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$\bullet$$\cdot$ An ideal gas at 4.00 atm and 350 $\mathrm{K}$ is permitted to expand adiabatically to 1.50 times its initial volume. Find the final pressure and temperature if the gas is (a) monatomic with $C_{p} / C_{V}=\frac{5}{3},$ (b) diatomic with $C_{p} / C_{V}=\frac{7}{5} .$

Physics 101 Mechanics

Chapter 15

Thermal Properties of Matter

Temperature and Heat

The First Law of Thermodynamics

The Second Law of Thermodynamics

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University of Washington

Hope College

McMaster University

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this is checked. Her 15 problem number 60 were given an ideal gets of initial pressure, or four atmospheres had an initial temperature off 350 Calvin's and this gas is permitted to extend that theoretically, um, to 1.5 times off its initial volume. So if we call the initial bombing to be high, then the final volume is one ahead and 1/2 times of this initial value. So we're asked to find a final pressure in final temperature if it's a rescue, final pressure and final temperature. If it's a long atomic guess with the ratio of CP over CV five over three. Now we're gonna have to remember hell by our equations or modified when we have an idiomatic process. So when we have an idea, very process um, initial pressure times the usual long will this guess to the power of the gamma, which is basically this ratio that is given to us proposed to the final thresher times the final volume of gamma right and also we have the measure temperature times the issue Bollywood Get on Linus. One people's final temperature times the final volume to the power of family spoke, So we'll be taken advantage of these two equations in order to answer in the final pressure and the final temperature in order. Now, let's start reference. Proceed with the final pressure. It's finding the final pressure. Then if you divide both sides of the situation by the final volume to power camera. But as you can see, the right hands up Hey, insides, they're gonna cancel out. So the final country equals initial pressure. Initial volume. I'm a volume to the back power of getting it right. So the initial pressure is given to us in sport at most three years. And then we have the initial volume. Even though we don't know the exact numerical values Freeze. Just knowing the spray show is enough, right instead of the final because of 1.5 initial to the power Gamma. In this case, our gamma is five over three. And finish your volumes, they're gonna cancel each other out. So inside distraction, we have one of the 1.5. So if we compute the final pressure, we find out that it's 2.4 atmospheres now. Yet there's a second part to part A. So we're now to determine the final temperature. We're gonna make use of this equation on the next page. So again, the initial temperature initial volume gamble minus one equals five. Mila again with minus one. Since we're after the final temperature list, divide bold sized by this term because you can cancel out. So the final temperature in terms of Mr Temperature is the initials or re finals gamma lines. One initial temperature is was given to us as 350 Calvin's. And again, we're going to take advantage of the ratio of the fighter volume, to be sure. Well, you and we have CP over. CV has five over three minus money, right? So if you compute the final temperature, then it is 267 Calvin. So as you can see, the gas is cooled down. That something expected for the process. There's expansion, I guess. Now import be we're gonna follow the same recipe break. Only difference is that now the ratio it's a doctor got CP over CBS's going to seven over five. But again, in order to get to the final pressure, I'm using the same equation that I drive on first page. So usual final Palin Gamma Initial pressure was four atmospheres and then I have the initial 15 times being official 7.5 actually, is again cancel. So you're gonna find out the fighter pressure to be 2.27 atmospheres again? I'm going to get to the final temperature I'm getting is the same equation of that I drive off here. So two pair of gamma like this one. It's 350 Calvin the initial 1.5 times the initial seven over five minus one. The initials are gonna cancel out, so we find its final temperature to be 298.

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