Finding how high a column of water in meters can be with the density of water and mercury

Question

I'm struggling to understand what the question is asking me and what concepts it is trying to test... 

 

If the density of water is 1.00g/mL and the density of mercury is 13.6g/mL, how high a column of watr in meters can be supported by standard atmospheric pressure? By 1 bar? 

Answer 1

To answer this question you need to first start with the definition of pressure:

 

P = F/A  (1)

 

where P is the pressure, F is the force exerted on the liquid column and A is the area. According to Newton's second law:

 

F = mg  (2)

 

where m is the mass of the liquid column and g is the acceleration of gravity. Substituting equation (2) into (1) yields equation (3):

 

P = mg/A  (3)

 

now multiply both numerator and denominator of (3) by the height of the liquid column that is raised as a result of the atmospheric pressure P:

 

P = mgh/Ah (4)

 

but the product of the area and height of a liquid column is simply the volume of the column, thus equation (4) becomes:

 

P = mgh/V  (5)

 

finally, we recognize that the density of the liquid column is defined as m/V; so equation (5) takes the final form of equation (6):

 

P = dgh   (6)

 

where d is the density. In your problem, regardless of the identity of the liquid, the pressure is always the same (1 bar), so the right side of the last equation must be equal for both water and mercury:

 

d1gh1 = d2gh2  (7)

 

or

 

d1h1 = d2h2  (8)

 

At this point you must first determine what the height of the mercury barometer is at 1 bar (100,000 N/m2) using equation (6):

 

h = P/dg = (100,000 kg/ms2)/[(13600 kg/m3)(9.8 m/s2)] = 0.75 m

 

where the units were changed to SI units (1 N = 1 kg/ms2). At this point you have everything you need to solve for the height of a water column using equation (8).

 

Hope this was of help.

Answer 2

This relates to how the first barometers to measure atmospheric air pressure were made.

A long glass tube with one end open and the other end sealed was completely filled with mercury, then it was stood on end with the open end in a container filled with mercury.  (You can google "mercury barometer".)

 

If the tube is relatively short, it will still be filled with mercury all the way to the top.  If it's long enough, the column of mercury will not reach the top.  Why?  The air pressure on the surface of the mercury pool that the tube sits in is what supports the mercury. When the weight of a 1 cm^2 column of mercury is greater than the air pressure per cm^2, it can't be supported by that pressure.  The top of the tube will have an almost perfect vacuum, with only a tiny amount of mercury vapor in it.

 

So what does all of this mean?  The height of a column of water or mercury that can be supported by air pressure depends on, and is proportional to, the density of the water or mercury.

 

Now, to your problem. How you're supposed to get the answers depends on what units are used to define 1 bar of pressure. Your textbook or lecture notes should give you the value and units of a bar. If you're supposed to look it up, you'll find a variety of different combinations, including 750.06 mm of Hg or 1019.72cm of H₂O. That would actually give you the answers directly.   

750.06 mm ≈ 0.75 m of Hg

1019.72 cm ≈ 10.20 m of water

 

If you're supposed to calculate everything, you'll need to use 1 bar = 1x10⁵ N/m². Then you'll have to look up the gravitation constant and calculate the weight of a column of mercury or water. Note that grams and kilograms are units of mass, not weight, which is a force.  The unit of force in the metric system is the Newton. This is confusing because we measure weight in pounds, which is a force. Most of the rest of the world measures "weight" in kilograms, which is really mass. I don't think they expect you to go through that, but maybe they do.

 

If you can look up enough information about what you're supposed to use as the basic information to do the solution, add a comment to this and I'll help you.

How is 1 bar defined for you?

If it's mm Hg, just multiply that by 13.6 to get the height for a column of water.

If it's a force per unit area:

We can calculate the force (weight) of Hg using the gravitation constant and the given density, then go from there.