# Two engineers, a physicist, and an orbital mechanist can't figure this out

## Homework Statement

Friend's fiancee gave this problem on a final she was administering and isn't sure how the answer key is right:

Boyle's law describes the inverse variation between the volume of a gas and the pressure, y, exerted on it. A balloon with a volume of 2.25 liters is sealed in a bell jar, and air is pumped out. As the air is pumped out, the balloon expands. Which equation could be used to calculate the pressure exerted on the balloon?

## Homework Equations

Boyle's Law: P1V1=P2V2=k (presumably, this isn't given in the problem)

## The Attempt at a Solution

The question is worded as is, we've argued about specific meanings for like half an hour

Answer key says choice B) y=2.25/k

Everyone and their mother says C) y=k/2.25

I mean, by dimensional analysis alone it should be C, right?

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Well, if you assume that the volume after the air is pumped out is known, then B] is the right answer. Let the atmospheric pressure be $P_a$ and the final pressure and final volume be $P_f$ and $V_f$. Then:

$$(2.25)P_a = P_f~V_f$$

Hence,

$$P_f = \frac{(2.25)P_a}{V_f} = \frac{2.25}{k}$$

Do note here, that 'k' is not the constant you mentioned earlier i.e. $P_1 V_1 = P_2 V_2 = k$, as you said that it isn't given in the problem.

Also, unless the units of 'k' are given, nothing can be said about how the equation stands dimensionally. IMHO, this question seems quite incomplete.

See, we went down that train of thought, EXCEPT that 2.25*k is also an option. Why couldn't it be that?

If you can prove that Pa>Vf in terms of magnitude, I'll buy that the problem just sucks in that it wants you to assume k is greater than 1, and is otherwise ok

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If you can prove that Pa>Vf in terms of magnitude...
What you are essentially saying is that:

$$P_a - V_f > 0$$

Look at the operation on the left hand side. This operation is not dimensionally correct. You cannot subtract volume from Pressure.

And yes, the problem does suck since you have no idea what 'k' is given. If 'k' is said to be independent of the final volume, then my answer is wrong. Here, again, I do consider that the initial pressure is known. In our case, we have two unknowns [final pressure and final volume], but one equation only. Which leaves with either finding another equation, or knowing one of the variables, which is why I assumed that the final volume is known.

Again take the case as, we take the equation to be:

$$\frac{2.25}{k}$$

also,

$$\frac{p}{2.25}$$

is a solution, where, $p = 6.025 / k$. Now, both of them are constants, and even the second solution is right. In this way you could come up with infinite solutions involving an arbitrary constant and the value '2.25'. Which is why this question is an epic fail.

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