Bulk Modulus Problem - Two solutions?

[najd]

Homework Statement

Question:
When water freezes it expands about 9%. What pressure increase would occur inside your automobile engine if the water froze. The bulk modulus of the ice is 2x10⁹N/m².

Homework Equations

B = - Δp/(ΔV/V)

The Attempt at a Solution

My 1st solution:
Volume was originally V. Then increased by 9%, so that Vf = 0.09V + V = 1.09V.
ΔV/V = 0.09.
Δp = 0.09*2x10⁹= 180MPa.

Answer should be 165MPa.

My 2nd solution:
I tried treating it in reverse.
Volume was originally 1.09V. Then decreased by 9% to V.
Surprisingly, ΔV/V = 0.09V/1.09V.
Δp = (0.09*2x10⁹)/1.09 = 165MPa.

Can someone please explain the error in my first attempt?

 

[Mapes]

Hi najd, welcome to PF!

The engine is not forcing the ice to expand from V to 1.09V. The engine is forcing the ice, which would otherwise expand on its own from V to 1.09V, to remain at V. This is equivalent to applying pressure to reduce the volume from 1.09V to V. Does this make sense?

 

[najd]

Hey! Thanks!

So you're saying my second approach is correct because the pressure caused by the engine, which can only hold V of water, is preventing the ice from expanding to 1.09V, whereas if it the engine weren't there, the ice would expand normally.

Okay. I understand. Thank you!

Hmm, okay, another situation. Let's pretend that water of volume V is inside a balloon which has no effect except for occupying the water. As it freezes, it will expand by 9%, right? The balloon is NOT stopping it from expanding because it's merely occupying it. If I were to calculate the pressure done by the water on the balloon, would my first attempt at solving the problem be correct?

 

[Mapes]

Hmm, okay, another situation. Let's pretend that water of volume V is inside a balloon which has no effect except for occupying the water. As it freezes, it will expand by 9%, right? The balloon is NOT stopping it from expanding because it's merely occupying it. If I were to calculate the pressure done by the water on the balloon, would my first attempt at solving the problem be correct?

Not really, because the balloon's volume doesn't increase by 9%.

 

[najd]

I think you misunderstood me because I forgot to mention that the balloon's initial volume is V as well. The water of volume V was filling it, entirely, so the balloon's volume would increase by 9%, too.

 

[Mapes]

I think you misunderstood me because I forgot to mention that the balloon's initial volume is V as well. The water of volume V was filling it, entirely, so the balloon's volume would increase by 9%, too.

Certainly, but you asked whether you could apply your first approach to the balloon. The volume of the balloon material (rubber, for example) doesn't increase by 9%. So you couldn't take the bulk modulus of the balloon material and argue that the pressure on the balloon material is $\Delta P=-B(\Delta V/V)$. The pressure that a stretched balloon exerts on its contents is a different type of calculation.