Why is specific volume calculated like this here?

Question

In a solved example in Cengel's Thermodynamics text,

As they have stated, we don't know if the refrigerant is in compressed liquid form. In the case that it is, it's not necessary that the volume the liquid will occupy will be the entire 80L vessel. But that is what they've assumed when making the calculation

They, what feels like to me, first implicitly assumed that it is in the saturated mixture phase, checked if this value lies between $v_f$ and $v_g$ and used that to confirm that it is. This seems off to me.

EDIT: To diagrammatically clarify my confusion

As can be seen here, $v_{avg}$ is a changing quantity. It's value can vary along the horizontal line from $v_f$ to $v_g$

However, if it is defined as being equal to $V/m$, where V represents the volume of a rigid vessel, then we can see that since neither V nor m change, that this quantity would be fixed.

Here's my speculation: During this step of the derivation,

V, which equals the sum, doesn't actually represent the tank volume.

Answer 1

The specific volume they have calculated is the average value for the contents of the vessel, weighted in terms of mass fraction of liquid and mass fraction of vapor.

Answer 2

Let $m_L$ represent the mass of liquid in the tank at equilibrium, $m_V$ represent the mass of vapor in the tank at equilibrium, $v_L$ represent the specific volume of the liquid, $m_V$ represent the volume of vapor, and V represent the tank volume. Then $$m_Lv_L+m_Vv_V=V$$If x_V represents the mass fraction of vapor, then $$m_L=m(1-x_V)$$and $$m_V=mx_V$$So, combining these three equations, we have: $$(1-x_V)v_L+x_Vv_V=\frac{V}{m}$$Does this make sense to you and does it answer your question?