## Volume Continuity

(Temperature-Pressure Differential at Tg)

At the glass transition we can assume two-phase equilibrium, that is, the pure-component molar volumes in the liquid (rubbery) and glassy state should be identical:

d*V*_{glass}(*T*,*p*) = d*V*_{liquid}(*T*,*p*)

and

*V _{glass}(T,p) = V_{liquid}(T,p)*

At equilibrium, the pressure and the volume are the same in both phases. Hence

(∂*V*_{glass} / ∂*T*)_{p} d*T* + (∂*V*_{glass} / ∂*P*)_{T} d*P*
= (∂*V*_{liquid} / ∂*T*)_{p} d*T* + (∂*V*_{liquid} / ∂*P*)_{T} d*P*

The temperature and pressure coefficient of the molar volume can be written in terms of volumetric thermal expansion and isothermal compressibility, respectively:

(∂*V* / ∂*T*)_{P} = *V* *α*

(∂*V* / ∂*P*)_{T} = -*V* *β*

Thus

*V*_{glass} *α _{glass}* d

*T*+

*V*

_{glass}

*β*d

_{glass}*P*=

*V*

_{liquid}

*α*d

_{liquid}*T*+

*V*

_{liquid}*β*

*d*

_{liquid}*P*

A two-phase equilibrium of a pure material has only one degree of freedom. Therefore, the temperature and pressure changes are not independent
on the glass transition phase boundary. Since *V*_{glass}
= *V*_{liquid}, the differential (∂*V* / ∂*P*)_{T}
along the glass-liquid boundary is