Thermodynamics problem - filling a bottle from reservoir

[Petrushka]

I've been grappling with this problem for days, but can't produce the answer.

The problem can be summarised as follows:

We have a bottle containing a certain quantity of a perfect gas whose ratio of specific heat capacities is gamma. The pressure in the bottle is doubled by the admission of more gas from a high pressure reservoir in which the temperature and pressure remain constant. Initially the temperatures in the bottle and in the reservoir are the same, and heat transfer during the process is negligible.

Using only the 1st law, determine the fraction by which the mass of gas in the bottle increases.

I know the answer, and did come up with a method of producing the answer, but I was unconvinced by it.

Any help/hints/prodding in the right direction is appreciated.

 

[russ_watters]

What is your answer and method?

 

[Petrushka]

The answer is 1/gamma.

Method:

Q=0.

-W = U2 - U1

W = pV (where V is the volume of the bottle)

U1 + pV = U2
H1 = U2

=> (initial mass) * Cp * T1 = (final mass) * Cv * T2

T1 = T2 (not sure about whether T1 does actually equal T2) thus (Final mass) / (Initial Mass) = Cp / Cv = 1/gamma

 

[siddharth]

Petrushka, I don't think that's the right way to do it.

The pressure inside the bottle is continously changing, right? So the work done will not be PV.

In fact, here is how I would attempt it. First, you need to use the first law for a control volume. From that, you'll get

$$\frac{dE_{cv}}{dt} = \dot{Q} - \dot{W} + \dot{m_i}h_i$$

(Note that here $\dot{W}$ is the boundary work (like expansion, etc) and is 0 in this problem)

And the conservation of mass gives

$$\frac{dm_{cv}}{dt} = \dot{m_i}$$

substitute this in your first relation, and you'll get

$$\frac{d(m_{cv}\hat{U}_{cv})}{dt} = (h_i)\frac{dm_{cv}}{dt}$$

Here, I'm stuck, and I think more info needs to be given in the question. For example, if the temperature is assumed to be constant, then from the ideal gas law, you can get

$$\dot{P}V=\dot{m_{cv}}RT$$

$$\dot{m_i} = \frac{\dot{P}V}{RT}$$

So if you substitute in terms of $\dot{P}$ and integrate, and substitute C_p to find H, you should be able to find the fraction by which the mass changes.