2.0 g of helium at an initial temperature of 300 K interacts ther...

Question

2.0 g of helium at an initial temperature of 300 K interacts thermally with 8.0 g of oxygen at an initial temperature of 600 K.c. How much heat energy is transferred, and in which direction?

Accepted Answer

Hey, everyone. Let's go through this practice problem. 3.2 g of neon at T equals 20 degrees Celsius comes into contact through a conducting boundary with 4.2 g of nitrogen at T equals 320 degrees Celsius. Calculate the amount of heat that flows between the gasses to establish thermal equilibrium and state the direction of flow. Option A 363 jewels nitrogen to neon. Option B 363 Jews Neon to nitrogen. Option C 668 Jews Neon to nitrogen and option D 668 Jews Nitrogen to Neon. This is a problem that can be kind of frustrating because there are a lot of steps to it. But even though there are so many steps, conceptually, it's not that bad. The fundamental concept here is the I is a fact that when we have two bodies that are touching each other, they will eventually achieve thermal equilibrium where both bodies have reached the same temperature as one another. The process that we'll use for solving this problem will first involve taking the temperatures that were given and using them with our thermodynamic equations to find the initial energies of both the neon and the nitrogen. Then we will use those energies to determine the total energy of this system. Then we'll use that total energy to find what the final temperature of the combined system must to be. Then we'll use that final temperature to figure out what the individual final energies are of both the neon and the nitrogen. And then we'll use those final energies see how they compare to the initial energies that we calculated earlier. And then it'll become clear how the energy is flowing. So yeah, a lot of steps, but fortunately none of them are individually difficult. It's just a lot of work that is required to take us there. But let's start with step one, which is finding the initial energies of both the neon and the nitrogen. So first off neon neon is mo atomic and recall that the energy of a mo atomic molecule is equal to three divided by two, multiplied by the number of moles multiplied by the ideal gas constant multiplied by the temperature. In this case, since we're looking at initial energies, that would be the initial temperature. In order to actually do anything with this equation. One thing we'll want to establish right away is N the number of moles for each of the gasses. So the number of moles for neon. Well, we're given the mass 3.2 g, which means that we can determine the number of moles by dividing this by the molar mass of night of a of neon. If we look this up in a periodic table, we see that the molar mass of neon is 20.18 g for one mole. So putting that into a calculator tells us that the number of moles of a neon for this problem is equal to about 0. 8573 moles. So let's use that with the energy equation we discussed earlier to find the amount of energy initially present in the neon. So three halves multiplied by 30.18573 moles multiplied by the ideal gas constant. 8.314 Jews per mole. Kelvins multiplied by the initial temperature and the initial temperature of the neon is given as 20 degrees Celsius. So converting that into Kelvins, that's 20.273 Kelvins. And if we put that into a calculator to find the energy, that's an initial energy of about 0.4 jewels. Now we'll do the exact same thing for the nitrogen, except since nitrogen is diatomic, we'll be using five halves instead of three halves because that's the formula for the energy in a diatomic gas. So five halves then multiplied by the number of moles. So let's find the number of moles again, just like we did with the neon. So the number of moles of nitrogen is equal to the mass of the nitrogen 4.2 g. And again, we divide this by the molar mass of nitrogen. You look this up in a periodic table. You might find that the molar mass of nitrogen is about 14 g. But because nitrogen is diatomic, it's actually going to be closer to 28.1 g because there are two atoms in the nitrogen molecule, the 8.1 g. And this is a number of moles of about 0. moles. So we go back to our energy equation five, divided by two, multiplied by 20.149946 moles multiplied by the ideal gas constant. 8.314, multiplied by the initial temperature of the nitrogen, which is given as 328 degrees Celsius or plus 273. Kelvins put this into a calculator and there's an initial energy of the nitrogen of about 1848.2 jules. Now that we have the initial energy of both the neon and the nitrogen, we can find the total energy for the entire system by adding these two together. So 579.4 jewels plus 1848.2 jewels put that into a calculator. And we find a total energy of about 2427.6 Jews. And due to the law of conservation of energy, we should expect that this total amount of energy will remain the same throughout the course of the problem. Now that we have the total amount of energy in the system, we can use the fact that there is thermal equilibrium in the final state of the process to determine what that final temperature is. After all, the total energy is also going to be equal to the sum of the final energy for both the neon and the nitrogen. So E total is also going to be equal to three halves multiplied by the number of the moles of the, of the neon multiplied by the ideal gas constant, multiplied by the final temperature plus five halves multiplied by the number of moles of the nitrogen gas multiplied by the ideal gas constant multiplied by the same final temperature. So to emphasize that these are the same, I'm going to factor out T sub F as well as the R and the one half just in the name of simplification. So the total energy is also equal to one half multiplied by the ideal gas constant, multiplied by the final temperature multiplied by, in parentheses, three multiplied by the number of moles of the neon plus five multiplied by the number of moles of the nitrogen. So now if we want to find the final temperature in both, both molecules, well, it's solve for T sub F by dividing both sides of the equation by one half, R three N N E plus five N N. This is some fairly simple algebra. And what we find is that the final temperature is equal to two multiplied by the total energy divided by the ideal gas constant R multiplied by, in parentheses, three multiplied by the number of moles of the neon plus five multiplied by the number of moles of the nitrogen. So to find out what that final temperature is, let's just take the total energy that we determined earlier 2427.6 Jews and plug that in for e tote here. And then we'll take the number of moles for both neon and nitrogen that we determined on the left in red way early on in the problem as well. And plug all those numbers into this equation that we've just determined. And what we find is that the final temperature for both the nitrogen and the neon is equal to about 476. Kelvins. And now that we have the final temperature, we can use that temperature in the energy equations that we used way earlier on in the problem, the very beginning of the problem to find the amount of energy that exists in both the nitrogen and the neon after the process has taken place. So let's start with nitrogen. Let's start with neon once again. So the final energy for the neon is equal to and we use the same equation we discussed earlier. So three halves multiplied by the ideal gas constant multiplied by temperature. In this case, the final temperature says three halves multiplied by the number of moles of the neon. So that's 0.158573 moles multiplied by the ideal gas constant. 8.314 Jews per mole. Kelvin multiply by the final temperature of 476.5 Kelvins. And we find that neon has a final qu amount of energy of about 942.3 Jews. Now let's do the same thing for the nitrogen. So, e nitrogen final is equal to five halves multiplied by N RT or five halves multiplied by 50.1499, 46 moles multiplied by the same ideal gas constant multiplied by the same temperature 76.5 Kelvins. We put that into a calculator. Then we find a final quantity of energy for the nitrogen of about 1485.1 Jews. So now that we have now we have both the initial and final energies for both the neon and the nitrogen. So if we want to figure out which direction the energy is moving, let's sub, let's take the final energy for each of the two molecules and subtract their initial molecules to see what the change is. So again, we're starting with the neon sets the final energy in the neon minus the initial energy in the neon. So that's gonna be equal to 942.3 Jews minus the initial energy we found for the neon, which as we discussed way earlier is equal to 579.4 Jews. We put that into a calculator and we find an energy change of about 363 Jews. So the neon gained 363 jewels in the process. If you want, we can do the same exact thing with the neon or with the nitrogen. So the final energy in the neon in the nitrogen minus the initial energy in the nitrogen s equal to 1485.1 Jews minus the initial amount of energy in the nitrogen, which we discussed is 1848. Jews put this into a calculator and we find a value of approximately negative 363 Jews. So the quant so the magnitude is the same, understandably that makes sense as per the law of conservation of energy. And what's interesting is that the change for neon is positive, meaning it gained energy, but the change for nitrogen is negative, meaning it lost energy. So the answer to this problem is that the energy flowed from nitrogen to neon. So there's an energy change of 363 jewels in the direction of nitrogen to neon. So that is our answer to this problem. And if we look at our multiple choice options, we can see that option. A agrees with our solution 363 Jews Nitrogen to Neon. So option A is the correct answer to this problem and that is it for this problem. I hope this video helped you out. If you feel you need more practice, please check out some of our other tutoring videos which will give you more experience with these types of problems. But that's it for now. I hope you all have a lovely day. Bye bye.

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2.0 g of helium at an initial temperature of 300 K interacts thermally with 8.0 g of oxygen at an initial temperature of 600 K.c. How much heat energy is transferred, and in which direction?

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