Manometers and barometers measure pressure by balancing an unknown pressure against the weight of a liquid column.
The key formula is the hydrostatic pressure relation:
\[
\begin{aligned}
\Delta P &= \rho gh.
\end{aligned}
\]
Here \(\rho\) is the liquid density, \(g\) is gravitational field strength, and \(h\) is the vertical height difference between liquid levels.
1. Hydrostatic pressure difference
A liquid column of height \(h\) creates a pressure difference:
\[
\begin{aligned}
\Delta P &= \rho gh.
\end{aligned}
\]
The height must be measured vertically. If the tube is slanted, the length along the tube is not the correct \(h\).
2. Barometer
A barometer measures atmospheric pressure. In a mercury barometer, the top of the tube is almost vacuum, so the atmosphere supports the mercury column:
\[
\begin{aligned}
P_{\mathrm{atm}} &= \rho_{\mathrm{Hg}}gh.
\end{aligned}
\]
For mercury, \(\rho_{\mathrm{Hg}}\approx13600\ \mathrm{kg/m^3}\). A height of about \(0.760\ \mathrm{m}\) gives approximately one atmosphere.
\[
\begin{aligned}
P_{\mathrm{atm}}
&= (13600)(9.81)(0.760) \\
&\approx 1.01\times10^5\ \mathrm{Pa} \\
&\approx 101.3\ \mathrm{kPa}.
\end{aligned}
\]
3. Open U-tube manometer
In an open U-tube manometer, one side is connected to the unknown pressure and the other side is open to the atmosphere.
If the unknown pressure is higher than atmospheric pressure, then
\[
\begin{aligned}
P_{\mathrm{unknown}}
&= P_{\mathrm{atm}}+\rho gh.
\end{aligned}
\]
If the unknown pressure is lower than atmospheric pressure, then
\[
\begin{aligned}
P_{\mathrm{unknown}}
&= P_{\mathrm{atm}}-\rho gh.
\end{aligned}
\]
The gauge pressure relative to atmosphere is
\[
\begin{aligned}
P_{\mathrm{gauge}} &= \pm\rho gh.
\end{aligned}
\]
4. Closed-end manometer
In a closed-end manometer, the closed side is often a near vacuum. If the closed side is vacuum, then the measured pressure is absolute:
\[
\begin{aligned}
P_{\mathrm{unknown}} &= \rho gh.
\end{aligned}
\]
If the closed side has a nonzero trapped pressure \(P_{\mathrm{closed}}\), then
\[
\begin{aligned}
P_{\mathrm{unknown}}
&= P_{\mathrm{closed}}+\rho gh.
\end{aligned}
\]
5. Differential manometer
A differential manometer compares two pressures, usually labeled \(P_A\) and \(P_B\).
In the simplest educational case with one manometer liquid and a vertical height difference \(h\),
\[
\begin{aligned}
P_A-P_B &= \pm\rho gh.
\end{aligned}
\]
The sign depends on which side has the higher pressure.
6. Solving for column height
If pressure is known and the required column height is needed, rearrange:
\[
\begin{aligned}
\Delta P &= \rho gh.
\end{aligned}
\]
\[
\begin{aligned}
h &= \frac{\Delta P}{\rho g}.
\end{aligned}
\]
A denser fluid gives a smaller height for the same pressure. This is why mercury barometers are much shorter than water barometers.
7. Altitude correction
Atmospheric pressure decreases with altitude. A common troposphere approximation is
\[
\begin{aligned}
P(z)
&= P_0\left(1-\frac{Lz}{T_0}\right)^{5.25588},
\end{aligned}
\]
where \(P_0\) is sea-level pressure, \(z\) is altitude, \(L\approx0.0065\ \mathrm{K/m}\), and \(T_0\approx288.15\ \mathrm{K}\).
This model is useful for ordinary altitudes but is still an approximation.
8. Worked example: mercury barometer
A mercury barometer reads
\[
\begin{aligned}
h &= 0.760\ \mathrm{m}.
\end{aligned}
\]
Use
\[
\begin{aligned}
\rho_{\mathrm{Hg}} &= 13600\ \mathrm{kg/m^3},\\
g &= 9.81\ \mathrm{m/s^2}.
\end{aligned}
\]
Then
\[
\begin{aligned}
P
&= \rho gh \\
&= (13600)(9.81)(0.760) \\
&= 101396\ \mathrm{Pa} \\
&\approx 101.4\ \mathrm{kPa}.
\end{aligned}
\]
This is very close to standard atmospheric pressure, \(101.325\ \mathrm{kPa}\), so the reading is about \(1\ \mathrm{atm}\).
9. Formula summary
| Instrument / goal |
Formula |
Meaning |
| Pressure difference |
\(\Delta P=\rho gh\) |
Pressure represented by a liquid height difference |
| Barometer |
\(P_{\mathrm{atm}}=\rho_{\mathrm{Hg}}gh\) |
Atmospheric pressure balances the liquid column |
| Open manometer, higher unknown pressure |
\(P_{\mathrm{unknown}}=P_{\mathrm{atm}}+\rho gh\) |
Unknown pressure exceeds atmospheric pressure |
| Open manometer, lower unknown pressure |
\(P_{\mathrm{unknown}}=P_{\mathrm{atm}}-\rho gh\) |
Unknown pressure is below atmospheric pressure |
| Closed-end manometer |
\(P_{\mathrm{unknown}}=P_{\mathrm{closed}}+\rho gh\) |
Closed side may be vacuum or trapped gas |
| Height from pressure |
\(h=\Delta P/(\rho g)\) |
Find column height needed for a pressure difference |
10. Assumptions
- The manometer liquid is at rest.
- The density of the manometer liquid is constant.
- The height difference is measured vertically.
- Capillary effects are ignored.
- For simple U-tube formulas, the connected gases have negligible density compared with the manometer liquid.
- Altitude correction is an atmospheric approximation, not a local weather forecast.
Key idea: a pressure difference is converted into a height difference in a liquid column, and the conversion is \(\Delta P=\rho gh\).
