14 g of nitrogen gas at STP are adiabatically compressed to a pressure of 20 atm. What are (d) the compression ratio Vₘₐₓ/Vₘᵢₙ?

Video transcript

Hey, everyone in this problem, 2.5 g of H two gas at room temperature and pressure undergoes adiabatic compression to 15 atmospheres. We're asked to express the final volume in terms of the initial volume. OK. So we're gonna have VF, this equals us something multiplied by V I. We have four answer choices. Option A VF is equal to 6.92 V I. Option BVF is equal to 0.145 V I. Option CV F is equal to 15 V I and option DVF is equal to 0.0667 V I. We're gonna start, we're gonna write down the information we've been given and the problems, the first thing we're given is the mass and that's 2.5 reps. Yeah, we're dealing with H two. Then we have room temperature and room pressure. Now room pressure tells us that our initial pressure P I is going to be one atmosphere and we're told that we undergo compression until we have 15 atmospheres. So the final pressure PF is going to be 15 atmospheres. OK? So we have some information about the pressures and we wanna figure out a relationship between the volumes. Now, one key piece of information that we're given in this problem is that this is an adiabatic compression. And because this is an adiabatic compression recall that we have the following relationship between pressure and volume piv I to the exponent gamma is equal to PFVF to the exponent gamma. OK. We know both pressures. This is gonna allow us to relate V I and VF the only thing we need to do first is figure out what this gamma value is, but we can do that. OK. Recall that gamma is going to be CP divided by CV, the ratio of heat capacity. OK. So CP at constant pressure, CV at constant volume, we have H two, we know that this is a diatomic gas and so CV is going to be equal to five halves R and again, because we're dealing with a die atomic gas. Now CP what is CP? Well, we have H two, we can treat this as an ideal gas. So CP is just going to be CB plus R five halves are plus R gives us seven halves R, OK? Now we have CP and CV, we can take the ratio to get that gamma value that we wanted. So gamma is going to be seven halves, our divided by five halves are the RS and the divided by twos will cancel out and we're left with seven fifths or 1.4 per gallon. OK? So we have our gamma value, we know these pressures. Let's get back to our equation. And we can find this relationship between our volumes. Substituting in what we have, we have one atmosphere multiplied by V I to the exponent, 1.4 is equal to 15 atmospheres multiplied by VF to the exponent 1.4. Now oftentimes when we do these calculations, we convert our pressure into standard pressure units of pa scalp, we left them as atmospheres here. Just because in the next line, we're gonna divide one by the other, the units are gonna divide out anyway. OK. So whether we converted them both to pascals or left them in atmospheres, when we divide them, the units are gonna divide out and we're just gonna get the ratio which is the same no matter what unit we're looking at. OK. So let's go ahead and do that. We're going to divide both sides of our equation by 15 atmospheres because we want to try to isolate VF when we do that, we have the VF to the exponent, 1.4 is going to be equal to 1/15 of the I to the exponent, 1.4. OK. Again, we wanna get VF by itself right now, we have an exponent of 1.4. How do we undo that? Or we can raise it to an exponent of one divided by 1.4. If we do that, we get that VF is going to be equal to 1/15 of V I to the exponent 1.4 all to the exponent one divided by 1.4. This exponent we can apply to both terms. So VF is gonna be 1/15 to the exponent one divided by 1.4. And then we left with just vi a 1.4 divided by 1.4 gives us an exponent of one. So now we've got VF in terms of V I, the last thing we have to do is simplify 1/15 to the exponent one divided by 1.4. And when we work this out on our calculator that gives us a value of 0.1445. So we have VF is equal to 0.1445 multiplied by V I. And that is the final solution we were looking for we were looking for in equation four BF in terms of V I, let's go back to our answer choices, compare what we found and we're gonna round and when we do, we can see that our answer matches with option BVF is equal to 0.145 V I thanks everyone for watching. I hope this video helped you in the next one.