# How is formula for converting pressure from mmHg to Pa derived?

Today my younger brother asked me from where does the 1 Pa = 0.00750061683 mmHg formula for mercury barometer come. He needs a way to derive it, or an academic source which can be cited.

After doing some calculations we got the formula for a standard U-tube manometer: $P=\frac{h_2}{h_1}P_0$ where $P_0$ is atmospheric pressure, $P$ is pressure being measured, $h_1$ is height is mercury column exposed to atmospheric pressure and $h_2$ is height of the column exposed to pressure being measured.

The problem is that in the case of a barometer, the $h_2$ is exposed to vacuum and I don't know how to use that.

I've searched the Internet, and got countless sites which explain how a mercury column barometer works, but I was unable to find a site which explains which forces are acting there and how the number was derived. To make things even worse, none of the physics books I have access to have a detailed explanation.

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## 1 Answer

If the height difference between the mercury level in the two arms is $h$ (it's called $\Delta h$ on the figure), then

$$P_1 - P_2 = h\rho g$$

where $P_1,P_2$ are the pressures in both wings (called $P,P_{\rm ref}$ on the figure). One of them is the measured atmospheric pressure. The two pressures are being subtracted because the air pushes the liquid from the two sides in two opposite directions. You may also move $P_2$ to the right hand side, so that the two sides exactly express the pressure in both directions (to be specific, you may think about forces acting on a special separator inserted to the point $B$ at the bottom of the figure - most of the mercury cancels, only the height difference doesn't).

The basic-school formula $h\rho g$ for the pressure may be derived as the force of the mercury column per unit area of the base. The mass is $V\rho = A h\rho$, the force is $g$ times larger i.e. $A h \rho g$, and the force per unit area is therefore $h\rho g$ because $A$ cancels. My derivation is only valid for "cylindrical" shapes but the $h\rho g$ formula is actually true for any shape - the pressure only depends on the depth $h$ beneath the surface.

Restricting our attention to the pressure and height differences only, it's clear that $h=1$ millimetre of mercury corresponds to the pressure difference:

$$\delta P = h \rho g = 0.001 \,{\rm m} \times 13,595.1\, {\rm kg}/{\rm m}^3 \times 9.80665\,{\rm m}/{\rm sec}^2 = 133.332 \,{\rm Pa}$$

The inverse relationship is 1 Pascal is equivalent to $1/133.332 = 0.0075006$ mmHg. The exact values of the densities are a little bit conventional - the densities depend on temperature and pressure and the gravitational acceleration depends on the place. In the past, 1 mmHg wasn't needed that accurately. In the modern era, we define 1 mmHg by your relationship, and 1 Pa is much more accurately defined in terms of "fundamental physics".

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Thanks a lot! The 15 character limit and 15 second limit is idiotic. – AndrejaKo Apr 9 '11 at 19:28
@AndrejaKo The minimum character limit is there to filter out comments that just add noise, such as "Thanks a lot!". Upvotes and Accepts should be thanks enough. – deadly Mar 5 at 11:56
@deadly Except I've had numerous situations where just a few characters would be sufficient. Also don't assume that I don't know to accept and upvote. – AndrejaKo Mar 5 at 19:52
@AndrejaKo I was attempting to explain the rationale behind the minimum character requirement, not impugning your ability to accept and upvote. – deadly Mar 5 at 22:36
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