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Changes in Volume: Bulk Modulus

Changes in Volume: Bulk Modulus

An object will be compressed in all directions if inward forces are applied evenly on all its surfaces as in the figure below. It is relatively easy to compress gases and extremely difficult to compress liquids and solids. For example, air in a wine bottle is compressed when it is corked. But if you try corking a brim-full bottle, you cannot compress the wine—some must be removed if the cork is to be inserted.

The reason for these different compressibilities is that atoms and molecules are separated by large empty spaces in gases but packed close together in liquids and solids. To compress a gas, you must force its atoms and molecules closer together. To compress liquids and solids, you must actually compress their atoms and molecules, and very strong electromagnetic forces in them oppose this compression.

A cube with area of cross section A and volume V zero is compressed by an inward force F acting on all surfaces. The compression causes a change in volume delta V, which is proportional to the force per unit area and its original volume. This change in volume is related to the compressibility of the substance.

An inward force on all surfaces compresses this cube. Its change in volume is proportional to the force per unit area and its original volume, and is related to the compressibility of the substance.

We can describe the compression or volume deformation of an object with an equation. First, we note that a force “applied evenly” is defined to have the same stress, or ratio of force to area \(\frac{F}{A}\) on all surfaces. The deformation produced is a change in volume \(\Delta V\), which is found to behave very similarly to the shear, tension, and compression previously discussed. (This is not surprising, since a compression of the entire object is equivalent to compressing each of its three dimensions.) The relationship of the change in volume to other physical quantities is given by

\(\Delta V=\frac{1}{B}\frac{F}{A}{V}_{0},\)

where \(B\) is the bulk modulus (see this table from the previous lesson), \({V}_{0}\) is the original volume, and \(\frac{F}{A}\) is the force per unit area applied uniformly inward on all surfaces. Note that no bulk moduli are given for gases.

What are some examples of bulk compression of solids and liquids? One practical example is the manufacture of industrial-grade diamonds by compressing carbon with an extremely large force per unit area. The carbon atoms rearrange their crystalline structure into the more tightly packed pattern of diamonds. In nature, a similar process occurs deep underground, where extremely large forces result from the weight of overlying material. Another natural source of large compressive forces is the pressure created by the weight of water, especially in deep parts of the oceans. Water exerts an inward force on all surfaces of a submerged object, and even on the water itself. At great depths, water is measurably compressed, as the following example illustrates.

Example: Calculating Change in Volume with Deformation: How Much Is Water Compressed at Great Ocean Depths?

Calculate the fractional decrease in volume (\(\frac{\Delta V}{{V}_{0}}\)) for seawater at 5.00 km depth, where the force per unit area is \(5\text{.}\text{00}×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}N/{m}^{2}\) .

Strategy

Equation \(\Delta V=\frac{1}{B}\frac{F}{A}{V}_{0}\) is the correct physical relationship. All quantities in the equation except \(\frac{\Delta V}{{V}_{0}}\) are known.

Solution

Solving for the unknown \(\frac{\Delta V}{{V}_{0}}\) gives

\(\frac{\Delta V}{{V}_{0}}=\frac{1}{B}\frac{F}{A}.\)

Substituting known values with the value for the bulk modulus \(B\) from see this table from the previous lesson,

\(\begin{array}{lll}\frac{\Delta V}{{V}_{0}}& =& \frac{5.00×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}}{2\text{.}2×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}}\\ & =& 0.023=2.3%.\end{array}\)

Discussion

Although measurable, this is not a significant decrease in volume considering that the force per unit area is about 500 atmospheres (1 million pounds per square foot). Liquids and solids are extraordinarily difficult to compress.

Conversely, very large forces are created by liquids and solids when they try to expand but are constrained from doing so—which is equivalent to compressing them to less than their normal volume. This often occurs when a contained material warms up, since most materials expand when their temperature increases. If the materials are tightly constrained, they deform or break their container. Another very common example occurs when water freezes. Water, unlike most materials, expands when it freezes, and it can easily fracture a boulder, rupture a biological cell, or crack an engine block that gets in its way.

Other types of deformations, such as torsion or twisting, behave analogously to the tension, shear, and bulk deformations considered here.


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