How do you convert pressure units like millimeters of mercury, centimeters of water, and inches of mercury into units like newtons per meter squared without resorting to a table of pressure conversion factors?
The correct answer and explanation is :
To convert pressure units like millimeters of mercury (mmHg), centimeters of water (cmH₂O), and inches of mercury (inHg) to units like newtons per meter squared (N/m² or pascal, Pa), we need to understand the relationship between the units of pressure in these systems and the SI base units of force and area.
Step 1: Understand Pressure Units
Pressure is defined as force per unit area:
[
P = \frac{F}{A}
]
where (P) is the pressure, (F) is the force, and (A) is the area.
The pressure in different units (like mmHg, cmH₂O, and inHg) corresponds to the weight of a column of liquid exerting force over an area. Therefore, to convert between these units and pascals (N/m²), we need to calculate the force exerted by the column of liquid and then divide it by the area.
Step 2: Conversion from Height to Pressure
The pressure exerted by a column of liquid is given by the formula:
[
P = \rho g h
]
where:
- (P) is the pressure,
- (\rho) is the density of the liquid,
- (g) is the acceleration due to gravity (approximately 9.81 m/s²),
- (h) is the height of the liquid column.
Step 3: Convert Units
- Millimeters of Mercury (mmHg):
- Density of mercury: (\rho_{\text{Hg}} = 13,595 \, \text{kg/m}^3)
- The height (h) in mm is converted to meters by dividing by 1000.
- Pressure in pascals: (P = 13,595 \times 9.81 \times h_{\text{mmHg}} / 1000).
- Centimeters of Water (cmH₂O):
- Density of water: (\rho_{\text{H}_2\text{O}} = 1000 \, \text{kg/m}^3)
- The height (h) in cm is converted to meters by dividing by 100.
- Pressure in pascals: (P = 1000 \times 9.81 \times h_{\text{cmH}_2\text{O}} / 100).
- Inches of Mercury (inHg):
- Density of mercury: (\rho_{\text{Hg}} = 13,595 \, \text{kg/m}^3)
- Height (h) in inches is converted to meters by multiplying by 0.0254.
- Pressure in pascals: (P = 13,595 \times 9.81 \times h_{\text{inHg}} \times 0.0254).
Step 4: Conclusion
Using the formula (P = \rho g h) and converting the height appropriately for each unit, you can convert from pressure units like mmHg, cmH₂O, and inHg to pascals (N/m²) without referring to a table of conversion factors. The key is knowing the density of the fluid and the gravitational constant, then adjusting the height measurement to the correct SI units.