Specific Heat - Internal Energy Solids

I have a question about the relation between the specific heat and the internal energy in solids or incompressible liquids.

What are the limitations for using the following relation for calculating the internal energy in solids:

$u = C_v \cdot T$

where $C_v$ is the specific heat in kJ/kg/K and $T$ is the temperature.

By analyzing this equation it looks like one would get the absolute value of internal energy. Is that right? I know that this relation works for ideal gases and that the differential $du = C_v \cdot dT$ makes sense in thermodynamics.

I have checked the equation and it works very well for water at room temperature, but what are the limitations?

• for a small enough temperature range and away from phase transition $C_v$ is nearly independent of temperature – hyportnex Sep 21 '16 at 18:05
• For water, the heat capacity is a function of temperature. Also, how did it work in the transition across the freezing point? – Chet Miller Sep 22 '16 at 13:05
• See this as a starter. Also, phase changes require a change in heat content and different phases have different specific heats. – Carl Oct 18 '18 at 15:49