After Chapter 4.1, you will be able to:
Fluids are characterized by their ability to flow and conform to the shapes of their containers. Solids, on the other hand, do not flow and are rigid enough to retain a shape independent of their containers. Both liquids and gases are fluids. The natural gas (methane) that many of us use to cook flows through pipes to the burners of our stove and ovens, and the air that we breathe flows in and out of our lungs, filling the spaces of our respiratory tract and the alveoli.
Fluids and solids share certain characteristics. Both can exert forces perpendicular to their surface, although only solids can withstand shear (tangential) forces. Fluids can impose large perpendicular forces; falling into water from a significant height can be just as painful as falling onto a solid surface.
All fluids and solids are characterized by the ratio of their mass to their volume. This is known as density, which is a scalar quantity and therefore has no direction. The equation for density is
where ρ (rho) represents density, m is mass, and V is volume. The SI units for density are
but you may find it convenient to use
or
both of which may be seen on the MCAT. Remember that a milliliter and a cubic centimeter
are the same volume. A word of caution: students sometimes assume that if the mL and
the cm3 are equivalent, then so must be the liter and the m3. This is absolutely not the case; in fact, there are 1000 liters in a cubic meter.
For the MCAT, it is important to know the density of water, which is
The weight of any volume of a given substance with a known density can be calculated by multiplying the substance’s density by its volume and the acceleration due to gravity. This is a calculation that appears frequently when working through buoyancy problems on Test Day:
The density of a fluid is often compared to that of pure water at 1 atm and 4°C, a
variable called specific gravity. It is at this combination of pressure and temperature that water has a density of
exactly
The specific gravity is given by
This is a unitless number that is usually expressed as a decimal. The specific gravity can be used as a tool for determining if an object will sink or float in water, as described later in this chapter.
If an object’s density is given in
, its specific gravity is simply its density as a dimensionless number. This is because
the density of water in
is 1.
Pressure is a ratio of the force per unit area. The equation for pressure is
where P is pressure, F is the magnitude of the normal force vector, and A is the area. The SI unit of pressure is the pascal (Pa), which is equivalent to the newton per square meter
Other commonly used units of pressure are millimeters of mercury (mmHg), torr, and
the atmosphere (atm). Millimeters of mercury and torr are identical units. The unit
of atmosphere is based on the average atmospheric pressure at sea level. The conversions
between Pa, mmHg, torr, and atm are as follows:
If you ever forget the units of a variable, you can derive them from equations. You
know that pressure equals force over area. Because you know the units of force (N)
and area (m2), you can solve for the base units of pascal by plugging these units into the equation:
Pressure is a scalar quantity, and therefore has a magnitude but no direction. It is easy to assume that pressure has a direction because it is related to a force, which is a vector. However, note that it is the magnitude of the normal force that is used. No matter where one positions a given surface, the pressure exerted on that surface within a closed container will be the same, neglecting gravity. For example, if we placed a surface inside a closed container filled with gas, the individual molecules, which are moving randomly within the space, will exert pressure that is the same at all points within the container. Because the pressure is the same at all points along the walls of the container and within the space of the container itself, pressure applies in all directions at any point and, therefore, is a scalar rather than a vector. Of course, because pressure is a ratio of force to area, when unequal pressures are exerted against objects, the forces acting on the object will add in vectors, possibly resulting in acceleration. It’s this difference in pressure that causes air to rush into and out of the lungs during respiration, windows to burst outward during a tornado, and the plastic covering a broken car window to bubble outward when the car is moving. Note that when gravity is present, this also results in a pressure differential, which we will explore with hydrostatics later in this chapter.
At this very moment, countless trillions of air molecules are exerting tremendous pressure on our bodies, with a total force of about 2 × 105 N! Of course, we don’t actually feel all this pressure because our internal organs exert a pressure that perfectly balances it.
Atmospheric pressure changes with altitude. Residents of Denver (5280 feet above sea level) experience atmospheric pressure equal to 632 mmHg (0.83 atm), whereas travelers making their way through Death Valley (282 feet below sea level) experience atmospheric pressure equal to 767 mm Hg (1.01 atm). Atmospheric pressure impacts a number of processes, including hemoglobin’s affinity for oxygen and the boiling of liquids.
Absolute (hydrostatic) pressure is the total pressure that is exerted on an object that is submerged in a fluid. Remember that fluids include both liquids and gases. The equation for absolute pressure is
where P is the absolute pressure, P0 is the incident or ambient pressure (the pressure at the surface), ρ is the density of the fluid, g is acceleration due to gravity, and z is the depth of the object. Do not make the mistake of assuming that P0 always stands for atmospheric pressure. In open air and most day-to-day situations P0 is equal to 1 atm, but in other fluid systems, the surface pressure may be higher or lower than atmospheric pressure. In a closed container, such as a pressure cooker, the pressure at the surface may be much higher than atmospheric pressure. This is, in fact, exactly the point of a pressure cooker, which allows food to cook at higher temperatures. This is because the increased pressure raises the boiling point of water in the food, thus reducing the cooking time and preventing loss of moisture.
A useful way to remember the two parts of the absolute pressure equation is to think of diving into a swimming pool. At the surface of the water, the absolute pressure is usually equal to the atmospheric pressure (P0). But if you dive into the pool, the water exerts an extra pressure on you (ρgz), in addition to the surface pressure. You feel this extra pressure on your eardrums.
When you check the pressure in your car or bike tires using a device known as a gauge, you are measuring the gauge pressure, which is the difference between the absolute pressure inside the tire and the atmospheric pressure outside the tire. In other words, gauge pressure is the amount of pressure in a closed space above and beyond atmospheric pressure. This is a more common pressure measurement than absolute pressure, and the equation is:
Note that when P0 = Patm, then Pgauge = P – P0 = ρgz at a depth z.