A monatomic gas is adiabatically compressed to ⅛ of its initial volume. Does each of the following quantities change? If so, does it increase or decrease, and by what factor? If not, why not?

c. The thermal energy of the gas.

Question

A monatomic gas is adiabatically compressed to ⅛ of its initial volume. Does each of the following quantities change? If so, does it increase or decrease, and by what factor? If not, why not?

c. The thermal energy of the gas.

Accepted Answer

Hey, everyone in this problem, the volume of a sample of helium changes idiomatically from V knot to 1.5 vno where V knot is the initial volume, we're asked to determine if the thermal energy of the helium sample increases or decreases. And by what factor A we're told that if there is no change, we need to state why we have four answer choices. Option A, it decreases by a factor of 0.237. Option B, it decreases by a factor of 0.763. Option C it does not change because it is an adiabatic process. And option D it increases by a factor of 1.5. So let's think about what we're told and we know that V not the initial volume is just gonna be equal to V. Not NVF, the final volume is gonna be equal to 1.5 multiplied by V nine. This is an antibiotic process which tells us that the change in internal energy or that change in thermal energy is gonna be equal to negative of the work. And this is just gonna be equal to N multiplied by CV multiplied by delta T and where delta T is the final temperature minus the initial temperature. So this is what we have for an audio process. And so for thinking about the change in thermal energy, this is delta U value. And the thing that's really changing in this adiabatic process is the temperature A N is gonna stay constant from the beginning to the end. CV is gonna stay constant from the beginning to the end. And so we want to think about how the temperature is changing and how is the final temperature related to the initial temperature. So let's think about what else we have when dealing with adiabatic processes. And we have this relationship between temperature and volume. And so we can say that the initial temperature T knot multiplied by the initial volume V knot to the exponent gamma minus one is gonna be equal to the final temperature TF multiplied by the final volume. VF to the exponent gamma minus one. And we know V NA and via or at least we know the relationship between the two of them. If we can find gamma, that'll allow us to get a relationship between TN and TIA. Now gamma is related to this CV value. So in this case, helium, we know is a mono atomic gas because it's a mono atomic gas. This tells us that this CV value is going to be equal to three halves are now in order to calculate gamma, remember gamma is the ratio of heat capacity. So we have the heat capacity at constant volume here, but we need the heat capacity at constant pressure as well. Now, we can treat helium as an ideal gas which then tells us CP that heat capacity at constant pressure will just be CV plus R. So in this case, it will be five halves R and this will allow us to calculate gamma as that ratio CP divided by CV, which is going to be five thirds. OK. So we have this gamma value now that we can use in our exponent. So let's fill in everything we know. We have the initial temperature T knot multiplied by the initial volume V not to the exponent five thirds minus one. This is gonna be equal to the final temperature TF multiplied by the final volume which we know is 1.5 multiplied by the initial volume V knot all to the exponent five thirds minus one. Hey, let's simplify as much as we can. So we get T knot V knot to the exponent. Two thirds is equal to TF and we can break up, we have 1.5 V knot to an exponent. We can break that up and write it as 1.5 to the exponent two thirds multiplied by V knot to the exponent two thirds. And you can see that that V knot to the exponent two thirds term will divide out now and we have it on both sides. So we get that the initial temperature T knot is going to be equal to the final temperature multiplied by 1.5 to the exponent two thirds. And what we're gonna do is actually write this the other way we wanna write the final temperature in terms of the initial temperature. So we're gonna divide by 1.5 to the exponent two thirds. The final temperature TF is going to be the initial temperature divided by 1.5 to the exponent two thirds. And this is gonna be equal to about 0.763 T. OK. So the final temperature is going to be less than the initial temperature and the temperature is decreasing, the temperature is decreasing. We can already start to think about OK. The thermal energy will likely decrease as well. OK. So that's what we expect to happen. Now, let's get back to that equation. That delta U equation I recall that we had delta U is gonna be C or sorry, N CV multiplied by delta T. So using what we know now this is gonna be N multiplied by CB multiplied by 0.763 T knot minus T dot A delta T is the final temperature minus the initial temperature. Our final temperature is 0.763 the initial temperature. And so we get delta U is N CV multiplied by negative 0.237 T knot. Or we can write this as negative 0.237 multiplied by N CV T knot, right? So our change in thermal energy delta U is negative, that means that the thermal energy is decreasing. OK. So we definitely know that we're looking at one of those answer choices that has a decrease. Now, the question is by how much, right? Remember that delta U is the change and not the final. OK. So our energy, our thermal energy is changing by some amount by this 0.237 multiplied by by what would have been our initial thermal energy. And so the decrease is by this factor of 2.37. All right. So let's go back to our answer. Choices. We know that we're decreasing. OK? So we're already down to option A or B. And then we found that that decrease is by a factor of 0.237. And so the correct answer here is going to be option A. Thanks everyone for watching. I hope this video helped see you in the next one.

A monatomic gas is adiabatically compressed to ⅛ of its initial volume. Does each of the following quantities change? If so, does it increase or decrease, and by what factor? If not, why not? c. The thermal energy of the gas.

Verified Solution

Key Concepts

Adiabatic Process

Thermal Energy

Ideal Gas Law