Submarines and pressure

Question

An experimental research submarine with a volume of 15,000 liters has an internal pressure of 1.2 atm. If the pressure of the ocean breaks the submarine forming a bubble with a pressure of 250 atm pushing on it, how big will that bubble be?

Answer 1

Hi Rachel!

 

It looks like we are meant to make a few assumptions in this problem:

 

1.  The air in the submarine is an ideal gas

2.  No molecules or moles of air escape the bubble after the submarine breaks

3.  The temperature at the depth of the submarine stays constant during this process

 

If this is the case, then we can use the ideal gas law.

 

PV = nRT

 

where

 

n = number of moles of gas

R = ideal gas constant

T = temperature

 

P = pressure

V = volume

 

I always like to start, when using this law, what the constant quantities are, and gather them on one side of the equation.  Given the assumptions stated above, during this process, the entire right side of this equation is constant.  This means that the product of P and V will be constant during this process.  This, the product of P and V at any point in the process equals the product of P and V at any other point.  We can write this mathematically as:

 

P₁V₁ = P₂V₂

 

So, as long as we use the same units for P and V in each case, we can relate P times V at one point in the process to P times V at any other point.  So, before the submarine breaks, we have:

 

P₁V₁ = (1.2 atm)(15,000 L)

 

Then, after the submarine has broken, we have:

 

P₂V₂ = (250 atm)V₂

 

with V₂ being the volume of the resulting air bubble that we are trying to solve for.  Then we can use:

 

P₁V₁ = P₂V₂ -->  (1.2 atm)(15,000 L) = (250 atm)(V₂)

 

This can be solved for V₂ in liters.

 

Now, due to buoyancy and other factors, this bubble would not last long as is.  It would float up, be subject to reduced pressure, expand, have shearing forces act on it, experience changes in temperature, and all sorts of things.  But we would need much more information to deal with that.  I think this idealized picture of what happens right after the submarine breaks is what they are looking for.

 

I hope this helps!  Let me know if you have other questions about this or similar topics.