# Bulk modulus -- liquids

#### Prerequisite

As with solids, the bulk modulus, $B$, quantifies how much pressure must be applied to change the volume of a liquid:

$$B = \frac{\Delta p}{\Delta V/V}$$

Here, $Δp$ is the change in pressure necessary to cause a fractional change in volume, $ΔV$ is the volume change, and $V$ is the original volume. This quantity helps us understand how liquids respond to pressure changes, and is also important as it describes how easily a pressure wave, such as sound, will travel through a liquid. The large the value of $B$, the greater pressure is needed to produce the same fractional change of volume.

Liquids are typically more compressible than solids, though far less compressible than gases.

But the liquid we have to deal with most often in biology is water (and water with things in it). Water is not very compressible and, in fact, through our discussions of the motion of fluids, we treat water as if it were incompressible. Let's see how good this model for water is.

The bulk modulus for water is 1.96 x 10^{9} Pa. This is a lot smaller than that of many solids, for example 160 x 10^{9} Pa for stainless steel. However, the bulk modulus for bone, which is only 15 x 10^{9} Pa, so the compressibility of water is less than a factor of ten smaller than that of biological solids.

By considering how pressure and volume change with depth, we can use the bulk modulus to see how the density of water changes with depth and whether this is significant. Pressure increases with depth in a liquid because of the weight of the liquid pressing down from above. (See the webpage Archimedes' Principle.) For water, pressure increases 101.3 kPa (1atm ~ 100 kPa = 10^{5} Pa) for every 10 m of depth. Therefore, at a depth of 100 m, the increase in pressure ($Δp$) will be 1,013 kPa (10 times the pressure at the surface). As a result of this pressure increase, the water volume ($ΔV/V$) will decrease. Using on the equation above, we can calculate that this fractional change in volume will be $ΔP/B = 5.2 \times 10^{-4}$ — 5 parts in 10,000 or about 0.05%. At a depth of 100 m or 300 ft! As a result of this decrease in volume, the density of water will increase from 1000 kg m^{-3} to 1000.52 kg m^{-3}. Therefore, water density increases only slowly with depth in a lake or the ocean. For most of our considerations, we can treat water as if it is incompressible.

This enables animals to remain robust and survive at different altitudes and particularly different depths. Though some animals retain a modest gas volume which can vary in size with pressure, others avoid this problem by using incompressible liquids or solids of lower density. One of the more unusual ways of holding air in the body is found in a cuttlefish. It is named for its cuttlebone, a foam-like, incompressible structure which holds gases at varying pressure. Other organisms actually modify the chemical composition of their bodies to adjust their density and so avoid using a compressible volume of air for buoyancy compensation. Some jellyfish exclude sulfate from their tissues. Squids replace sodium ions with the lighter ammonium. Others accumulate low density lipids or waxes. Though these substantive changes take time to accumulate, they are more stable, and are insensitive to large pressure changes.

Julia Gouvea, Kerstin Nordstrom, & Joe Redish 8/21/13

Last Modified: November 6, 2023