Question about a mercury barometer with an imperfect vacuum

In summary, the mercury barometer will measure a surface pressure of 29.72 inches Hg when the true pressure is 29.80 inches Hg.
  • #1
guitarstorm
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Homework Statement



Homework question for a graduate level atmospheric science course:

A mercury barometer of height h has an imperfect vacuum above its mercury column so
that it measures a surface pressure of 29.80 inches Hg when the true surface pressure is
29.90 inches Hg, and it measures a surface pressure of 29.72 inches Hg when the true
surface pressure is 29.80 inches Hg.

What will this barometer measure when the true pressure is 29.7 inches Hg?

Homework Equations



p=rho*g*z

The Attempt at a Solution



This question seems easy but I just cannot seem to get to an answer. I've tried using the hydrostatic equation to set up some sort of ratio, but in the end I get 2 unknowns and I can't figure out how to fix it.

I set up the equations for the first 2 cases as such:

p_actual = g((rho_mercury*z_mercury)+(rho_air*z_air))

So, for the first case:

29.90 in = 101,253 Pa = (9.81 m/s^2)(13,594 kg/m^3)(29.80 in = 0.7569 m) + (9.81 m/s^2)(1.2 kg/m^3)(z_air)

And then I solved for z_air, getting 26.758 m.

Doing the 2nd case, I got z_air = 21.747 m.

I then calculated dp/dz for the two cases as such:

(p_actual-p_mercury)/(z_air-z_mercury)

So, for the first case:

315 Pa/(26.758 m - 0.7569 m )= 12 Pa/m

And I got pretty much the same value for the 2nd case, so I'm assuming I have to use this somehow... But for the third case, I end up with two unknowns: the height of the mercury, and the pressure measured by the mercury barometer, and I don't know what to do from here. Any help would be greatly appreciated... Thanks!
 
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  • #2
Let's forget about numbers for the time being and let's set up an equation balancing pressures.

pairHggh1Hggh10 (h1=29.80", h10=29.90").

Now use the ideal gas law to write [itex]p_{air}=\frac{NkT}{A(h-h_1)}[/itex] where A is the cross-sectional area of the column. Replace in the pressure balance equation. Write a second such equation for the second set of given pressures. Once you do this, observe that you have a system of two equations and two unknowns, h and NkT/A. Find them and use them in the third pressure balance equation.

On edit, I add that the assumption here is that the temperature stays constant.
 
Last edited:
  • #3
Thanks for the help!

The final equation I got was:

(3.67/(0.7679-h)) + 133357.14h=100591

Putting it into Mathematica, I got h=0.7525 m=752.5 mm=29.63 in., which makes sense...

However, I tried solving it algebraically as a quadratic equation and did not get that answer... I got h=0.039 or 1.483... Maybe it was just a calculation error somewhere.
 

FAQ: Question about a mercury barometer with an imperfect vacuum

1. How does an imperfect vacuum affect a mercury barometer?

An imperfect vacuum can affect the accuracy of a mercury barometer by allowing air molecules to enter the tube and interfere with the mercury column. This can result in incorrect measurements of atmospheric pressure.

2. How do you determine if the vacuum in a mercury barometer is imperfect?

The vacuum in a mercury barometer can be tested by tapping the barometer and observing the mercury column. If the column fluctuates or does not return to its original level, it is a sign of an imperfect vacuum.

3. Can an imperfect vacuum be fixed in a mercury barometer?

It is difficult to fix an imperfect vacuum in a mercury barometer. Repeatedly tapping the barometer may help to remove some air molecules, but it will not create a perfect vacuum. It is best to replace the barometer with a new one.

4. Why is an imperfect vacuum in a mercury barometer a problem?

An imperfect vacuum can cause inaccurate readings on the barometer, which can lead to incorrect measurements of atmospheric pressure. This can affect weather forecasting and other scientific data that relies on precise pressure readings.

5. How does the accuracy of a mercury barometer with an imperfect vacuum compare to one with a perfect vacuum?

A barometer with a perfect vacuum will provide more accurate readings compared to one with an imperfect vacuum. The presence of air molecules in the tube can cause fluctuations in the mercury column, leading to less precise measurements of atmospheric pressure.

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