Ideal Gas Compression Homework: T(cylinder) = 298K

[w3390]

Homework Statement

A compressed cylinder holds nitrogen gas at room temperature. How cold is the gas that is escaping from the cylinder?

T(cylinder) = 298 K
P(cylinder) = 1.034e7 Pa
n = 1

Homework Equations

PV = nRT

The Attempt at a Solution

So since I was not given either the volume of the container or its final volume, I assumed that the initial and final volumes were approximately the same. This may be a fatal flaw to my line of reasoning.

Assuming the above is correct, I said:

P/V = constant , and therefore:

P1/T1 = P2/T2 , where P1 and T1 are the pressure and temperature in the cylinder.

I am not sure if I need the adiabatic exponent of f+2/2 in there somewhere or not. Also, I used P2 = 101.3 kPa and yes I put that into Pa before doing the algebra.

Without the adiabatic exponent, I went through the algebra and found the temperature of the escaping gas is 2.94 K. This to me doesn't seem like a reasonable answer since that is really cold. However, I do not have a good understanding of what is reasonable since I do not know much about gas temperatures.

Any comments on whether or not this is correct or flawed would be greatly appreciated.

 

[w3390]

In addition to above, there are two equations that I know of but neither of them contain both pressure and volume. They are:

VT^(f/2) = constant

PV^(f+2/2) = constant

I don't know how to get P and T together.

 

[vela]

Look at it as an adiabatic expansion. That'll give you a relationship between two of the variables. With that relationship along with the ideal gas law, you can solve for the final temp.

 

[w3390]

Okay, I rearranged the equation for adiabatic expansion to get that P^(1-gamma)*T^(gamma) = constant.

Going through the algebra for this, I get that the temperature of the gas being released is about 79 K.

The only thing that is still throwing me off a bit is that the problem explicitly states that the calculation should be for one mole of gas. However, nowhere in my calculations is that value relevant. This makes me think I'm not including everything.

Can anybody confirm this number as correct? Thanks for the advice vela.

 

[paranormal]

Your approach is correct, but there are a few things to consider. First, assuming the initial and final volumes are the same may not be a valid assumption. It is possible that the gas is being compressed in the cylinder, meaning the final volume would be smaller than the initial volume.

Secondly, the adiabatic exponent is only necessary if the process is adiabatic, meaning there is no heat exchange with the surroundings. In this case, the problem does not state whether the process is adiabatic or not, so it is safe to assume it is not.

Finally, the temperature you calculated (2.94 K) is indeed very cold, but it is not unreasonable. Nitrogen gas can be liquefied at temperatures below 77 K, so it is possible for the escaping gas to reach such low temperatures. It is always important to check if the result makes sense in the context of the problem, and in this case, it does.