# What is the term for the amount of volume change at a phase boundary?

Apologies for the extremely basic question, but I would like to look up the amount by which the volume (or equivalently, density) of various substances changes when they freeze. However, I don't know the correct term for this, and Google isn't helping much.

To clarify, I'm asking about the vertical line segment on the plot below (from Wikipedia). If the $y$ axis was internal energy then the height of this line segment would be called "latent heat of fusion", and I could easily look it up for any given substance. I would like to know the equivalent term for the discontinuous volume (or density) change, in whichever units are most standard.

I realise that unlike latent heat, this will depend quite a bit on the pressure. However, I would hope that it would be tabulated for atmospheric pressure at least.

Phase transition can be distinguished by analysing their chemical potential, such classification has been introduced by Ehrenfest. For a given transition, lets call it $\phi$ transition:

$$\ce{X_{\alpha} \rightleftharpoons X_{\beta}}$$

You can assess transition molar specific volume in the following way:

$$\Delta_{\phi}V = v_{\beta} - v_{\alpha} = \left(\frac{\partial \mu_{\beta}}{\partial p}\right)_{T} - \left(\frac{\partial \mu_{\alpha}}{\partial p}\right)_{T}$$

Thus, by modelling your potential in term of pressure and temperature, you can assess molar volume change of a phase transition. This is the key to solve your question. In real life application, we use diagrams and tables to assess it. But building a model is always valuable for comprehension.

In a similar way, molar entropy is given by:

$$\Delta_{\phi}S = s_{\beta} -s_{\alpha} = \left(\frac{\partial \mu_{\beta}}{\partial T}\right)_{p} - \left(\frac{\partial \mu_{\alpha}}{\partial T}\right)_{p} = \frac{\Delta_{\phi}H}{T_\phi}$$

Because $\Delta_{\phi}V$ and $\Delta_{\phi}H$ are not null for first order (in term of first derivative) transition, slope difference is not null and therefore potential slopes are different both side of phase transition when Gibbs parameters vary ($p$ and $T$). Thus there are discontinuities in specific volume, enthalpy and entropy when a first order transition occurs.

Ehrenfest did a classification of phase transition based on this criterion which led to: first order, second order (smooth, no slope discontinuity) and $\lambda$ (sharper) transitions regarding how such chemical potential discontinuity happens.

This discontinuity also impact heat capacity $c_p$. In case of water, whose phase transitions can be assumed as a first order, when a phase transition occurs, discontinuity leads to a virtually infinite heat capacity, explaining why fusion or vaporisation happens at constant temperature (when pressure is held constant). This is why we need an extra term called latent heat (temperature independent) which contrast with sensible heat (temperature dependent).

You will find detailed explanation (models, diagram and tables) of those phenomenons in Atkins, Physical Chemistry, Chapters 2-7.

• I greatly appreciate the effort you've gone to to give a detailed answer. Unfortunately I know all this stuff already! All I'm asking for is the name of $\Delta_\phi V$, so that I can search online for tables of its values. – Nathaniel Mar 9 '14 at 10:56
• You will rarely encounter tables for $\Delta_\phi V$ but specific volume $v_\gamma$ (molar or mass) for a given phase $\gamma$ are tabulated or plotted. Look for an Handbook of Chemistry or Thermodynamic Handbook for Air or Water. NIST Chemistry Handbook has numerous substance and phase transition data. – jlandercy Mar 9 '14 at 17:34