Mathematical challenge wrt diving

[ClayJar]

If PV = constant, R = constant, and T changes, n must too, right?

If PV is a constant, yes.

Of course, in this problem, P and T are variable, while V and n are constant. (R, of course, *is* a constant.)

 

[Blackwood]

Since diving physics problems almost always deal with comparisions, there's almost no need to deal with R. (Plus, if you were solving for, say, the number of moles of air in a tank, it'd probably not be ideal to use the ideal gas law anymore. )

Aye.

^thanks

 

[Blackwood]

If PV is a constant, yes.

Of course, in this problem, P and T are variable, while V and n are constant. (R, of course, *is* a constant.)

Ratio-ing Ps and Vs assumes the product PV is constant.

 

[ClayJar]

Ratio-ing Ps and Vs assumes the product PV is constant.

Assuming you're effectively referring to Boyle's Law, I agree that is a specific case of an application of the ideal gas law. If you were saying something about the particular problem at the top of this thread, I don't follow you.

The generalized form of the equation relating two sets of conditions using the ideal gas law is:

P₁V₁/(n₁T₁) = P₂V₂/(n₂T₂)

And of those variables, the assumptions I'm working with are V₁=V₂ and n₁=n₂. Are you assuming something other than that?

 

[Mad Scientist]

The only numbers given are for initial temperature, initial pressure, and final pressure. The problem asks you to solve for the final temperature "using the universal gas law". Given the problem statement at the beginning of this thread, the problem boils down to a simple case of Amonton's Law.

The simple case of a named "Law" is not the case here. This problem is asking us to solve 2 unknowns using 1 equation. Temperature is changing in response to a pressure increase inside a constant volume. How is the pressure increasing? That pressure increase must be comming from outside the system (a compressor maybe) so as P increases has to increase. The only constants in the problem is the constant (R) and the volume. The problem is flawed and is unsolvable with the information given.

 

[Mad Scientist]

the assumptions I'm working with are V₁=V₂ and n₁=n₂. Are you assuming something other than that?

If you assume n₁=n₂ how is the pressure changing?

Volume is constant
Pressure is changing
Temperature is changing in response to pressure changing

PV=nRT

If Volume is constant and Pressure doubles and everyting else is constant Temperature doubles.
If Volume is constant and Pressure doubles and Temperature does not double than something else is not a constant....

 

[ClayJar]

The simple case of a named "Law" is not the case here. This problem is asking us to solve 2 unknowns using 1 equation.

Incorrect. The problem is asking for final temperature given an initial temperature and both initial and final pressures.* Don't make it more complex than it needs to be.

Temperature is changing in response to a pressure increase inside a constant volume. How is the pressure increasing?

You're thinking about it back-to-front. *Pressure* is increasing as a result of an unspecified increase in temperature. You can increase the temperature by merely adding energy to the system. For example, leave a black tank in the sun; you will note the pressure increases as the temperature increases.

That pressure increase must be comming from outside the system (a compressor maybe) so as P increases has to increase.

The cause of the pressure increase is indeed from outside the system (such system being defined as the gas inside the chamber). That cause is the addition of energy (in the form of heat). That heat energy manifests as an increase in temperature, which, in this simple case of a named "Law", also shows up as a proportional increase in absolute pressure.

The only constants in the problem is the constant (R) and the volume. The problem is flawed and is unsolvable with the information given.

It is perfectly valid to assume that both V and n are constants, as the pressure and temperature increases can be directly attributed to the addition of energy to the fixed quantity of gases in the fixed volume of the chamber. The problem is indeed solvable, and by your own words, is "simple". You have merely fallen afoul of your assumption that the addition of gas is necessary to have the manifest increase in pressure, which I hope I have explained to be not necessarily the case.

If you assume n₁=n₂ how is the pressure changing?

The system is not adiabatic.

*Edit: Originally said "temperatures" instead of "pressures"... sorry.

 

[Mad Scientist]

Incorrect. The problem is asking for final temperature given an initial temperature and both initial and final temperatures.

Don't make it more complex than it needs to be.You're thinking about it back-to-front. *Pressure* is increasing as a result of an unspecified increase in temperature. You can increase the temperature by merely adding energy to the system. For example, leave a black tank in the sun; you will note the pressure increases as the temperature increases.The cause of the pressure increase is indeed from outside the system (such system being defined as the gas inside the chamber). That cause is the addition of energy (in the form of heat). That heat energy manifests as an increase in temperature, which, in this simple case of a named "Law", also shows up as a proportional increase in absolute pressure.

So you are saying there is a external temperature increase that is increasing the pressure inside the chamber that is causing the temperature increase from the pressure increase the question is asking about? The question asks for the temperature increase due to pressure increase not what temperature is necessary to increase the pressure.

It is perfectly valid to assume that both V and n are constants, as the pressure and temperature increases can be directly attributed to the addition of energy to the fixed quantity of gases in the fixed volume of the chamber. The problem is indeed solvable, and by your own words, is "simple". You have merely fallen afoul of your assumption that the addition of gas is necessary to have the manifest increase in pressure, which I hope I have explained to be not necessarily the case.

Actually it is not a valad assumption because it does not give you the correct answer. If you assume n₁=1 and n₂=2.2 (calculating back from the answer the book gives you) the correct answer can be derived (yeah I know it is cheating using the answer to find the answer) showing that the assumption of n being constant is a bad assumption.

The system is not adiabatic.

What does that have to do with n₁=n₂? Adiabatic has to do with the transfer of heat (not temperature) not increasing the number of moles of a substance (n).

 

[OHGoDive]

1475°F is not roughly twice the temperature of 750°F. It is approximately 1.6 times the temperature of 750°F. When you deal with ratios, you must use absolutes. 1475°F is 1935°R, and 750°F is 1210°R. Dividing 1935°R by 1210°R gives 1.599.

Why must you use absolutes? Well, is 10°F "negative 1 times" -10°F?

You are, of course, correct. Geez, deja vu to all those C-'s in college chemistry. :shakehead

 

[ClayJar]

Oops. Sorry about the transcription error. "The problem is asking for final temperature given an initial temperature and both initial and final pressures."

So you are saying there is a external temperature increase that is increasing the pressure inside the chamber that is causing the temperature increase from the pressure increase the question is asking about? The question asks for the temperature increase due to pressure increase not what temperature is necessary to increase the pressure.

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).

Actually it is not a valad assumption because it does not give you the correct answer. If you assume n₁=1 and n₂=2.2 (calculating back from the answer the book gives you) the correct answer can be derived (yeah I know it is cheating using the answer to find the answer) showing that the assumption of n being constant is a bad assumption.

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.

If one wants to solve the completely different problem of what the ratio of n₁ to n₂ would be, that's great (and I myself entertained myself by doing just that earlier in the thread), but it's not relevant to the solution of the problem as stated, which I reiterate is a *very* simple problem written solely in pressure and volume.

What does that have to do with n₁=n₂? Adiabatic has to do with the transfer of heat (not temperature) not increasing the number of moles of a substance (n).

Your question to me was "If you assume n₁=n₂ how is the pressure changing?" My answer was that the system was not adiabatic. What I meant by that was that if you assume that the quantity of gas was constant, you can increase the kinetic energy of the gas in the system by adding heat. This increased kinetic energy manifests as increases in the temperature and the pressure of the system. Does that explain more fully what I was saying?