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One mole of an ideal gas takes 9.5 R T1 amount of heat on the path shown: Find the molar heat capacity at constant volume: Cv=?3V.4R/35R/7SR/23R/2...

Question

One mole of an ideal gas takes 9.5 R T1 amount of heat on the path shown: Find the molar heat capacity at constant volume: Cv=?3V.4R/35R/7SR/23R/2

One mole of an ideal gas takes 9.5 R T1 amount of heat on the path shown: Find the molar heat capacity at constant volume: Cv=? 3V. 4R/3 5R/7 SR/2 3R/2



Answers

A mole of ideal monatomic gas at $0^{\circ} \mathrm{C}$ and $1.00 \mathrm{atm}$ is warmed up to expand isobarically to triple its volume. How much heat is transferred during the process?

Good day in this question we will be. It's all being for the thermal energy in terms of volumes and temperature. So you see the first love thermodynamics where the day you equals you. My nose work. We have different differential of delta. It was the U minus T W. We have D two equals B tv loss. And she we tell daddy, where did you over? T is equal to N R tv over coffee plus. And see we did the over tea. So having this in integral we have Q over the the sequel to NR Ln free F. Or V. I plus and C V Ln D F over D I. We're in Is equal to one more. Therefore U. Is equal to N. R. Sorry, are the Ln free F over V I plus? C V L N D F Over the I C V. See, thank you.

Hello students in discussion. We have to calculate the every smaller heat capacity that is C V. Constant volume. So see we have race. We have to determine okay, when two moles of each guest are mixed. Okay so we can right here that C V embrace. Mhm. It is equal to and one Cv one plus and two Cv to denver by and one plus and two. Okay so and one is two and C. V. One which is given as three by two. R plus and two which is two and C. V. Two which is five by two are for the second guessed the verb two plus two. So after solving the average city it is opted as to our. Okay so from the given options we can say that option B. It is the correct answer for this problem. Okay so using this formula we can identify or calculate the average value of the city. Okay so if we want to calculate the average value for the sippy then simply replace the city with the C. P. Okay thank you.

Since the expansion is a diabetic. Therefore, there is no hate flow into our out off the gas, which means that Q is equal to zero now, using the first law of thermodynamics that's given by question 15 point one. We calculated the temperature change its according to this equation. Then you changing Internal energy is equal to heat loss. Our gain with zero here minus work done by the guests on There you is three over two and our tell Daddy that end is the number off walls and is the Gascon Stern and has a value 8.314 jewels, far more Forget it on N is given Toby 8.5 months Over here on that is a call do cubes. He was also a substitute that so this is it will do negative. W on. This means that Delta T has the change in temperature. Is acquittal negative to over tree walked on by the gas over number off more Einstein's gas constant. And here w is work done by the gas hands that's supposed to do and has a value of 8300 Jews. So now we substitute this revalues over here and define the temperature change to be negative. 78 0.3. So we see that there is a decrease in temperature. So because of this negative science and see that the temperature change that is it falling temperature on that among the fall is indeed going treacle.

So recall from Chapter five that the average kinetic energy of a system is going to be equal to three halves times the number of moles, times the gas, constant times, Champ temperature. Now, ah, One thing to keep in mind here is that any time you have a change So let's say we have a change in the energy. I can go ahead and write here that this is, uh, the energy of the system is just the energy of the particles rather the kinetic energy and the energy or the same thing. So this would then be three halves times. The number of moles doesn't change, so there's nothing changing. There are is a constant times, the change in temperature. So this says this tells me that the change in the internal energy of the system is dictated by the change in the temperature. Right. So ah, one thing to note here, if we have a system where the volume doesn't change, the volume is held constant. Another way to write that is that the volume cherry Rather, the change in volume is equal. Zero recall that work is equal to negative pressure times the change in volume in this instance, work is just equal to zero. So from there we have another version of an equation that involves energy. And that's heat plus work. Since this is zero, we can just say it's equal to heat. So, um, I'm gonna then substitute dis equation over here. What you find is that the heat are Q is equal to three halves in our delta T for a system where the pressure is changing, but the volume is held constant. I'm gonna rewrite this slightly to say that it is n times three halves are times the change in temperature. Now, this might look a little similar to something we've seen before. We've seen something that looked like, Ah, we've seen Q is equal to M c Delta T right where m is mass, See is the specific heat. Ah, we've also seen this in terms of N. C. Moeller, heat times, Delta T. And this may be looks a little bit more familiar where Theo ends match up, but then this'll So this quantity here is actually serving as the heat capacity. So one thing we can then right is that see so the heat can Moeller he capacity at constant volume. Excuse me. Constant volume is equal to three halves are And that kind of makes sense. Because if you look at the units of our units of our is really equal Teoh, it wants you. There's a few different ways to write our But when we're talking about it in terms of kinetic energy of gases, we write the units as jewels per no Kelvin, which looks oddly similar to the specific heats or the Moeller heats that we've been dealing with. So it kind of makes sense that this would be our specific or rather, our Moeller heat capacity for a system at constant volume. Now we do the same thing if we have a system at constant pressure um, recall. So I'm gonna rewrite this again. Change and internal energy is equal to heat plus work which really work, remember, is minus pressure times change in volume. So here we do actually have a change in volume and the pressure is held constant so we can't get rid of that work term. You have to keep that around. But we do know from before that the change in energy Onley depends on the temperature of our system. So three halves in our delta T that's from the very first slide. So I'm gonna set these two parts equal to each other and I'll get three halves in our delta. T is equal to Q. Whatever the heat is minus P Delta V. Okay, so this might not immediately seem to be helpful, but recall that P. Delta V we could talk about this in terms of the ideal gas equation. So let's do ideal gas. So pressure times volume is equal to n times are times t Ah. I have a system where only the volume and the temperature are changing so I can rewrite this as p Delta V is equal to end our does a T. Because those were the only two pieces that are changing. I'm holding the number of moles constant, and I'm holding the pressure constant. So this bit here, I can just substitute back into this part here. So what we get at the end of the day is three halves in our delta T is equal to Q ah, minus the portion that I plugged in before. So, in our delta t that comes from the ideal gas law and we're rearrange this to get Cuba by itself and I'll get Ah, in times five halves are delta t where I've just sort of changed the ordering of things slightly and again. You sort of noted, noticed this pattern where this is acting as a heat capacity. Specifically, this is the capacity he capacity Moeller heat capacity for a set pressure where I'm only allowing the volume in the temperature to change come. And so one thing you can actually write. So if we have let me write this in a different color to make that a little bit easier to see the difference. So change our Moeller. He capacity at constant pressure is equal to five halves. Our ah, and really, if you want to get down to it, it's the molar. Heat capacity at constant pressure is equal to the Mueller capacity at constant volume, plus our because CV, we already figured out, is going to be Three halves are so the two are interrelated


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