Manometer
A manometer measures gas pressure by balancing the gas against a vertical liquid column; the observed height difference represents a pressure difference generated by the liquid’s weight.
Principle of hydrostatic balance
In a static liquid, pressure increases with depth. Two points at the same vertical level within the same continuous manometer fluid have equal pressure; this equality links the measured height difference to a pressure difference between the two arms.
The fundamental relation is:
\[ \Delta P = \rho \cdot g \cdot \Delta h \]
where \(\rho\) is the manometer fluid density, \(g\) is gravitational acceleration, and \(\Delta h\) is the vertical difference between the liquid levels (measured along the direction of gravity).
Open-end and closed-end configurations
Open-end manometer: one arm is open to the atmosphere, so the reference pressure is \(P_{\text{ref}} = P_{\text{atm}}\). The gas pressure is \[ P_{\text{gas}} = P_{\text{atm}} \pm \rho \cdot g \cdot \Delta h \]
Closed-end (vacuum) manometer: one arm is sealed and ideally contains vacuum, so the reference pressure is approximately zero. The gas pressure becomes \[ P_{\text{gas}} \approx \rho \cdot g \cdot \Delta h \] and the result is an absolute pressure.
Sign convention (when “+” or “−” applies)
The sign follows the physical comparison between the gas and the reference pressure: the side with the higher pressure pushes the manometer liquid down in that arm and makes the liquid level higher in the opposite arm.
- Gas pressure greater than reference: liquid level lower on the gas side; \(P_{\text{gas}} = P_{\text{ref}} + \rho \cdot g \cdot \Delta h\).
- Gas pressure less than reference: liquid level higher on the gas side; \(P_{\text{gas}} = P_{\text{ref}} - \rho \cdot g \cdot \Delta h\).
Visualization: U-tube manometer reading
Worked example (open-end mercury manometer)
A gas flask is connected to a mercury manometer whose other arm is open to the atmosphere. The mercury level is lower on the gas side, and the height difference is \(\Delta h = 72\,\text{mm}\). The atmospheric pressure is \(P_{\text{atm}} = 742\,\text{mmHg}\).
The observed geometry corresponds to \(P_{\text{gas}} > P_{\text{atm}}\), so:
\[ P_{\text{gas}} = P_{\text{atm}} + \Delta h = 742\,\text{mmHg} + 72\,\text{mmHg} = 814\,\text{mmHg} \]
Conversions:
\[ 814\,\text{mmHg} \times \frac{1\,\text{atm}}{760\,\text{mmHg}} = 1.071\,\text{atm} \] \[ 1.071\,\text{atm} \times \frac{101.325\,\text{kPa}}{1\,\text{atm}} = 108.5\,\text{kPa} \]
Worked example (SI form with water)
A water manometer shows a height difference of \(\Delta h = 25.0\,\text{cm}\) between the two arms. Using \(\rho = 1000\,\text{kg}\cdot\text{m}^{-3}\) and \(g = 9.81\,\text{m}\cdot\text{s}^{-2}\):
\[ \Delta P = \rho \cdot g \cdot \Delta h = 1000 \cdot 9.81 \cdot 0.250 = 2452.5\,\text{Pa} = 2.45\,\text{kPa} \]
Units and common equivalences
| Pressure unit | Useful equivalence | Notes for manometer readings |
|---|---|---|
| atm | \(1\,\text{atm} = 101325\,\text{Pa} = 101.325\,\text{kPa}\) | Standard reference for gas law calculations. |
| mmHg | \(760\,\text{mmHg} = 1\,\text{atm}\) | Directly usable as \(\Delta h\) only when the manometer fluid is mercury and \(\Delta h\) is measured in mm. |
| torr | \(760\,\text{torr} = 1\,\text{atm}\) | Often treated numerically similar to mmHg in introductory chemistry contexts. |
| bar | \(1\,\text{bar} = 100000\,\text{Pa}\) | Common engineering unit; conversions typically pass through Pa or kPa. |
Practical considerations
- Fluid choice: high-density fluids (mercury) keep \(\Delta h\) manageable; low-density fluids (water or oil) require larger height differences for the same \(\Delta P\).
- Reading the meniscus: consistent reference (top or bottom of the meniscus) matters; mercury has a convex meniscus, water typically concave.
- Gauge vs absolute pressure: open-end manometers naturally compare to \(P_{\text{atm}}\) (gauge relative to atmosphere); closed-end manometers approximate absolute pressure.
- Inclined manometers: the same physics applies, but the vertical height component \(\Delta h\) is the relevant quantity.
Common pitfalls
- Sign confusion: the lower liquid level marks the higher pressure side; mismatching this rule flips “+” and “−”.
- Mixing length and pressure units: a height difference in mm is not a pressure in mmHg unless mercury is the fluid and the conventional mmHg scale is intended.
- Density mismatch: \(\rho\) must match the manometer fluid at the measurement temperature; large temperature changes slightly alter \(\rho\) and thus \(\Delta P\).
Reliable manometer results follow from a clear reference pressure (\(P_{\text{atm}}\) or vacuum), a consistent sign convention tied to the observed liquid levels, and unit handling consistent with \(\Delta P = \rho \cdot g \cdot \Delta h\).