# 3.1: Introduction to Two-Phase Equilibria

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There is more than one way in which the principle of conservation of mass for single component systems can be stated. One attractive form is^{1};

*the mass of a body is constant*

however, we often express this idea in the *rate form* leading to an equation given by

\[\begin{Bmatrix} \text{time rate of} \\ \text{change of the} \\ \text{mass of a body}\end{Bmatrix} = 0 \label{1}\]

The principle of conservation of mass is also known as the *axiom* for the conservation of mass. In physics, one uses the word axiom to describe an accepted principle that cannot be derived from a more general principle. Axioms are based on specific experimental observations, and from those *specific observations* we construct the *general statement* given by Equation \ref{1}.

As an example of the application of Equation \ref{1}, we consider the motion of the cannon ball illustrated in Figure \(\PageIndex{1}\).

Newton’s second law requires that the force acting on the cannon ball be equal to the time rate of change of the linear momentum of the solid body as indicated by

\[\mathbf{f}= \frac{d}{dt} \left(m\mathbf{v}\right) \label{2}\]

We now apply Equation \ref{1} in the form

\[\frac{dm}{dt} =0 \label{3}\]

to find that the force is equal to the mass times the acceleration.

\[\mathbf{f} = m\frac{d\mathbf{v}}{dt} = m \mathbf{a} \label{4}\]

Everyone is familiar with this result from previous courses in physics and perhaps a course in engineering mechanics.