00:01
We have a gas at 200 kelvin, and we want to double its root mean square velocity.
00:10
So we want to know how much do we need to change the temperature by to do this? well, if you remember for gas, the root mean square velocity goes something like this 3kt over m, something like a square root of this.
00:25
So if we want to double the velocity, we need to quadruple the temperature, right? because we want to basically want a two to end up outside of the square root so we need a four inside the square root so the temperature needs to be quadrupled so we need to change the temperature to 800 kelvin for this to happen so that would be like part a and if we do that at a constant volume as the problem states what is going to be the increase in pressure so the initial pressure is 60 ,000 pascal's i believe so six times 10 to the fourth pascal's if we quadruple the temperature for an ideal gas remember we've got pv equals nr t right if we make this increase this by a factor of four while keeping the volume constant then the pressure is also going to increase by a factor of four so the final pressure is just going to be four times this which is 2 .4 times 10 to the fifth pascals and then lastly the question asked at this final temperature what is the average force transfer to the walls of a one meter cued container by a single molecule colliding with the wall so the pressure is the force divided by pressure is the force divided by the area right and so if we multiply the pressure by the area then we're going to get our force and so the force in this case or the area if each of these is one square meter we're told that it's a cubic container so all the sides are the same length and we're doing this for a single molecule and to figure out basically we need to know how many molecules are going to be in this first off because we want to do this for a single molecule so we need to find out what n is or better yet we could write this as n k t this is another way of writing this and technically we'd rather know what this is this is the number of molecules and so the initial pressure is, or sorry, the final pressure is 2 .4 times 10 to the 5th pascal.
02:41
So if we substitute in some of these numbers times the volume, which we said was one cubic meter, that's the left -hand side of this equation.
02:55
This is the number of molecules times boltzman's constant times the final temperature, which you said was 800 kelvin.
03:01
Right and so basically if we do 2 .4 or 240 ,000 times one is going to be 2 .4 from the fifth joules if we divide this by 800 kelvin and also divide it by boltzman's constant which is like 1 .38 times 10 to the negative 23rd and this is like joules per kelvin something like that this will tell us the number of molecules that we have.
03:36
And let's see.
03:38
What do we get when we do that? divided by 800 divided by 1 .38 times 10 to the negative 23rd...