How Do You Calculate Initial Pressure and Final Volume in Adiabatic Expansion?

[JeffEvarts]

TL;DR

I have a compressed pure gas at a specific temperature and volume. It suddenly (adiabatically) expands until it's at ambient pressure and a specific temperature, T2. Given: T1, V1, T2, and P2, I want to find P1 and V2.

I have a compressed pure gas at a specific temperature and volume. (T1, V1) It suddenly (adiabatically) expands until it's at ambient pressure and a specific temperature. (P2, T2). Given: T1, V1, T2, and P2, I want to find P1 and V2.

There's a great example in wikipedia which is almost exactly this case, with compression rather than expansion, and I can't seem to generalize the math. Grr.

Here it is: Example of adiabatic compression

It starts with room temp air and compresses it into 1/10 the volume.

So far, so good. High school physics: k = P1V1/T1 = P2V2/T2. From that, I can figure out what the ratio of the temperature and pressure will be: V goes down by a factor of 10, T/P varies by the same factor.

The math on the page goes farther than that, though, and gives individual values for P and T. That's where I get lost. It feels like (as a college grad in a technical major) I should be able to do this algebra, but I keep getting nonsense at the end.

Can anyone offer me a pair of formulas to solve for P1 and V2 in whatever "common" case you like?

So just so we're all on the same page, here are some approximate values I'm working with:
P1 = ? // something quite large. 10atm? more?
T1 = 263K // -10C
V1 =1L

P2 = 101kPa // 1 atm
T2 = 193K // -80C
V2 = ? //something quite large. 10L? more?

NB: The gas is "pure" CO2 so the "7/5" exponent may be inappropriate.

Thank you for your attention,
-Jeff Evarts

 

[Chestermiller]

If you are familiar with the first law of thermodynamics, how would you apply the first law to this problem? Please present the first -law equation you derive.

 

[JeffEvarts]

ChesterMiller:

First law of thermodynamics: Energy is conserved. If I COULD apply it, I would, but that's why I asked the question. Now I'm off the "unanswered questions" queue, and no one will follow up. Please (really!) answer my question if you can, or I'll repost to get back into the queue.

Sorry, that sounded snippy. It's been over 25 years since my last physics class, so I guess I'm not up for a rhetorical teaching moment. I was just hoping for a couple quick formulas.

 

[Chestermiller]

ChesterMiller:

First law of thermodynamics: Energy is conserved. If I COULD apply it, I would, but that's why I asked the question. Now I'm off the "unanswered questions" queue, and no one will follow up. Please (really!) answer my question if you can, or I'll repost to get back into the queue.

Sorry, that sounded snippy. It's been over 25 years since my last physics class, so I guess I'm not up for a rhetorical teaching moment. I was just hoping for a couple quick formulas.

As you well know, that's not the way we do things around here. But, OK, just this once I'll make an exception. $$\Delta U=nC_v(T_2-T_1)=Q-W=-P_2(V_2-V_1)$$ Can you figure out what to do next?

 

[JeffEvarts]

As you well know, that's not the way we do things around here. But, OK, just this once I'll make an exception. $$\Delta U=nC_v(T_2-T_1)=Q-W=-P_2(V_2-V_1)$$ Can you figure out what to do next?

I'm betting the Ts are temperatures and the Vs are volume. I was going to guess the P was pressure, but there's only one of them, so that's probably wrong. The U, n, C, Q, and W are complete mysteries .

ChesterMiller, you seem to want blood from me... here it is, from my open vein:

It's been 25 years since my last physics class. I have basically NO IDEA what that formula means. What do you want me to say? I'm useless as a classical physicist? I can't figure this out without help from others? True on both counts. You're a "Mentor" and "Moderator": awesome and knowledgeable, I am a member of the public: pathetic and ignorant.

Now that I have properly abased myself, will you render me assistance?

I figured that a place called PhysicsForums would be an appropriate forum to ask a question about physics, and I'm still hoping it is.

My original question was pretty straightforward:

I have a compressed pure gas at a specific temperature and volume. It suddenly (adiabatically) expands until it's at ambient pressure and a specific temperature, T2. Given: T1, V1, T2, and P2, I want to find P1 and V2.

Can anyone offer me a pair of formulas to solve for P1 and V2 in whatever "common" case you like?

Is it too much to hope that you or someone else will answer it?

Failing that, will someone point me to an appropriate forum to get an answer to a question like this?
-Jeff[/quote]

 

[Chestermiller]

What you are specifically asking is contrary to what we can offer at Physics Forums. We are here to help people who are learning a subject overcome their obstacles, and also to help them to solve problems they are working on (provided they are learning the fundamentals and are making an effort to understand them).

If you are trying to learn thermodynamics from a course and/or textbook, we're here for you. But we are not here to provide a comprehensive tutorial on a subject. So, if you want help with this problem, I suggest you start by learning the first few chapters of a good thermodynamics text, like Fundamentals of Engineering Thermodynamics by Moran et al.

We are just not going to give you the solution to your problem with no effort on your part. I hope you understand and accept these principles of how Physics Forums works.[/QUOTE]

 

[JeffEvarts]

So because I'm a professional person with a programming job (and thus spend my study time on algorithms, operating systems, and languages), people who know the answer to my physics question will not answer it. That's a shame. When someone asks me "how do I reboot my cellphone", I usually just tell them, rather than asking them to buy and read a book on Android. Different strokes for different folks, I guess.

Cheers,
-Jeff

 

[Chestermiller]

$$\frac{T_2}{T_1}=\frac{1+(\gamma-1)(P_2/P_1)}{\gamma}$$
Happy now? Was it really worth getting banned over?

The only unknown in this equation is $P_1$.