Pressure is a fundamental physical quantity defined as the force applied perpendicular to a surface divided by the area over which that force is distributed. The standard unit for measuring pressure is the pascal (Pa), equivalent to one newton of force per square meter. Calculating the total pressure in a system is necessary across many fields, including chemistry, engineering, and meteorology. Total pressure represents the sum of all individual pressures acting upon a system or object. This concept allows for accurate prediction of system behavior, whether dealing with a mixture of gases or a submerged object.
Calculating Total Pressure in Gas Mixtures
The total pressure exerted by a mixture of non-reactive gases is calculated using Dalton’s Law of Partial Pressures. This law states that the total pressure is the sum of the partial pressures of each individual gas component within the mixture, represented by the formula \(P_{total} = P_1 + P_2 + P_3 + …\). Each gas behaves independently, contributing pressure based on its amount, the container volume, and the temperature. For example, if two gases exert pressures of \(1.5\) atmospheres (atm) and \(0.5\) atm, the total pressure is \(2.0\) atm.
The partial pressure of a single gas can also be calculated using its mole fraction, which is the ratio of the moles of that gas to the total moles of all gases in the mixture. The partial pressure (\(P_i\)) is found by multiplying the mole fraction (\(\chi_i\)) by the total pressure (\(P_{total}\)), resulting in the formula \(P_i = \chi_i P_{total}\). This relationship is useful for calculating individual pressures when the total pressure is known, such as determining the partial pressure of oxygen in the atmosphere. For instance, at sea level, where atmospheric pressure is about \(760\) millimeters of mercury (mmHg) and oxygen makes up roughly \(21\%\) of the air, the partial pressure of oxygen is approximately \(159.6\) mmHg.
Calculating Total Pressure in Fluids
When dealing with liquids or other fluids, the total pressure at a specific depth is determined by the weight of the fluid column above that point. This component of pressure is known as hydrostatic pressure, and it increases linearly with depth. The total pressure at any point submerged in a fluid is the sum of the pressure acting on the surface of the fluid and the hydrostatic pressure.
If the fluid is open to the air, the pressure acting on the surface is the atmospheric pressure (\(P_{atm}\)). The hydrostatic pressure created by the fluid itself is calculated using the formula \(P = \rho g h\). Here, \(\rho\) (rho) is the fluid’s density, \(g\) is the acceleration due to gravity (typically \(9.8 \text{ m/s}^2\) on Earth), and \(h\) is the depth of the fluid column.
The total pressure (\(P_{total}\)) at a depth \(h\) is therefore expressed as \(P_{total} = P_{atm} + \rho g h\). This equation highlights that the pressure experienced by a scuba diver includes both the water pressure and the pressure of the atmosphere pushing down on the surface. For example, the total pressure calculation is important in engineering design for submarines, dams, and structures exposed to fluid at depth.
Converting Between Absolute and Gauge Pressure
The total pressure calculated in a system is often referred to as absolute pressure (\(P_{absolute}\)), which is the pressure measured relative to a perfect vacuum, or zero pressure. Most common pressure-measuring devices, however, report a value called gauge pressure (\(P_{gauge}\)), which is the pressure measured relative to the surrounding atmospheric pressure. Gauge pressure indicates how much a system’s pressure is above or below the local atmospheric pressure.
The conversion between these two types of measurements is straightforward and allows engineers and scientists to use practical gauge readings for total pressure calculations. The relationship is given by the simple additive formula: \(P_{absolute} = P_{gauge} + P_{atmospheric}\). Atmospheric pressure (\(P_{atmospheric}\)) is the pressure exerted by the air around the system, which is approximately \(14.7\) pounds per square inch (psi) at sea level.
This conversion is necessary because many physical laws, such as the ideal gas law, require the total pressure relative to a vacuum for accurate results. For example, if a tire gauge reads \(30\) psi, the absolute pressure inside the tire is \(30 \text{ psi} + 14.7 \text{ psi}\), resulting in \(44.7\) psi. A negative gauge pressure indicates a vacuum, or a pressure below the surrounding atmosphere.