Boyle's Law [mmHg]: Solving Atmospheric Pressure with a Mercury Barometer

[Magma828]

A simple mercury barometer consists of a vertical glass tube sealed at its upper end containing a column of mercury. The space between the top of the mercury column and the upper end of the tube is a vacuum. A mercury column height of 760 mm is equivalent to atmospheric pressure 1.0x10⁵ Pa.

A school experiment to find atmospheric pressure befotre the days of modern health legislation was as follows. A length of air was trapped in a capillary tube sealed at one end by a thread of mercury. When the tube was held horizontally, the length of the trapped air column was 82 mm and the length of the thread of mercury was 39 mm.

When the tube was held vertically with the open end upwards, the air column was squashed to 78 mm.

Find the value of atmospheric pressure in mm of mercury (mmHg).


We're given this statement (Boyle's Law):
The volume of a fixed mass of gas is inversely proportional to the pressure exerted on it, provided the temperature is kept constant.

I imagine this constructs an equation like this: p₁V₁ = p₂V₂

I can't really relate this to the question though, it's probably something really simple I'm missing.

 

[mgb_phys]

What is the pressure on the air (in the sealed part of the tube) in the two cases?

 

[Magma828]

What is the pressure on the air (in the sealed part of the tube) in the two cases?

That's all the information it gives, sorry.

There are a few rogue questions in our textbook because they are published so quickly for the changing syllabus, so it appears it hasn't been very well checked. Maybe this is another one of those questions, where they didn't give needed information?

 

[mgb_phys]

No they give you all the information you need - it's a very simple and elegant experiment.
i was asking you to think about pressure = force / area, what's the difference in the force and hence the pressure on the trapped air in the two cases/

 

[gamer_x_]

This is a really cool question. I was stumped for a few minutes before I started thinking about what additional effects are present when the capillary tube is vertical that were not there before. Do you remember in which direction pressure acts?

 

[mgb_phys]

it's a very neat experiment. it's a pity you can't do it anymore because if you allowed a few grams of mercury into schools billions would instantly be killed.

 

[Magma828]

No they give you all the information you need - it's a very simple and elegant experiment.
i was asking you to think about pressure = force / area, what's the difference in the force and hence the pressure on the trapped air in the two cases/

Well the area and force will be the same in both, won't they? It doesn't actually give the surface area of the tube, and the force provided by the weight of the mercury is anyone's guess.

I'm not having difficulty understanding how the experiment works, I just don't know how to quantize everything.

 

[mgb_phys]

In the horizontal case the mercury is being pushed toward the sealed end of the tube by the outside atmosphere and the air is pushing back with an equal force. So the air trapped in the tube is under one atmosphere of pressure.

In the second case the mercury is pushing down with one atmosphere of pressure PLUS the weight of the blob of mercury.
So the air in the tube is under 1 atmosphere + X of pressure.
From boyles law we know that if the volume has dereased by 78/82 then the pressure must have gone up by 82/78 = 1.05.

if you work out the weight of the mercury you know this is (82/78-1) of an atmosphere you can work out what an atmosphere of pressure is.

hint; since we are talking about pressure (force/area) the width of the tube doesn't matter - can you see why?

hint2: the drawing is a little bad - you aren't expected to take into account the curved ends of the mercury bead.