A Tank full of Helium in space?

Classical Mechanics Level 5

A vessel of volume $\displaystyle V$ containing Helium( $\displaystyle He$ ) is floating in space.

Find the velocity $\displaystyle v$ (in $\displaystyle m/s$ )at which $\displaystyle He$ flows out of a small hole in the vessel.

Details and Assumptions:
$\bullet$ $\displaystyle He$ is a monoatomic gas.
$\bullet$ The vessel is thermally insulated .
$\bullet$ The outside pressure is $\displaystyle 0 atm$ .
$\bullet$ Velocity inside the vessel is negligible.
$\bullet$ Temp. of the vessel is $\displaystyle T = 1000K$
$\bullet$ $\displaystyle V = 1m^3$

The answer is 3223.74.

1 solution

Anish Puthuraya
Apr 2, 2014

Noting that the internal energy of the gas also changes during the process, we can prepare an extended version of the Bernoulli's equation.

$\frac{P}{\rho}+gh+\frac{v^2}{2}+\frac{U}{m} = K$

Since there is no change in the height of the fluid,

$\frac{P}{\rho} + \frac{v^2}{2} + \frac{U}{m} = K$

$\Rightarrow \frac{P}{\rho} + 0 + \frac{3}{2} \frac{kT}{m} = 0 + \frac{v^2}{2} + 0$

$\frac{RT}{M} + \frac{3}{2} \frac{RT}{M} = \frac{v^2}{2}$

$\frac{5}{2}\frac{RT}{M} = \frac{v^2}{2}$

$\Rightarrow v = \sqrt{\frac{5RT}{M}} = \boxed{3223.74m/s}$

Thanks a ton!

I had once got the same expression, but I took $M=4$ rather than $4 \times 10^{-3}$ , and got an altogether different answer. First, I got answer as $\displaystyle \sqrt{\frac{(f+2) RT}{M}}$ . But as I didn't consider units and got wrong answer in first try. Then, I tried another way round by putting $P=0$ , as you said velocity inside is $0$ , and without velocity there can't be pressure of course. Hence, then I got $\displaystyle \sqrt{\frac{fRT}{M}}$ , again not considering units. Then, at my final try, I took correct unit but applied the later result, and hence, all three tries were over :P

jatin yadav - 7 years, 2 months ago

Can you please tell how will the internal energy of the gas change?

Ishan Singh - 7 years, 2 months ago

I am not sure but I think the temperature changes. In the problem it is specified that temperature of the vessel is 1000K. I think he assumed temperature of surrounding to be 0 as is clear by his equation.(I may be wrong)

Pinak Wadikar - 7 years, 1 month ago

@Pinak Wadikar
@Jatin Yadav Don't you think the internal energy doesn't change because the gas isn't doing any work? It's free expansion i.e It's expanding against 0 pressure so change in Internal Energy must be zero? Is this reasoning correct?

A Former Brilliant Member - 6 years, 11 months ago

A Tank full of Helium in space?

The answer is 3223.74.

1 solution

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