8.2: Bulk Modulus and Related Measures
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The bulk modulus \(\mathbb{B}\) describes how a gas, liquid, or solid changes as it is compressed [103]. More specifically, bulk modulus per unit volume is the change in pressure required to get a given compression of volume,
\[\mathbb{B} = -\mathbb{V}\frac{\partial \mathbb{P}}{\partial \mathbb{V}} \label{8.2.1} \]
and bulk modulus is specified in the SI units of pascals or \(\frac{N}{m^2}\). The bulk modulus is greater than zero (\(\mathbb{B} > 0\)) even though there is a minus sign in Equation \ref{8.2.1} because volume shrinks when pressure is applied. Table \(\PageIndex{1}\) lists example bulk modulus values.
| Material | Bulk modulus \(\mathbb{B}\) in GPa | Thermal cond. \(\kappa\) in \(\frac{W}{m \cdot K}\) | Electrical cond. \(\sigma\) in \(\frac{1}{\Omega \cdot m}\) | Ref. |
|---|---|---|---|---|
| Diamond | 539 | 300 | \(1 \cdot 10^{-12} − 1 \cdot 10^{-2}\) | [104] |
| Stainless steel | 143 | 15.5 | \(1.3 \cdot 10^{6} − 1.5 \cdot 10^{6}\) | [105] |
| Graphite | 18.6 | 195 | \(1.6 \cdot 10^{4} −2.0 \cdot 10^{7}\) | [106] |
| Silicone rubber | 1.75 | 1.38 | \(1 \cdot 10^{-14} − 3.2 \cdot 10^{-12}\) | [107] |
Assuming constant temperature, the inverse of the bulk modulus \(\frac{1}{\mathbb{B}}\), is also called the isothermal compressibility [108]. There is a relationship between this compressibility and the permittivity \(\epsilon\) discussed in Chapter 2. If we take an insulating material and apply an external electric field, a material polarization is established, and energy is stored in this charge accumulation. The permittivity is a measure of the charge accumulation per unit volume for a given strength of external electric field, in units of \(\frac{F}{m}\). It is the ratio of the displacement flux density \(\overrightarrow{D}\) to the electric field intensity \(\overrightarrow{E}\).
\[\epsilon = \frac{|\overrightarrow{D}|}{|\overrightarrow{E}|} \label{8.2.2} \]
If we take a material and apply an external pressure, the material compresses and energy is stored in this compressed volume. The inverse of the bulk modulus per unit volume is a measure of the change in volume for a given external pressure
\[\frac{1}{\left( \frac{\mathbb{B}}{\mathbb{V}} \right)} = - \frac{\partial \mathbb{V}}{\partial \mathbb{P}} \label{8.2.3} \]
in units of \(\frac{m}{Pa}^3\). Both Equations \ref{8.2.2} and \ref{8.2.3} can be called constitutive relationships because they describe how a material changes when an external influence is applied.
Multiple other measures describe the variation of a gas, liquid, or solid, with respect to variation of a thermodynamic property. The specific heat describes the ability of a material to store thermal energy, and it has units \(\frac{J}{g \cdot K}\) [109, p. 98]. More specifically, the specific heat over temperature is equal to the change in entropy with respect to change in temperature [108]. It may be given either assuming a constant volume or assuming a constant pressure.
\[\text{Specific heat at constant volume} = C_v = \left. T \frac{\partial S}{\partial T} \right|_\mathbb{V} \nonumber \]
\[\text{Specific heat at constant pressure} = \left. T \frac{\partial S}{\partial T} \right|_\mathbb{P} \nonumber \]
The Joule-Thomson coefficient is defined as the ratio of change in temperature to change in pressure for a given total energy of the system
\[\text{Joule-Thomson coefficient} = \frac{\partial T}{\partial \mathbb{P}}, \nonumber \]
and it has units \(\frac{K}{Pa}\) [102, p. 685]. When a pressure is applied and overall energy is held fixed but entropy is allowed to vary, some materials cool and others heat. So, this coefficient may be positive, negative, or even zero at an inversion point. Additionally, the volume expansivity is defined as
\[\text{Volume expansivity} = \left. \frac{1}{\mathbb{V}} \frac{\partial \mathbb{V}}{\partial T} \right|_\mathbb{P} \nonumber \]
[108].