Consider a container like that shown in Figure 20.12, with n₁ mol...

Question

Consider a container like that shown in Figure 20.12, with n₁ moles of a monatomic gas on one side and n₂ moles of a diatomic gas on the other. The monatomic gas has initial temperature T₁ᵢ. The diatomic gas has initial temperature T₂ᵢ.a. Show that the equilibrium thermal energies are

Accepted Answer

Hi, everyone. In this practice problem, we are being asked to find the final thermal energy of helium and oxygen. We will have a thin barrier separating two compartments in a rigid insulator box with the right compartment holding an H E most of helium at a temperature of DH E. And the left compartment holds an O X most of oxygen at a temperature of the O X. The barrier prevents any mixing of the gasses and will be asked to find the final the energy of helium and oxygen in terms of N O X N H E and the total initial energy E to or E total. When they reach equilibrium, the options listed the different expressions for the final terminal energies of helium and oxygen. So, in order for us to tackle this problem, we want to notice that helium is a mono atomic gas with three degrees of freedom. Well, oxygen is a diatomic gas with five degrees of freedom. Therefore, the right compartment with oxygen will begin with an initial thermal energy, which will follow the following formula of E O X I for initial equals to five divided by two, multiplied by N O XX multiplied by R multiplied by T O X or N uh or five divided by two N RT. While the life compartment helium will begin with an initial thermal energy of E H E I or initial equals to three divided by two and H E RT H E just like. So, so the total thermal energy of the system is going to be the sum of the initial energies for the two compartments. Um assuming that there will be no energy loss because this is an insulating container. So because of that, the total initial energy or E total I will then be equal to five divided by two and O X RT O X plus three divided by two and H E RT H E. And um combining this two, we can then combine this into R divided by two of that multiplied by five and O X T O X plus three. Mh E T H E just like. So the barrier is going to be so thin that the helium and oxygen atoms can collide at the boundary. But without moving from one side to the other, the two gasses will interact until the equilibrium temperature T F is going to be reached. And at the F, the total thermal energy of the system or E to F will follow the expression of R multiplied by T F because the T is now going to be common divided by two multiply that with five N O X plus three N H E just like. So, so we wanna um equalize expression for the initial total energy and the final total energy to find an expression for the final temperature or T F. So let's do exactly that. Equalizing E total I equals to E total F we build and G are divided by two, multiplied by five N O X T O X plus three N H E T H E equals to RT F divided by two, multiplied by five N O X plus three N H E. We can cross out the R and the twos because they are both the same for both sides. And then taking or finding an equation for T F, we can rearrange this so that we will get an equation for T F by taking the five N O X three N H E to the other side. So we will have five N O X T O X plus three N H E T H E divided by five N O X plus three N H E. So then we wanna notice that, that the E total or E P O T is listed in the options given for the final thermal energy for helium and oxygen here. So you wanna incorporate that somehow. And the way you want to do that is to buy rearranging the E total initial formula here so that we can essentially substitute the five N O X T O X and the plus three and H E T H E with E total here. So let's start by um doing exactly that. So we are going to rearrange this formula right here so that we will have five N O X T O X plus three N H E T H E to then be equal to E total multiplied by two divided by R. So we wanna substitute that into our equation here so that we will have a final equation for T F to then be equal to two E total divided by R multiplied by five N O X plus three N H E just like so awesome. So now that we have uh obtained this equation for T F, we can then actually find the final thermal energy of oxygen and the final thermal energy of helium when boat reaches equilibrium. And the way we want to tackle that is to by uh employing the formula for E ox E O X final, which is going to be exactly the same as the energy of oxygen initial. The difference is just the T is now going to be T F instead of P O X here. So let's do that. We have the uh E O X final to then be five divided by two, multiplied by N O X multiplied by R multiplied by T F. And then we can substitute RT F value here or T F expression. So we will get five divided by two, multiplied by N O X multiplied by R multiplied by two in total, divided by R multiplied by five N O X plus three N H E. We can cross out the R here and then we can cross out the twos as well. So that the final E O X uh F equation or expression will then come out to be five multiplied by A N O X multiplied by E total divided by five N O X plus three N H E just like. So now moving on to E H E final, similarly, this will then be equal to three divided by two and H E multiplied by R multiplied by T F which is the same as the initial helium energy with the difference only being in the T H E equals to T F. So then we can uh substitute our T F expression so that we will get uh three divided by two, multiplied by N H E multiplied by R multiplied by T F which is the expression here, two E total divided by R multiplied by five N O X plus three N H E. And then we can cross out the R S crossed out the twos. And that will give us the uh final uh the energy of helium equation to then be three N E multiplied by E total, multiple uh divided by five N O X plus three N E just like. So, so there we have the final thermal energy for oxygen and the final thermal energy for helium. And for oxygen, it will be five N O X E total divided by five N O X plus three N H E. And for helium, it will be three N H E E total divided by five N O X three N H E. And these two uh final thermal energy expressions will correspond to option B and R answer choices. So option B will be the answer to this particular practice problem and that will be it for this video. If you guys still have any sort of confusion, please feel free to check out our adolescent videos on similar topics available on our website. But other than that, that will be it for this one. Thank you.

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Consider a container like that shown in Figure 20.12, with n₁ moles of a monatomic gas on one side and n₂ moles of a diatomic gas on the other. The monatomic gas has initial temperature T₁ᵢ. The diatomic gas has initial temperature T₂ᵢ.a. Show that the equilibrium thermal energies are

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