# 1.6.2.1: Bulk Modulus of Mixtures

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

In the discussion above it was assumed that the liquid is pure. In this short section a discussion about the bulk modulus averaged is presented. When more than one liquid are exposed to pressure the value of these two (or more liquids) can have to be added in special way. The definition of the bulk modulus is given by equation (34) or (35) and can be written (where the partial derivative can looks as delta $$\Delta$$ as $\partial{V} = \frac{V\partial{P}}{B_T} \cong \frac{V \Delta P}{B_T}$ The total change is compromised by the change of individual liquids or phases if two materials are present. Even in some cases of emulsion (a suspension of small globules of one liquid in a second liquid with which the first will not mix) the total change is the summation of the individuals change. In case the total change isn't, in special mixture, another approach with taking into account the energy-volume is needed. Thus, the total change is $\partial{V} = \partial{V_{1}} + \partial{V_{2}} + \cdot \cdot \cdot + \partial{V_{i}} \cong \Delta{V_{1}} + \Delta{V_{2}} + \cdot \cdot \cdot + \Delta{V_{i}}$

Substituting equation (45) into equation (46) results in $\label{intro:eq:DeltaVexpli} \partial V = \dfrac{V_1\,\partial P }{{B_T}_1} + \dfrac{V_2\,\partial P }{{B_T}_2} + \cdots + \dfrac{V_i\,\partial P }{{B_T}_i} \cong \\ \dfrac{V_1\,\Delta P }{{B_T}_1} + \dfrac{V_2\,\Delta P }{{B_T}_2} + \cdots + \dfrac{V_i\,\Delta P }{{B_T}_i} \,\,$ Under the main assumption in this model the total volume is comprised of the individual volume hence, $\label{intro:eq:v-vi} V = x_1 \, V + x_1 \,V + \cdots + x_i \, V$ Where $$x_1$$, $$x_2$$ and $$x_i$$ are the fraction volume such as $$x_i= V_i/V$$. Hence, using this identity and the fact that the pressure is change for all the phase uniformly the previous equation can be written as $\label{intro:eq:v-vi1} \partial V = V \,\partial P \left( \dfrac{x_1}{{B_T}_1} + \dfrac{x_2}{{B_T}_2} + \cdots + \dfrac{x_i}{{B_T}_i} \right) \cong \\ V\, \Delta P \left( \dfrac{x_1}{{B_T}_1} + \dfrac{x_2}{{B_T}_2} + \cdots +\dfrac{x_i}{{B_T}_i} \right)$ Rearranging it yields $\label{intro:eq:newBTstart} v\, \dfrac{\partial P}{\partial v} \cong v\, \dfrac{\Delta P}{\Delta v} = \dfrac{1}{ \left( \dfrac{x_1}{{B_T}_1} + \dfrac{x_2}{{B_T}_2} + \cdots + \dfrac{x_i}{{B_T}_i} \right) }$ This equation suggested an averaged new bulk modulus $\label{intro:eq:BTmixDef} {B_T}_{mix} = \dfrac{1}{ \left( \dfrac{x_1}{{B_T}_1} + \dfrac{x_2}{{B_T}_2} + \cdots + \dfrac{x_i}{{B_T}_i} \right) }$ In that case the equation for mixture can be written as $\label{intro:eq:BTmix} v\, \dfrac{\partial P}{\partial v} = {B_T}_{mix}$