The rms speed of the molecules in 1.0 g of hydrogen gas is 1800 m...

Question

The rms speed of the molecules in 1.0 g of hydrogen gas is 1800 m/s.c. 500 J of work are done to compress the gas while, in the same process, 1200 J of heat energy are transferred from the gas to the environment. Afterward, what is the rms speed of the molecules?

Accepted Answer

Hey, everyone. Let's go through this practice problem. 2.5 g of nitrogen moves at an R MS speed of 1150 m per second. The gas expands doing 925 joules of work to the surroundings and absorbs 650 jews of heat from the surroundings. Calculate the R MS speed of the nitrogen molecules. After this process, option, a 1210 m per second. Option B 1090 m per second. Option C 787 m per second and option D 1510 m per second. First, let's discuss the equations that we want to use. The problem is asking for the R MS speed of the molecules. Recall that the formula for the R MS speed of a gas is equal to the square root of three multiplied by the Boltzmann constant K multiplied by the temperature divided by the mass. More specifically M is the mass in kilograms of the molecule. Unfortunately, for us, using this equation requires that we know the final temperature of the gas after the process. But we aren't given any temperatures. We are however told about the energy changes in the system. So we can use the formula relating thermal energy to temperature. Of course, this is different for different types of molecules. But recall that for a diatomic molecule, which this is because it's nitrogen, the thermal energy is equal to five divided by two, multiplied by the number of moles multiplied by the ideal gas constant multiplied by the temperature. So the approach we're going to use has a lot of steps to it. But conceptually, it's pretty simple, we're going to use the initial R MS speed. The problem gives us use that initial speed with the first equation to find the initial temperature. Then we're going to use that initial temperature in the second equation to find the initial thermal energy. Then we're going to use the information. The problem gives us about the energy changes to find the final thermal energy. And then we're going to work backwards. We're going to use the final thermal energy to find the final temperature and then use the final temperature with the first equation to get the final R MS speed. All right. So let's begin by finding an equation for the initial temperature in terms of the R MS speed. We'll take the first equation we wrote and just algebraically solve it for T. So the initial temperature and, and we can solve for T by squaring both sides of the equation multiplying both sides of the equation by M and dividing both sides by three K. And so we find that the initial temperature is equal to the mass of the molecule multiplied by the square of the R MS speed divided by three, multiplied by the Boltzmann constant. Before we can plug numbers into this equation, we first need to find the mass of one of a molecule in kilograms. So we know we're looking at nitrogen and you can look it up in a periodic table. The nitrogen has an atomic mass about 28 atomic mass units. This is because nitrogen on its own has about 14. But we're looking at a diatomic molecule since it's a nitrogen molecule and nitrogen molecules are diatomic. So we multiply 14 by two and that's 28. But we need to convert this into kilograms in order to use it in our equation. So we convert to equate to kilograms by multiplying by 1.661 multiplied by 10 to the power of negative 27 kg per atomic mass units. And that tells us that the mass of the molecule is equal to about 4.6508, multiplied by 10 to the power of negative 26 kg. Now let's put this into our T sub I equation to find the initial temperature. So the mass is 4.6508 multiplied by 10, the power of negative 26 kg. And this is being multiplied by the square of the initial R MS speed 1150 m per second. And this is being divided by three multiplied by the Boltzmann constant K, which is 1.38 multiplied by 10 to the power of negative 23 Jews per Kelvin. If we put this into a calculator, then we find an initial temperature of about 1485.7 Kelvins. Now we'll use the thermal energy equation to find the initial thermal energy of the gas. So as discussed earlier, that's five divided by two, multiplied by 10 rt or in other words, five divided by two by two, multiplied by the number of moles of the gas. We're not told the number of moles directly, but we're told we have 2.5 g of it. So we'll convert this into a number of moles by taking into account the molar mass of nitrogen, which as we talked about earlier is about 28 atomic mass units or in other words, about 28 g per mole. So one mole for 28 g, then this is multiplied by the ideal gas constant or 8. Jews per mole. Kelvins and then multiplied by the temperature which as we discussed, the initial temperature is 1485. Kelvins. If we put this into a calculator, then we find that the initial thermal energy is about 2756. jewels. All right. So that's the initial thermal energy of the gas. But now let's talk about what the final thermal energy will be. Let's discuss the change in energy. Oh, we're told him the problem that when the gas expands it does 925 Jews of work and absorbs 650 Jews of heat. So that means it loses 925 jewels due to the work it does, but it's absorbing or adding 650 jewels. So if we do that in a calculator, we find negative 275 jewels. So there is a net loss here. So the final thermal energy of the gas is going to be less Jews less than its initial. So the final thermal energy it's going to be equal to 200 or 2756.13 Jews minus 275 jewels which we find in the calculator as about 81.3 0.13 jewels. As we discussed earlier, we'll now do the reverse of what we did at the beginning of the problem. So we'll take this final thermal energy and use it to find the final temperature. And we can do that using the formula that we use to find the initial thermal energy in the first place where the thermal energy is equal to five divided by two, multiplied by N RT. So now if you want to solve this for t specifically if you want to solve this for T sub F temperature final, we divide both sides of the equation by five, divided by two and R. So what we find is that the final temperature is equal to two divided by five multiplied by the final thermal energy T H vinyl, divided by N R or more specifically 2/2 divided by five multiplied by 2481.13 jewels divided by the number of moles that we discussed earlier. 2.5 g multiplied by one mole over or divided by 28 g multiplied by the ideal gas constant, 8.314 jewels per mole. Kelvins. We put this into a calculator. We find a final temperature about 1337. Kelvins. Now we'll take that temperature and put it into the R MS speed formula. We talked about all the way back to the beginning of the video. You'll find the final R MS speed, the final R MS speed as we discussed earlier. Here's a subscript final for this is equal to the square root of three, multiplied by the Boltzmann constant or 1.38 multiplied by 10. The power of negative 23 Jews per Calvin multiplied by the final temperature. 1337.43 Kelvins all divided by the mass of the molecule or 4. multiplied by 10th. The power of negative 26 Kelvins. And if we put this into a calculator, then we find a final R MS speed of about 1090 m per second. So this is our answer to the problem. And if you look at our multiple choice options, this agrees with option B 1090 m per second. So option B is our answer to the problem and that's it for this video. I hope this video helped you out if you would if it did and you need more practice. Please consider checking out some of our other videos which will give you more experience with these types of problems. That's all for now. I hope you all have a lovely day. Bye bye.

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The rms speed of the molecules in 1.0 g of hydrogen gas is 1800 m/s.c. 500 J of work are done to compress the gas while, in the same process, 1200 J of heat energy are transferred from the gas to the environment. Afterward, what is the rms speed of the molecules?

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