The rms speed of molecules in a gas is 600 m/s. What will be the rms speed if the gas pressure and volume are both halved?

Question

Accepted Answer

Hey, everyone. So this problem is dealing with the root mean square velocity. Let's see what it's asking us. We're told that the root mean square speed of molecules in a gas is m per second. If the volume of that gas is suddenly doubled, what is the new R MS speed? Assuming that the pressure remains constant. Our multiple choice answers here are a 990 m per second. B 350 m per second, C 1400 m per second or D 700 m per second. Now, we, the first thing we can do here is recall that the relationship between pressure and our root mean squared speed including volume as well is this PV is equal to one third nm. So this uppercase V for volume and then this lower case V for speed. So V to the negative two. So that equation, if we recall that equation, we can also recall that the root means square speed, by definition is equal to the square root of V to the negative two. So VR MS squared then is equal to V to the negative two. And we can plug that into this initial equation. So we have P uppercase V for volume is equal to one third NMVR MS squared. And so if we were to isolate the root mean squared speed to the left-hand side of the equation, we have VR MS and this, we'll call this um case one is equal to three PV divided by NM. And so in case two, after the volume of the gas is doubled, the equation looks the same where we have VR MS squared. Sorry, those are squared is equal to three PV. Two supposed to be one divided by N multiplied by M. The two we know is simply three M er is simply two multiplied by the original volume or doubled. So that we can write that as three P multiplied by two V one divided by NM. And so if three P V one divided by nm equals VR MS squared, we are left with VR MS squared. For this case two is simply two multiplied by VR MS one squared. And so to solve for our new R MS, we'll take the square root of two multiplied by our initial root means squared speed of 700 m per second. And that quantity is squared. So the square root of two multiplied by 700 m per second squared, plug that into our calculator and we get 990 m per second. And that aligns with answer choice A. So that's all we have for this one, see you in the next video.

The rms speed of molecules in a gas is 600 m/s. What will be the rms speed if the gas pressure and volume are both halved?

Verified Solution

Key Concepts

Root Mean Square (RMS) Speed

Ideal Gas Law

Kinetic Theory of Gases