How to Find Air Pressure in a Vessel Being Evacuated Over Time?

[Saitama]

Homework Statement

Find the pressure of air in a vessel being evacuated as a function of evacuation time t. The vessel volume is V, the initial pressure is p_o. The process is assumed to be isothermal, and the evacuation rate equal to C and independent of pressure.
Note: The evacuation rate is the gas volume being evacuated per unit time, with that volume being measured under the gas pressure attained by that moment.

Homework Equations

The Attempt at a Solution

By the ideal gas equation,
pV=nRT
$$\frac{dp}{dt}=\frac{RT}{V}\frac{dn}{dt}$$
$$\frac{dp}{dt}=\frac{RT}{MV}\frac{dm}{dt}$$
where m is the mass of gas in the vessel and M is the molar mass of the gas.

Now i am stuck here, i don't know what to do next.

 

[TSny]

The process removes C units of volume of gas each second. If C units of volume are removed at pressure P and temperature T, how many moles are removed?

 

[Saitama]

The process removes C units of volume of gas each second. If C units of volume are removed at pressure P and temperature T, how many moles are removed?

$\frac{PC}{RT}$ ?

 

[TSny]

$\frac{PC}{RT}$ ?

Yes. Use this for dn/dt in your diff eq. [Edit: think about the sign of dn/dt]

 

[Saitama]

Yes. Use this for dn/dt in your diff eq. [Edit: think about the sign of dn/dt]

$$\frac{dn}{dt}=-\frac{PC}{RT}$$
(Negative sign because the moles keeps on decreasing)
$$\frac{dp}{dt}=\frac{RT}{V}\frac{-PC}{RT}$$
$$\frac{dp}{dt}=\frac{-PC}{V}$$
Solving this, i get the right answer. Thanks a lot TSny!