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Problem 8 Medium Difficulty

$\textbf{Figure E19.8}$ shows a $pV$-diagram for an ideal gas in which its absolute temperature at $b$ is one-fourth of its absolute temperature at $a$. (a) What volume does this gas occupy at point $b$? (b) How many joules of work was done by or on the gas in this process? Was it done by or on the gas? (c) Did the internal energy of the gas increase or decrease from $a$ to $b$? How do you know? (d) Did heat enter or leave the gas from $a$ to $b$? How do you know?

Answer

(a) $$=0.125 \mathrm{L}$$
(b) $$=-57.0 \mathrm{J}$$
(c) For an ideal gas, $\Delta U=n C_{V} \Delta T$ so $U$ decreases when $T$ decreases. The internal energy of the gas decreases because the temperature decreases.
(d) For a constant pressure process, $Q=n C_{p} \Delta T . T$ decreases so $\Delta T$ is negative and $Q$ is therefore negative. Negative $Q$ means heat leaves the gas.

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Video Transcript

Okay, so this problem states that a certain figure shows a PV diagram for an ideal gas in which it's absolute temperature I'd be is one fourth. It's absolute temperature. A volume Does thie gas occupy a point B? How many jewels of work is done on the gas and this process? Was it done on our by the gas and the internal energy of the gas increase or decrease from A to B? How do you know? Did he enter Li the Astor? Maybe. How do you know? Okay, so starting with a So, actually, maybe I should go ahead and draw this this p me diagram since I'm going to be referring to it. So I'll start doing that. Hey, so cares my p access and my access and it you know, it's our leaders and up those fears. So neither of those air s I. There's so we'll need to remember to convert on dso This all occurs at a pressure of one point five atmosphere and were going from A to B and a war. Ah, half theater. Okay, And then this is and then we also know that t and B equals point two five t a. Okay, so first we're asked to find the volume of gas I might be. So to do that, we want to invoke the ideal gas law which states that P b because and Artie and then re arranging about to solve the volumes you can get that thie Cool's on Artie divided by piece, just dividing both sides by TV and here. We can see what happens to volume if we change the temperature. So if the temperature is multiplied by point two five and everything else stays the same on R and P on that, we know that the volume is going Teo, you multiply by point two five a cz Well, so the volume B is than equal two point two five of half leader And, um so let's see. So that's point one two five leaders or big. So now you know what the volume is? Point one to five liters. So I'll go ahead and label. That s so now on to the next question How many jewels of work was done by Ana got gas and the process was done by our on the gas. So groups to take her back. Helen right for Be so the work done by the gas thie integral of P Delta V. Um, if for a constant pressure, that's just peed. All to be a Delta Bi is the final nicely initial. Since we final as lesson be initial on, this is negative on DH. So therefore, the work done by the gas is negative. Therefore, the work done on the gas is positive. So there's work done on the gas to sort of compress it. So maybe it's easiest to say that there is work done on the cast. Okay, So for sea did the internal energy of the gas increase or decrease from a to B Internal energy is always equal to three house on our tea, um, and so the same r stay the same, but t stay the same. So yes, you did change and it particularly got smaller. So say you increased d'Oh d'Oh e cr for decreased. And, um did he under relieve the gas from a to B? How do you know? Okay, so d to get the sign of Q Q is equal Teo the internal energy plus the work done by the gas. So that's just rearranging in its usual form, which is internal energy use equal to Q minus the work done by the gas on DH. Then we can see that this told you this color. Maybe so. That's less than zero summarily said that the work done by us less than zero. Therefore, we know that Q is also less than zero. Um, so there is.

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