Question about a mercury barometer with an imperfect vacuum

guitarstorm

1. The problem statement, all variables and given/known data

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?

2. Relevant equations

p=rho*g*z

3. 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!

kuruman

Let's forget about numbers for the time being and let's set up an equation balancing pressures.

p_air+ρ_Hggh₁=ρ_Hggh₁₀ (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.

guitarstorm

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.