pulled out for all of our constants. And another way of writing that is that C is equal to VR MS divided by the square root of H. And so when we plug, sorry, and this is DH GT and the speed of the piston is equal to DH DT because that is by definition, how fast that height is changing or the how fast that height is changing is equal to how fast that piston is moving. So we'll call that the speed of the piston. And so when we put all of that together, the rate of change of our route means squared speed is going to be equal to VR MS divided by two H multiplied by V of the piston. And so when we look add our multiple choice answers, it aligns with answer choice D so D is the correct answer for this problem. Hope that helped. We'll see you in the next video.

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Key Concepts

Isobaric Process

Velocity Relationships

Length of the Cylinder