Question about a mercury barometer with an imperfect vacuum

[guitarstorm]

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!

 

[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 p_{air}=\frac{NkT}{A(h-h_1)} 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.