Mathematical challenge wrt diving

[boat sju]

The pressures and temperatures can be expressed in any units you'd like, but they must be on absolute scales. You can do the problem in Kelvins or degrees Rankine, and you can use any units of pressure, as long as the values are absolute and not gauge.

I was away from this thread so long that it may be redundant to say this now but :

you're right . . . Kelvin or Rankine.

 

[tnfireman]

Uuuuuuuuuhhhhhhhh 1

 

[ianw2]

Interesting. Once the argument gets going here, the original post or question just goes right out the window!

To review:

From the book Diving Science I got the following challenge, which I am not able to solve:
An explosion occurs in a hyperbaric chamber that is at a depth of 100 FSW. If the ambient temperature in the chamber before the explosion was 72 degrees F, and the pressure gauge on the outsideber shows that the chamber pressure immediately increased to 650 FSW. How high did the temperature rise in the chamber?

This is one of those problems that ignores much of reality and reduces everything to the minimum to prove a point. In this case, pressure and temperature are related.
The Volume is the internal volume of the chamber. Unless there is significantly more information to be given, we MUST assume that the volume before and after the explosion, is the same. V is a constant.

n and R must also be treated as constants for this system.
So, as ClayJar has posted: T₂ =T₁ * P₂ / P₁

The increase in temperature comes from and EXPLOSSION inside the chamber. We all know that the explosion is going to produce a significant change in the molecular composition of the gasses in the chamber. We all know that, assuming the gases before the explosion and after the explosion are ALL "Ideal Gasses", we would need molecular weights, etc. to completely solve the PV = nRT equation.

Quit being obtuse!

Its a simple problem. It was created by coming up with a simple "scenario" and reducing the elements to the bare minimum to prove a point. Good Grief!

T₁ is 72ºF = 295k
P₁ is 100FSW = 400 kPa
P₂ is 650FSW = 2044 kPa

therefore,

T₂ is 1507k or 2253ºF.

Incidentally, the problem can be worked backward holding the 750ºF, the initial temperature and pressure. In this case, the increase in pressure is equivalent to a depth of just 275FSW.

Clearly, the answer given in the book is wrong.

 

[ClayJar]

Quit being obtuse!

Hehe, well, I for one am hardly acute diver.

(Incidentally, although it is likely insignificant given the precision of the problem, what conversion factors did you use for pressure? The standard US assumption of 33fsw = 1ata gives me absolute pressures of 408 kPa and 2097 kPa, respectively, for gauge pressures of 100 fsw and 650 fsw.)

 

[ianw2]

Hehe, well, I for one am hardly acute diver.

Good one, Clay!

Actually, the comment was directed at another poster who seemd to refuse to re-read the original question. There wasn't a thing there about adding anything from outside the chamber.

 

[Mad Scientist]

Surely, you understand that ideal gas equations are *reversible*! The temperature is merely a measure of the kinetic energy of the matter that makes up the gas. The pressure is merely a measure of the force that matter exerts on the walls of the vessel, which is directly related to the energy of the matter that makes up the gas. (After all, pressure is caused by the particles bouncing off the walls, to put it simply.) They're related not by one being the cause and the other being an effect; they're related because they're both manifestations of the same thing (that being the energy of the particles that make up the gas).It is not cheating to work backward from known conditions to find unknown values, however, the problem shows no reason to assume that it was meant to ask anything other than what it quite clearly asked. That the posted "correct" answer was nothing of the sort has no bearing on the process of solving the problem as stated.

Your assumption that heat causes a pressure change is not a wrong assumption, but where is the heat coming from? With your assumption you can not find the right answer. The answer you are getting is the temperature the chamber needs to be heated to to get the final pressure.

My assumption is that the increase in pressure causes the increase in temperature (you even said it the equation is *reversible*!). I make this assumption for two reasons:
1 the question states P1 to P2 (pressure change), T1 to T? (affected temperature)
2 because chambers are pressurized not heated to change pressure (the original question makes no reference to heating the chamber either).

Now while you get a incorrect answer to the problem, and I cant get an answer to the problem because my assumption makes too many unknowns. I could go off and calculate compression ratios of gasses at various rasing temperatures curves and come up with a number for the increase in the number of moles but like you said this was suppose to be a "simple" problem.

 

[ianw2]

Your assumption that heat causes a pressure change is not a wrong assumption, but where is the heat coming from?

The original post said:
An explosion occurs in a hyperbaric chamber

It was written in plain English and clearly understood by all who read it, with one exception. The increase in pressure was clearly due to the explosion in the chamber.

The whole point of the problem in the book was to point out that there is a direct relationship between pressure and temperature in a closed system. As the temperature goes up, so does the pressure. Clearly, you missed that, too.

It can also be used to illustrate the need to work these types of equations in absolute temperatures instead of artificially created scales.

My assumption is that the increase in pressure causes the increase in temperature

And, based on the wording, in English, in the original problem statement, your assumption has nothing to do with this problem. The temperature rise is due solely to the pressure generated by the explosion. Pray tell, why else would the statement have been made?

(you even said it the equation is *reversible*!).

No. re-read what I wrote. I did NOT say the equation was reversible. I said I worked it backward from the given information and the "answer" supplied by the text. There is a very big difference.

I make this assumption for two reasons:
1 the question states P1 to P2 (pressure change), T1 to T? (affected temperature)
2 because chambers are pressurized not heated to change pressure (the original question makes no reference to heating the chamber either).

Again, assumption No. 2 is contrary to the problem statement If the question stated that a sudden influx of gas caused the pressure to soar, you would be correct. Unfortunately, the problem statement clearly intended the increase in pressure to be due to the explosion.

Since no other information was given about the conditions, we MUST assume that the constants and variable other than P and T in the Ideal Gas Equation are meant to be considered as identical at both states of the chamber. Otherwise, there is no solution and the problem is a complete waste of time, effort, paper and ink.

Now while you get a incorrect answer to the problem, and I cant get an answer to the problem because my assumption makes too many unknowns. I could go off and calculate compression ratios of gasses at various rasing temperatures curves and come up with a number for the increase in the number of moles but like you said this was suppose to be a "simple" problem.

Everyone else understand that this is a simple problem designed, albeit poorly, to illustrate a concept to students.

Since you claim to be a "rocket scientist", you should have been exposed to a significant level of education. You should have run across many examples of a pedantic problem which, when examined from a practical standpoint, is patently absurd. You should have been exposed to teaching techniques that are intended to give students an simplistic example of a concept and mathematical methods to analyze that connect.

My BS Chemistry, MS Biochemistry and all the self-education to achieve my license to practice Land Surveying in California have certainly exposed me to such text book problems. Indeed, I have even created a few during my days as a TA.

The problem is not meant to be a perfect statement about a real world condition. It is meant as an illustration. It is meant as a teaching aide to students. It is an example of reduction to the absurd to prove a point. It is solvable.

It is also unfortunate that the answer given by the text was not checked and verified by others before appearing in print.

 

[ClayJar]

Your assumption that heat causes a pressure change is not a wrong assumption, but where is the heat coming from? With your assumption you can not find the right answer. The answer you are getting is the temperature the chamber needs to be heated to to get the final pressure.

And that heat comes from what was described as an "explosion". Is this even disputed?

My assumption is that the increase in pressure causes the increase in temperature (you even said it the equation is *reversible*!). I make this assumption for two reasons:
1 the question states P1 to P2 (pressure change), T1 to T? (affected temperature)
2 because chambers are pressurized not heated to change pressure (the original question makes no reference to heating the chamber either).

If that is your assumption, you are displaying a fundamental lack of understanding. If the system is adiabatic, you need only consider the application of the ideal gas law:

P₁V₁/(n₁T₁)=P₂V₂/(n₂T₂)

You are assuming:

1. V is constant
2. P₁, P₂, and T₁ are defined by the problem
3. T₂ is defined by the answer
4. n is the variable for which you are solving

I find this last assumption completely untenable. If you assume n is variable and that the problem is unsolvable, why not assume that V is also variable? Why not say that there is also no data for whether any nuclear reactions may have occurred in the chamber?

My assumptions are:

1. V and n are constant (as no data for either is given)
2. P₁, P₂, and T₁ are defined by the problem
3. T₂ is the variable for which you are solving

This is the most basic set of assumptions possible, and the problem is indeed solvable. Assuming that anything not given is a constant is proper scientific method. (For example, who would argue with assuming there are no nuclear reactions changing the number of atoms in an ideal gas physics problem?)

Now while you get a incorrect answer to the problem, and I cant get an answer to the problem because my assumption makes too many unknowns. I could go off and calculate compression ratios of gasses at various rasing temperatures curves and come up with a number for the increase in the number of moles but like you said this was suppose to be a "simple" problem.

Mad Scientist, it seems we are at an impasse. I believe that it is readily apparent that this is a simple case of a trivial problem with a posted answer which is simply incorrect. You believe that it is a complex problem which is not only very poorly written but also insufficiently constrained.

If I may be frank, your assumptions are preposterous. If one were to work in the manner in which you have illustrated, no elementary physics problem would be solvable. If you had not seen the "answer", how would you have approached the problem?

 

[Mad Scientist]

If I may be frank, your assumptions are preposterous. If one were to work in the manner in which you have illustrated, no elementary physics problem would be solvable. If you had not seen the "answer", how would you have approached the problem?

Be as frank as you would like, that is the purpose of an open forum discussion.

How would I solve this problem? Simple, I would start with P₁/T₁=P₂/T₂ same as everyone else here has done. Got the answer 2000+ degrees. I would check my answer if I could. Since we can see the answer about 750 degrees, I would try to figure out what was different between what I did and what the author of the problem did. Not assume the book is wrong and I am right.

As a lot of people have done in this thread, I go back and find out that n needs to change to get the correct answer (which with out the answer, the problem is unsolvable). In my mind this is the difference between plug and chug get your number move on and truely understanding what it is you are doing and how it applies.

 

[ClayJar]

I would[...] [n]ot assume the book is wrong and I am right.

As a lot of people have done in this thread, I go back and find out that n needs to change to get the correct answer (which with out the answer, the problem is unsolvable). In my mind this is the difference between plug and chug get your number move on and truely understanding what it is you are doing and how it applies.

Given such a simple problem and such a blatantly incorrect result, I *would* assume the book is wrong. In fact, to assume anything else would seem exceedingly misguided. If it were a more complex problem, I may investigate more deeply, but this is hardly complex. This is like finding the wrong answer to a long division problem. You don't try to figure out how to use square roots to make the values match; you just say, "Oh, look at that. The book's got the answer wrong."

My issue with arbitrarily choosing to make n variable is that there is no justification for choosing that particular value as your additional variable. If you choose to discard the assumption that the problem is constant in both V and n, why is it that you have chosen to have V remain a constant? Without explaining that, you cannot claim to be showing true understanding of what you are doing.

Creatively inventing an unnecessarily complex problem from a simple description is hardly a desirable trait. While you can "plug and chug" values without understanding, solving a simple problem in a simple, straightforward manner is not plug and chug. It's just a simple problem. (Knowing how to multiply two numbers is "plug and chug", but that hardly means it's wrong.)