Question

Using the Ideal Gas Law from the kinetic theory, develop an expression that predicts the pressure in the vacuum chamber after the pump has been running for a specific amount of time i.e. as a function of time.

Answer 1

As long as the pump is on, molecules from the vacuum chamber are pulled into the pump and then pushed out into the surrounding outside the chamber. And so, the pressure in the vacuum chamber will decrease with time.

According to the ideal gas law in the vacuum chamber is
   
  
And the change in the number of gas molecules inside the pump is related to the volumetric flow rate as follows
  
where,
     is the pumping speed.
From the conservation of gas molecule number, we get
  


  
  
   
  
  
  
  
where,
  P_0 is the pressure at time t = 0. and s is the pumping speed, V_{ch} is the volume of the chamber and t is the time spent.