Rearranging the Combined Gas Equation

Here we are going to see how to rearrange the

combined

gas

equation

so that you can solve for any of these variables. Now sometimes this can be a bit complicated and requires a lot of steps to get to the end point. What I'm going to do here is start with two examples with a shortcut, a trick that makes things faster and easier. You may not be familiar with this trick. This might be new to you, so I'll start with that. and then I'm going to do two examples at the end using the more traditional method of doing a lot of multiplication and division so we can cancel things out back and forth, so let's start from the beginning.

I'm going to show you how I am. we are going to solve for t2 using the shortcut two things to keep in mind whenever we rearrange

equation

s like this we need to solve for the variable we need to make sure that this variable is alone on one side of the equation and on the other What we need to make sure is that the variable is at the top of the fraction, so even if we get it alone but it's still at the bottom, that's not good, it has to be alone and at the top of the numerator, anyway. t2 solving for this guy, this is what we do for the shortcut we cross and multiply, which means we do this multiplied by this p1 times v1 times t2, which is this is equal to this multiplied together, so t1 times p2 times v2 now this is good because it immediately puts t1 and t2 outside the denominator off the bottom of the fraction, so we don't have to worry about that anymore now everything is up, so solving for t2, let's divide both sides by p1 and v1 to get rid of them from this divided side. by p1 by v1 and we have to do the same on the other side now we have p1 over p1 they cancel and we have v1 over v1 so they cancel and we are left with t2 equal to t1 times p2 times v2 divided by p1 times v1 let's do an example more with a shortcut here are two gas logins

combined

and this time let's solve for v2 I'm going to multiply again so I'm going to do p1 times v1 times t2 equals cross multiplication of this equals t1 times p2 times v2 now we want to get v2 by itself, so let's divide by p1 and by p2 so we can get rid of those on this side of the equation divided by t1 by p2 divided by t1 by p2 these guys are going to cancel T 1 over T 1 P 2 P 2 cancels out and we are left with p1 multiplied by v1 multiplied by t2 divided by t1 multiplied by P 2 equals v2.

You know, sometimes people get uncomfortable when we have what we're working out. on the right side because we're used to saying that v2 is equal to this and sometimes people don't like it when they say this is equal to v2, you know, that's totally fine, you can turn it around if you prefer, we can just write v2 equals p1 times v1 times t2 divided by t1 times p2 don't worry if you have the variable alone on one side but it happens to be the right side, just flip it over if you prefer to have the variable you are solving for B on the left, fine , those are the two examples where we used this cross multiplication method for the shortcut, if this makes you a little uncomfortable or maybe your teacher wants you to use a more traditional method, now we are going to solve two problems. where we do a series of multiplication and division steps, so I'm going to start by solving for P one, we want to get P 1 on its own, so the first thing I'm going to do is take this t1 off the bottom of the fraction and to do that I'm going to multiply both sides times t1 at the top, so now we have t1 and you know you can imagine this fraction line going through here now that I have t1 at the top and bottom. so we get rid of t1 on this side and we're left with p1 times v1 equals p2 times v2 times t1 divided by t2, okay, the next thing I want to do since I'm solving for P one is get rid of the v1 that's up here, so I'm going to divide both sides by v1 and now I add v1 to the end of the fraction there, so I have v1 over v1 and those cancel out and I'm left with p1 equals p2 times v2 times t1. divided by t2 multiplied by v1 ok one more example here oh and look I made a mistake here this should say this should say t1 there instead of t2 so that's what I want to solve.

I want to solve T 1 now, if you remember. I said before that we can never solve something if it is at the bottom of the fraction. Well, the first thing I have to do is get this guy off the bottom of the fraction. Well, how can I get it out of the bottom? of the fraction I can multiply t1 by both sides both sides by t1 and then we remove it from this side but it ends up at the top here so then we're good because now it's no longer at the bottom of the fraction so we rewrite this p2 multiplied by v2 multiplied by t1 divided by t2, okay, now I have it here and I need to separate it, so I need to move all these guys to the other side to get rid of t2. the denominator first, although it doesn't really matter what order you do this in, so I'll multiply both sides by t2 and now I have t2 at the bottom and t2 I multiplied at the top and now I have t2 times p1 times v1 equals p2 times v2 for t1 getting so close finally I can divide p2 by v2 to get them out of here to get t1 by itself do that on the other side P 2 / P 2 cancel V 2 over V 2 cancel and I'm going to do this in two steps.

Obviously I've gotten rid of these types so I can write it as this is equal to t1, but just to show you that I can reverse it, I'll write it as t1 is equal to t2 times p1 times v1 divided by p2 times v2 that's only if you prefer to have the variable only that you are solving on the left side just remember that you can always flip it this is how we rearrange the combined gas well now I did not do examples with these six variables, I only did it with four, but according to the techniques that I use at the beginning, the shortcut technique or this more traditional method, you should be able to figure out how to solve for any of these variables you need.